Ivan Di Liberti

[[!redirects Ivan di Liberti]]

Most of what I do is motivated by logic, foundations of mathematics and foundations of geometry. Concretely, my work is in categorical logic, syntax-semantics dualities, topos theory, general category theory and [[formal category theory]].

Academic life went as follows. Brno (Ph.D. under [[Jiří Rosický]]), Prague (Postdoc), Stockholm (Postdoc), Gothenburg (Postdoc, ongoing).

You can visit my [webpage](https://diliberti.github.io).

## Selected writings

A geometric account on the [[Scott adjunction]] and the duality between [[topoi]] and [[ionads]]:

* [[Ivan Di Liberti]], *Towards Higher Topology* [[arXiv:2009.14145](https://arxiv.org/abs/2009.14145)]

On geometric aspects of coherent topoi and their relationship to ultrastructures:

* [[Ivan Di Liberti]], *The geometry of coherent topoi and Ultrastructures* [[arXiv:2211.03104](https://arxiv.org/abs/2211.03104)]

On bipresentable 2-categories and their relations to logical doctrines.

 * [[Ivan Di Liberti]], Axel Osmond, *Bi-accessible and bipresentable 2-categories.* [[arXiv:2203.07046](https://arxiv.org/abs/2203.07046)]

On [[judgements]], [[natural deduction]] and [[dependent type theory]]:

* Greta Coraglia, [[Ivan Di Liberti]], *Context, Judgement, Deduction* [[arXiv:2111.09438](https://arxiv.org/abs/2111.09438)]

On the [[adjoint functor theorem]] in the context of [[lax-idempotent 2-monads]]:

* {#ALL23} [[Nathanael Arkor]], [[Ivan Di Liberti]], [[Fosco Loregian]], _Adjoint functor theorems for lax-idempotent pseudomonads_, [[arXiv:2306.10389](https://arxiv.org/abs/2306.10389)]

On [[formal category theory]]:

* [[Ivan Di Liberti]], [[Simon Henry]], [[Mike Lieberman]], [[Fosco Loregian]], _Formal category theory_, course notes (2017) [[pdf](https://tetrapharmakon.github.io/stuff/course_muni-formal.pdf), [[DLHLL-FormalCategoryTheory.pdf:file]]]

* [[Ivan Di Liberti]], [[Fosco Loregian]], _On the unicity of formal category theories_ [[arXiv:1901.01594](https://arxiv.org/abs/1901.01594)]

## Related entries

* [[judgement]]

* [[Isbell duality]]

* [[Gabriel–Ulmer duality]]

* [[Functorial Semantics  of Algebraic Theories]]

* [[adjoint functor theorem]]

* [[ionad]]

* [[Kan extension]]

* [[codensity monad]]

* [[concrete category]]


Ivan Fesenko

* [webpage](https://www.maths.nottingham.ac.uk/personal/ibf/)

## related $n$Lab entries

* [[higher arithmetic geometry]]

* [[adelic integration]]

* [[inter-universal Teichmüller theory]]

category: people


Ivan G. Todorov

A mathematician at University of Delaware. A different person from theoretical physicist [[Ivan T. Todorov]].

* [personal page](https://sites.google.com/view/ivan-g-todorov/home)

* [institute page](https://www.mathsci.udel.edu/people/faculty/todorov?uid=todorov&Name=Ivan%20Todorov)

category: people


Ivan Kolář

* [webpage](http://www.math.muni.cz/~kolar/)

## Selected writings

On [[differential geometry]] with focus on [[natural bundles]]:

* {#KolarSlovakMichor93} [[Ivan Kolář]], [[Peter Michor]], [[Jan Slovák]], _[[Natural operations in differential geometry]]_, Springer (1993) [[book webpage](http://www.emis.de/monographs/KSM/), [doi:10.1007/978-3-662-02950-3](https://link.springer.com/book/10.1007/978-3-662-02950-3)
[pdf](http://www.emis.de/monographs/KSM/kmsbookh.pdf)]

category: people

[[!redirects Ivan Kolar]]


Ivan Limonchenko

* [Mathematics Genealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=220154)

## Selected writings

On [[Calabi-Yau manifolds]] in [[SU-bordism theory]] (see [[Calabi-Yau manifolds in SU-bordism theory]]):

* [[Ivan Limonchenko]], [[Zhi Lu]], [[Taras Panov]], _Calabi-Yau hypersurfaces and SU-bordism_, Proceedings of the Steklov Institute of Mathematics 302 (2018), 270-278 ([arXiv:1712.07350](https://arxiv.org/abs/1712.07350))

* Georgy Chernykh, [[Ivan Limonchenko]], [[Taras Panov]], _$SU$-bordism: structure results and geometric representatives_, Russian Math. Surveys 74 (2019), no. 3, 461-524 ([arXiv:1903.07178](https://arxiv.org/abs/1903.07178))

category: people


Ivan Losev

According to wikipedia, __Ivan Losev__ is a Belarusian-American mathematician, specializing in representation theory, symplectic geometry, algebraic geometry, and combinatorial algebra.

* wikipedia

* L. Baulieu, A. Losev, [[Nikita Nekrasov]], _Chern-Simons and twisted supersymmetry in higher dimensions_, Nucl.Phys. B522 (1998) 82--104 arXiv:[hep-th/9707174](http://arxiv.org/abs/hep-th/9707174)
* [[Edward Frenkel]], A. Losev, [[Nikita Nekrasov]], _Instantons beyond topological theory I_, arXiv:[hep-th/0610149](https://arxiv.org/abs/hep-th/0610149); _Instantons beyond topological theory II_, arXiv:[hep-th/0610149](https://arxiv.org/abs/0803.3302); _Notes on instantons in topological field theory and beyond_, arXiv:[hep-th/0702137](https://arxiv.org/abs/hep-th/0702137)
* Ivan Losev, _Procesi bundles and [[symplectic reflection algebra]]s_, In: M. Hitrik, D. Tamarkin, B. Tsygan, S. Zelditch (eds), Algebraic and Analytic Microlocal Analysis. AAMA 2013. Springer Proc. in Math. & Stat. __269__; [arXiv:1501.00643](https://arxiv.org/abs/1501.00643)
* Ivan Losev, Ben Webster, _On uniqueness of tensor products of irreducible categorifications_, [arXiv:1303.1336](https://arxiv.org/abs/1303.1336)
* Barbara Bolognese, Ivan Losev, _A general introduction to the Hilbert scheme of points on the plane_, [pdf](https://web.northeastern.edu/iloseu/Barbara_14_complete.pdf)

category: people


Ivan Marin

* [MathematicsGenealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=56636)

## Selected writings

On the [[Knizhnik-Zamolodchikov equation]] in relation to [[braid group representations]]:

* [[Ivan Marin]], *Sur les représentations de Krammer génériques*, Annales de l'Institut Fourier, **57** 6 (2007) 1883-1925 [[numdam:AIF_2007__57_6_1883_0](http://www.numdam.org/item/AIF_2007__57_6_1883_0)]

On [[braid group representations]]:

* [[Ivan Marin]], *On the representation theory of braid groups*, Annales mathématiques Blaise Pascal, **20** 2 (2013) 193-260 ([arXiv:math/0502118](https://arxiv.org/abs/math/0502118), [dml:275607](https://eudml.org/doc/275607))

On the [[flat vector bundles]] corresponding to the [[Knizhnik-Zamolodchikov equation]]:

* [[Ivan Marin]], *Knizhnik-Zamolodchikov bundles are topologically trivial* ([arXiv:0809.3590](https://arxiv.org/abs/0809.3590))

category: people


Ivan Mirkovic

**Ivan Mirković** is a professor of mathematics at the University of Massachusets at Amherst (diploma from Zagreb and Ph. D. under D. Miličić in Utah) with interests in [[geometric representation theory]].

Selected links and papers:

* [homepage](http://www.math.umass.edu/~mirkovic)

* R. Bezrukavnikov, I. Mirković, _Representations of semisimple Lie algebras in prime characteristic and noncommutative Springer resolution_, [arXiv:1001.2562](http://arxiv.org/abs/1001.2562)

* R. Bezrukavnikov, I. Mirković, D. Rumynin, _Localization of modules for a semisimple Lie algebra in prime characteristic_, With an appendix by Bezrukavnikov and Simon Riche.  Ann. of Math. (2) __167__  (2008),  no. 3, 945--991, [math.RT/0205144](http://arxiv.org/abs/math/0205144)

* I. Mirković, K. Vilonen, _Geometric Langlands duality and representations of algebraic groups over commutative rings_, Ann. of Math. (2) __166__ (2007), no. 1, 95--143.

* I. Mirković, K. Vilonen, _Perverse sheaves on affine Grassmannians and Langlands duality_,  Math. Res. Lett. __7__ (2000),  no. 1, 13--24; [alg-geom/9703010](http://arxiv.org/abs/alg-geom/9703010)

* A. Braverman, D. Gaitsgory, M. Finkelberg, I. Mirković, _Intersection cohomology of Drinfeld's compactifications_, Selecta Math. (N.S.) __8__ (2002), no. 3, 381--418. and erratum in Selecta Math. (N.S.) __10__ (2004), no. 3, 429--430, [math.AG/0012129](http://arxiv.org/abs/math/0012129).

* S. Evens, I. Mirković, _Characteristic cycles for the loop Grassmannian and nilpotent orbits_,  Duke Math. J. __97__  (1999),  no. 1, 109--126, [doi](http://dx.doi.org/10.1215/S0012-7094-99-09705-3)

* J. Kamnitzer, _Mirković-Vilonen cycles and polytopes_, Ann. Math. __171__ (2010), no. 1, 245–294, [pdf](http://annals.princeton.edu/annals/2010/171-1/annals-v171-n1-p04-p.pdf)

* Croatian wikipedia [page](http://hr.wikipedia.org/wiki/Ivan_Mirkovi%C4%87)

category: people

[[!redirects Ivan Mirković]]


Ivan Panin

* [webpage](http://www.pdmi.ras.ru/~panin/)

* [MathematicsGenealogy page](http://genealogy.math.ndsu.nodak.edu/id.php?id=65017)

## Selected writings

On [[MSp]] and [[quaternionic oriented cohomology theory]]:

* [[Ivan Panin]], [[Charles Walter]], _Quaternionic Grassmannians and Pontryagin classes in algebraic geometry_ ([hal:00531725](https://hal.archives-ouvertes.fr/hal-00531725), [pdf](https://www.math.uni-bielefeld.de/LAG/man/405.pdf))

* [[Ivan Panin]], [[Charles Walter]], _Quaternionic Grassmannians and Borel classes in algebraic geometry_ ([arXiv:1011.0649](https://arxiv.org/abs/1011.0649))

* [[Ivan Panin]], [[Charles Walter]], _On the algebraic cobordism spectra $MSL$ and $MSp$_ ([arXiv:1011.0651](https://arxiv.org/abs/1011.0651))

* [[Ivan Panin]], _[Oriented theories and symplectic cobordism](https://www.esaga.uni-due.de/marc.levine/SpecialSemester/Panin/)_, Seminar

## Related topics

* [[motivic homotopy theory]]

* [[algebraic cobordism]]

category: people


Ivan Todorov

__Ivan T. Todorov__ is a Bulgarian theoretical and mathematical physicist, a specialist in quantum field theory, especially conformal field theory as well as representation theoretic methods in quantum field theory.

Disambiguation: not a mathematician [[Ivan G. Todorov]].

* userpage at [gScholar](https://scholar.google.com/citations?user=wZiNNXoAAAAJ)

## Selected writings

* N. N. Bogoliubov, A. A. Logunov, I. T. Todorov, *Introduction to axiomatic quantum field theory* (1975)

On [[braid group representations]] via [[monodromy]] of the [[Knizhnik-Zamolodchikov connection]] on [[conformal blocks]]:

* [[Ivan Todorov]], [[Ludmil Hadjiivanov]], *Monodromy Representations of the Braid Group*, Phys. Atom. Nucl. **64** (2001) 2059-2068; Yad.Fiz. **64** (2001) 2149-2158 [[arXiv:hep-th/0012099](https://arxiv.org/abs/hep-th/0012099), [doi:10.1134/1.1432899](https://doi.org/10.1134/1.1432899)]

A version of [[coherent state]]s for [[quantum group]] $SU_q(3)$ is for the first time inrtoduced in

* H. Sazdjian, Y. S. Stanev, I. T. Todorov,
$SU(3)$-coherent state operators and invariant correlation functions and their quantum group
counterparts, J. Math. Phys. 36 (1995) 2030--2052.

* L. K. Hadjiivanov, R. R. Paunov, I. T. Todorov, _$U_q$-covariant oscillators and vertex operators_, J. Math. Phys. __33__, 1379 (1992) [doi](https://doi.org/10.1063/1.529713)

category: people

[[!redirects Ivan T. Todorov]]


Ivan Vinogradov

Ivan Vinogradov (Иван Матвеевич Виноградов) was a Russian mathematician working at the Steklov Mathematical Institute in Moscow, known for his numerous contributions to [[analytic number theory]].

He got his PhD degree around 1915 at Saint Petersburg University, supervised by Yakov Uspensky (also published as James Uspensky).

He was the editor-in-chief of the [[Encyclopaedia of Mathematics]].

category: people

[[!redirects И. М. Виноградов]]
[[!redirects Иван Матвеевич Виноградов]]
[[!redirects Иван Виноградов]]
[[!redirects I. M. Vinogradov]]
[[!redirects I. Vinogradov]]


Iver Ottosen

* [personal page](https://people.math.aau.dk/~ottosen/)

* [institute page](https://orbit.dtu.dk/en/persons/iver-m%C3%B8lgaard-ottosen)

## Selected writings

On the [[ordinary cohomology]] of [[cyclic loop spaces]] ("[[string cohomology]]"):

* {#BokstedtOttosen05} [[Marcel Bökstedt]], [[Iver Ottosen]], *A spectral sequence for string cohomology*, Topology Volume 44, Issue 6, November 2005, Pages 1181-1212 ([arXiv:math/0411571](https://arxiv.org/abs/math/0411571), [doi:10.1016/j.top.2005.04.006](https://doi.org/10.1016/j.top.2005.04.006))

and specifically for [[complex projective spaces]]:

* [[Marcel Bökstedt]], [[Iver Ottosen]], *String cohomology groups of complex projective spaces*, Algebr. Geom. Topol. 7(4): 2165-2238 (2007). ([arXiv:math/0605754](https://arxiv.org/abs/math/0605754), [doi:10.2140/agt.2007.7.2165](https://projecteuclid.org/journals/algebraic-and-geometric-topology/volume-7/issue-4/String-cohomology-groups-of-complex-projective-spaces/10.2140/agt.2007.7.2165.full))

category: people


Ivo Dell'Ambrogio

Ivo Dell'Ambrogio is a _maître de conférences_ (associate professor) in pure mathematics, working with the Geometry and Topology team at the _Laboratoire Paul Painlevé_ of the _Université de Lille_.

* [webpage](https://math.univ-lille1.fr/~dellambr/)

## Related $n$Lab entries

* [[Spanier-Whitehead category]]

* [[Mackey functor]]

## Selected Publications and Preprints

* The spectrum of equivariant Kasparov theory for cyclic groups of prime order,
Joint with [[Ralf Meyer]], Preprint September 2020, 13 pages. Submitted.

* On the comparison of spans and bisets,
Joint with James Huglo,
Cahiers Topol. Géom. Differ. Catég. LXII-1, (2021) 61 - 104. pages.

* Green equivalences in equivariant mathematics,
Joint with [[Paul Balmer]], Math. Ann. (to appear); Preprint January 2020, 25 pages.

* Axiomatic representation theory of finite groups by way of groupoids,
In _Equivariant Topology and Derived Algebra_. Edited by S. Balchin, D. Barnes, M. Kedziorek, and M. Szymik. Cambridge University Press. (to appear); Preprint October 2019, 44 pages.

* Mackey 2-functors and Mackey 2-motives,
Joint with [[Paul Balmer]]

* (with [[Goncalo Tabuada]]), _A Quillen model structure for classical Morita theory and a tensor categorification of the Brauer group_ , arXiv:1211.2309 (2012). ([pdf](http://arxiv.org/pdf/1211.2309v1))

category: people
[[!redirects I. Dell'Ambrogio]]


Ivo Sachs

* [webpage](http://www.theorie.physik.uni-muenchen.de/cosmology/members/professors/sachs/index.html)

## Selected writings

On [[homotopy algebra]] in [[string field theory]]:

* [[Ivo Sachs]], _Homotopy Algebras in String Field Theory_, in: Proceedings of _[[Higher Structures in M-Theory 2018]]_ Fortschritte der Physik, Special Issue Volume 67, Issue 8-9 ([arXiv:1903.02870](https://arxiv.org/abs/1903.02870), [doi:10.1002/prop.201910013](https://doi.org/10.1002/prop.201910013))

## Related $n$Lab entries

* [[string field theory]]

category: people


Iwahori–Hecke algebra

* [[Hecke algebra]]

## Related concepts

* [[induced representation]]

## References

* Wikipedia, _[Iwahori-Hecke algebra](http://en.wikipedia.org/wiki/Iwahori%E2%80%93Hecke_algebra)_

[[!redirects Iwahori–Hecke algebra]]


Iwasawa algebra

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## Idea

The _Iwasawa algebra_, classically, is the completed group ring $\mathbb{Z}_{p}[[\Gamma]]$, where $\Gamma$ is the subgroup of $\Gal(\mathbb{Q}(\mu_{p^{}\infty})/\mathbb{Q})\cong \mathbb{Z}_{p}^{\times}$ which is isomorphic to $1+p\mathbb{Z}_{p}$. The Iwasawa algebra is also isomorphic to the power series ring $\mathbb{Z}_{p}[[T]]$.

## Related concepts

* [[Iwasawa theory]]

## References

* K. Ardakov, K. A. Brown, _Ring-theoretic properties of Iwasawa algebras: a survey_, Documenta Mathematica, Extra Volume: John H. Coates' Sixtieth Birthday (2006), 7-33 [link](http://www.math.uiuc.edu/documenta/vol-coates/ardakov_brown.html)

See also:

* Wikipedia, *[Iwasawa_algebra](http://en.wikipedia.org/wiki/Iwasawa_algebra)*

[[!redirects Iwasawa algebras]]


Iwasawa decomposition

## References

* Wikipedia, _[Iwasawa decomposition](http://en.wikipedia.org/wiki/Iwasawa_decomposition)_

[[!redirects Iwasawa decompositions]]


Iwasawa polynomial

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## Idea

As a construction in [[Iwasawa theory]], the __Iwasawa polynomial__ is a number theoretic analogue of the [[Alexander polynomial]].  (e.g. [CL 12, plage 3](#CL12))

The relation between the Alexander polynomial
and the Lefschetz (spectral) zeta function
is analogous to that between the Iwasawa polynomial
and the p-adic analytic zeta function. See ([Morishita 12, chapter 12](#Morishita12)).

> A generator of the characteristic ideal of (the Pontryagin dual of the Selmer's subgroup) $Sel(X_\inf ,\rho)^\ast$, called
the twisted Iwasawa polynomial, is an analogue of the twisted Alexander polynomial.

## References

* {#CL12} Chao Li, _Iwasawa theory_, lecture notes 2012 ([pdf](http://www.math.harvard.edu/~chaoli/tutorial2012/Lecture14.pdf))

* {#Morishita12} Masanori Morishita, _Knots and primes: an introduction to arithmetic topology_, Springer 2012

[[!redirects Iwasawa polynomials]]


Iwasawa theory

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>Compared with Kummer's criterion and class number formula, Iwasawa theory is finer in the point that it describes not only the class number, i.e. the order of the ideal class group, but also the action of the Galois group on the ideal class group. In fact, one could even say that the aim of Iwasawa theory is to describe Galois actions on arithmetic objects in terms of zeta values. [[Kato 06](#Kato06)]

## Idea

Iwasawa theory is the study of certain [[modules]] of [[arithmetic]] interest over the [[Iwasawa algebra]] $\mathbb{Z}_{p}[[\Gamma]]$, where $\Gamma$ is the [[subgroup]] of the [[Galois group]] $\mathrm{Gal}(\mathbb{Q}(\mu_{p^{\infty}})/\mathbb{Q})\cong (1+p\mathbb{Z}_{p})\times\mu_{p-1}$ [[isomorphism|isomorphic]] to $1+p\mathbb{Z}_{p}$. An example of such a module is the [[inverse limit]] of the $p$-part of class groups of [[cyclotomic fields]].

## The Main Conjecture of Iwasawa Theory

Let $A_{n}$ be the [[Sylow p-subgroup]] of the [[ideal class group]] of $\mathbb{Q}(\mu_{p^{n}})$. Let $X_{\infty}$ be the [[inverse limit]] of the $A_{n}$ under the norm maps. Then $X_{\infty}$ is a module over the Iwasawa algebra $\mathbb{Z}_{p}[[\Gamma]]$.

There is also an action of $\mathbb{Z}_{p}[[\Delta]]$ on $X_{\infty}$, where $\Delta=\mathrm{Gal}(\mathbb{Q}(\mu_{p})/\mathbb{Q})$. Let $\delta$ be a generator of $\Delta$ and let $a$ be such that $\delta(\zeta_{p})=\zeta_{p}^{a}$. Then $X_{\infty}$ breaks up into eigenspaces

$$X_{\infty}^{(i)}=\lbrace x\in X_{\infty}\vert\delta(x)=a^{i}x\rbrace$$

The _main conjecture of Iwasawa theory_ then says that for odd $i$, the characteristic ideal of $X_{\infty}^{(i)}$ is generated by a power series $g_{i}$ such that

$$g_{i}(v^{s}-1)=L_{p}(\omega^{1-i},s)$$

where $v$ is the topological generator of $1+p\mathbb{Z}_{p}$ which is the evaluation of the p-adic cyclotomic character on the topological generator of $\Gamma$ and $L_{p}(\omega^{1-i},s)$ is the Kubota-Leopoldt p-adic zeta function.

The main conjecture of Iwasawa theory was proved in [MazurWiles84](#MazurWiles84). It generalizes the _Herbrand-Ribet theorem_. The method of proof for the main conjecture of Iwasawa theory also follows similar ideas to the proof of the converse to Herbrand's theorem in [Ribet76](#Ribet76).

## Relation to Arithmetic Topology

Via the 3-manifold/number field analogy of [[arithmetic topology]], Iwasawa theory can be seen as the analog of Alexander-Fox theory (see [sec. 7 of Morishita](#Morishita)).

## References

* Romyar Sharifi, _Modular Curves and Cyclotomic Fields [pdf](https://swc-math.github.io/aws/2018/2018SharifiNotes.pdf)

* Ralph Greenberg (2001), _Iwasawa theory---past and present_, in Miyake, Katsuya, Class field theory---its centenary and prospect (Tokyo, 1998), Adv. Stud. Pure Math. 30, Tokyo: Math. Soc. Japan, pp. 335–385 ([ps file](https://www.math.washington.edu/~greenber/iwhi.ps)).

* Ralph Greenberg, _Topics in Iwasawa Theory_, ([online book](https://www.math.washington.edu/~greenber/book.pdf) in process of being written).

* {#Kato06} Kazuya Kato, _Iwasawa theory and generalizations_, ([ICM 2006 talk](http://www.icm2006.org/proceedings/Vol_I/18.pdf)).

* {#Morishita} Masanori Morishita, _Analogies between Knots and Primes, 3-Manifolds and Number Rings_, ([arxiv](http://arxiv.org/abs/0904.3399))

See also:

* Wikipedia, *[Iwasawa theory](http://en.wikipedia.org/wiki/Iwasawa_theory)*

The main conjecture of Iwasawa theory was proved in

* {#MazurWiles84}[[Barry Mazur]] and [[Andrew Wiles]], _Class fields of abelian extensions of $\mathbb{Q}$_, Invent. Math. 76 (1984), 179–330. MR0742853

The proof of the converse to Herbrand's theorem is in

* {#Ribet76}[[Ken Ribet]], _A modular construction of unramified p-extensions of $\mathbb{Q}(\mu_{p})$_, Invent. Math. 34 (1976), 151–162. MR0419403


Iwasawa-Tate theory

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## Idea

What has come to be called _Iwasawa-Tate theory_ is a method of expressing [[zeta functions]], [[L-functions]] and [[theta functions]] as [[adelic integrals]] over [[idele groups]] and deriving this way their fundamental properties such as their [[analytic continuation]], their [[functional equation]] and their [[Euler product]] form.

[[John Tate]] in his thesis ([Tate 50](#Tate50)) had generalized the notion of [[zeta function]] according to his advisor [[Emil Artin]] from "the sum over integral ideals of certain type of ideal character" to

> the [[adelic integral]] over the [[idèle group]] of a rather general weight function times the idèle character which is trivial on field elements. The role of Hecke's complicated theta-formulas for [[theta functions]] formed over a lattice in $n$-dimensional space of classical number theory can be played by a simple [[Poisson formula]] for general function of valuation vectors, summed over the discrete subgroup of field elements

[[Kenkichi Iwasawa]] in [Iwasawa 5x](#Iwasawa50) has rediscovered and extended this approach using the wider study of [[invariant integration]] on locally compact groups over ideles. It is distinguished from the related but different subject of [[Iwasawa theory]] (cf. [wikipedia](http://en.wikipedia.org/wiki/Iwasawa_theory))

## Related entries

* [[adelic integration]], [[locally compact group]], [[Fourier analysis]], [[zeta function]], [[L-function]], [[ring of adeles]], [[group of ideles]]

## References

The original articles are

* {#Tate50} [[John Tate]], _Fourier analysis in number fields, and Hecke's zeta-functions_, Princeton, May 1950, thesis; reproduced in _Algebraic Number Theory_ (Proc. Instructional Conf., Brighton, 1965) pp. 305–347, Acad, Press 1967
[MR0217026](http://www.ams.org/mathscinet-getitem?mr=0217026)

* {#Iwasawa50} [[Kenkichi Iwasawa]], _A note on functions_, Proc. ICM 1950, [link](http://www.mathunion.org/ICM/ICM1950.1) [MR0044534](http://www.ams.org/mathscinet-getitem?mr=0044534); _On the rings of valuation vectors_, Ann. Math. (II) 57:2 (Mar., 1953), pp. 331-356 [jstor](http://www.jstor.org/stable/1969863); Letter to [[Jean Dieudonne]] [link](http://books.google.com/books?ei=jyALTq-_L4nkiAL6zrHXAQ)

Reviews include

* {#Garrett11} [[Paul Garrett]], _Iwasawa-Tate on ζ-functions and L-functions_, 2011
([pdf](http://www-users.math.umn.edu/~garrett/m/mfms/notes_c/Iwasawa-Tate.pdf))

* cyclopaedia.info, _[Iwasawa-Tate theory](http://www.cyclopaedia.info/wiki/Iwasawa-Tate-theory-1)_

* Wikipedia, _[Tate's thesis](http://en.wikipedia.org/wiki/Tate%27s_thesis)_

  _Bibliographic fragments concerning Iwasawa-Tate theory_, ([pdf](http://www-users.math.umn.edu/~garrett/m/v/iwasawa-tate))

Further developments include

* [[David Wright]], _Twists of the Iwasawa-Tate zeta function_, Math. Zeitschrift 200:2, pp 209-231, 1989  ([[WrightIwasawaTate.pdf:file]])

[[!redirects Tate's thesis]]

[[!redirects Fourier analysis in number fields, and Hecke's zeta-functions]]
[[!redirects Fourier analysis in number fields and Hecke's zeta-functions]]

[[!redirects Fourier analysis in number fields, and Hecke's zeta-functions]]


J-B Vienney

I'm a French [[math]] PhD student at the University of Ottawa. My supervisor is [[Rick Blute]]. [[Phil Scott]] was also my supervisor.

I'm interested in the [[logic|logical]] approach to [[differentiation]] through [[differential linear logic]] and [[differential categories]]. I like trying to combine ideas from this field with the notion of [[graded modality]] to talk about ubiquitous functors in mathematics such as [[symmetric powers|symmetric]], [[exterior powers|exterior]], [[divided powers]] or [[homology |homology functors]].

More broadly, I'm driven by the idea of turning some algebraic concepts from [[mathematics]] into [[category theory]] and [[proof theory]] and make everything live together in the neatest way.

I like the philosophy of [[Ludwig Wittgenstein]] and think that it is a useful inspiration for research in mathematics and computer science.

## Projects

I wrote papers on:

- Graded differential linear logic and graded differential categories which index the [[exponential modality]] of differential linear logic or differential categories by a [[rig]]. The project includes graded kinds of [[Seely isomorphism]] (with [[Jean-Simon Lemay]]).
- A [[string diagrams|string-diagrammatic]] calculus for [[symmetric powers]] based on an algebraic characterization of symmetric powers in symmetric monoidal $\mathbb{Q}^{+}$-linear categories.

I'm working on:

- Several little projects related to graded differential categories and symmetric powers in additive symmetric monoidal categories.
- Higher-order tangent categories which are a categorical framework for higher-order tangent bundles similar to [[tangent bundle category|tangent bundle categories]].
- Hasse-Schimdt differential categories which replace usual differentiation by [[Hasse-Schmidt derivative|Hasse-Schmidt differentiation]] in the notion of differential category.

## Papers

*  [[Jean-Simon Lemay|Jean-Simon Pacaud Lemay]], [[J-B Vienney|Jean-Baptiste Vienney]], _Graded Differential Categories and Graded Differential Linear Logic_,  MFPS2023, [link](https://entics.episciences.org/paper/view/id/12290)
 * [[J-B Vienney|Jean-Baptiste Vienney]], _String diagrams for symmetric powers I: In symmetric monoidal $\mathbb{Q}_{\ge 0}$-linear categories_, preprint, [ArXiv](https://arxiv.org/abs/2308.02094)

category: people

[[!redirects Jean-Baptiste Vienney]]


J-homomorphism

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
#### Stable Homotopy theory
+--{: .hide}
[[!include stable homotopy theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _$J$-homomorphism_ is traditionally a family of [[group]] [[homomorphisms]]

$$
  J_i \;\colon\; \pi_i(O(n)) \longrightarrow \pi_{n+i}(S^n)
$$

from the [[homotopy groups]] of (the [[topological space]] underlying) the [[orthogonal group]] to the [[homotopy groups of spheres]].
This refines to a morphism of [[∞-groups]]

$$
  J \;\colon\; O \longrightarrow GL_1(\mathbb{S})
$$

from the [[stable orthogonal group]] (regarded as a [[group object in an (∞,1)-category|group object]] in $L_{whe} Top \simeq$ [[∞Grpd]])
to the [[∞-group of units]] of the [[sphere spectrum]], regarded as an [[E-∞ ring spectrum]].

By postcomposition, the [[delooping]] of the J-homomorphism

$$
  B J \;\colon\; B O \to B GL_1(\mathbb{S})
$$

sends real [[vector bundles]] to [[sphere bundles]], namely to [[(∞,1)-line bundles]] with typical [[fiber]] the [[sphere spectrum]] $\mathbb{S}$. See also at _[[Thom space]]_ for more on this.

The description of the image of the $J$-homomorphism in the [[stable homotopy groups of spheres]] was an important precursor to the development of [[chromatic homotopy theory]], which is used to explain the periodicities seen in the image of the J-homomorphism (see also [Lurie 10, remark 8](#Lurie)). See also at _[[periodicity theorem]]_.

## Definition

### On groups
 {#OnGroups}

+-- {: .num_defn #SphereAsCompactification}
###### Definition

For $n \in \mathbb{N}$ regard the $n$-[[sphere]] (as a [[topological space]]) as the [[one-point compactification]] of the [[Cartesian space]] $\mathbb{R}^n$

$$
  S^n \simeq (\mathbb{R}^n)^\ast
  \,.
$$

=--

+-- {: .num_remark #ActionOfOrthogonalOnSphere}
###### Remark

Since the process of [[one-point compactification]] is a [[functor]] on [[proper maps]], hence on [[homeomorphisms]], via def. \ref{SphereAsCompactification} the $n$-sphere inherits from the canonical [[action]] of the [[orthogonal group]] $O(n)$ on $\mathbb{R}^n$ an [[action]]

$$
  O(n) \times S^n \longrightarrow S^n
$$

(by [[continuous maps]]) which preserves the base point (the "point at infinity").

=--

For definiteness we distinguish in the following notationally between

1. the $n$-[[sphere]] $S^n \in Top$ regarded as a [[topological space]];

1. its [[homotopy type]] $\Pi(S^n) \in L_{whe} Top \simeq$ [[∞Grpd]] given by its [[fundamental ∞-groupoid]].

Similarly we write $\Pi(O(n))$ for the [[homotopy type]] of the [[orthogonal group]], regarded as a [[group object in an (∞,1)-category]] in [[∞Grpd]] (using that the [[shape modality]] $\Pi$ preserves [[finite products]]).

+-- {: .num_defn #AutoequivalencesOfnSphere}
###### Definition

For $n \in \mathbb{N}$ write $H(n)$ for the [[automorphism ∞-group]] of homotopy self-equivalences $S^n \longrightarrow S^n$, hence

$$
  H(n) \coloneqq Aut_{\infty Grpd^{\ast/}}(\Pi(S^n))
  \,.
$$

=--

+-- {: .num_remark }
###### Remark

The [[∞-group]] $H(n)$, def. \ref{AutoequivalencesOfnSphere}, constitutes the two [[connected components]] of the $n$-fold based [[loop space]] $\Omega^n S^n$ corresponding to the [[homotopy classes]] $\pm 1 \in \pi_n(S^n)$.

=--

+-- {: .num_defn #OrthogonalActionOnSphereOnHomotopyGroups}
###### Definition

Via the presentation of [[∞Grpd]] by the [[cartesian closed model structure|cartesian closed]] [[model structure on topological spaces|model structure on compactly generated topological spaces]] (and using that $S^n$ and $O(n)$ and hence their product are [[compact topological spaces|compact]]) we have that for $n \in \mathbb{N}$ the [[continuous function|continuous]] [[action]] of $O(n)$ on $S^n$
of remark \ref{ActionOfOrthogonalOnSphere}, which by [[cartesian closed category|cartesian closure]] is equivalently a homomorphism of [[topological groups]] of the form

$$
  O(n) \longrightarrow Aut_{Top^{\ast/}}(S^n)
  \,,
$$

induces a homomorphism of [[∞-groups]] of the form

$$
  \Pi(O(n)) \longrightarrow Aut_{\infty Grpd^{\ast/}}(\Pi(S^n))
  \,.
$$

This in turn induces for each $i \in \mathbb{N}$ homomorphisms of [[homotopy groups]] of the form

$$
  \pi_i(O(n)) \longrightarrow \pi_i(\Omega^n S^n) \simeq \pi_{n+i}(S^n)
  \,.
$$

=--

+-- {: .num_remark #OrthogonalActionOnSphereOnHomotopyGroups}
###### Remark

By construction, the homomorphisms of remark \ref{OrthogonalActionOnSphereOnHomotopyGroups} are compatible with  [[suspension]] in that for all $n \in \mathbb{N}$ the [[diagrams]]

$$
  \array{
    O(n) &\longrightarrow& Aut_{Top^{\ast/}}(S^n)
    \\
    \downarrow && \downarrow
    \\
    O(n+1) &\longrightarrow& Aut_{Top^{\ast/}}(S^{n+1})
  }
$$

in $Grp(Top)$ commute, and hence so do the diagrams

$$
  \array{
    \Pi(O(n)) &\longrightarrow& Aut_{\infty Grpd^{\ast/}}(\Pi(S^n))
    \\
    \downarrow && \downarrow
    \\
    \Pi(O(n+1)) &\longrightarrow& Aut_{\infty Grpd^{\ast/}}(\Pi(S^{n+1}))
  }
$$

in $Grp(\infty Grpd)$, up to [[homotopy]].

=--

Therefore one can take the [[direct limit]] over $n$:

+-- {: .num_defn #JHom}
###### Definition

By remark \ref{OrthogonalActionOnSphereOnHomotopyGroups}
there is induced a homomorphism

$$
  J_i \;\colon\; \pi_\bullet(O) \longrightarrow \pi_\bullet(\mathbb{S})
$$

from the [[homotopy groups]] of the [[stable orthogonal group]]
to the [[stable homotopy groups of spheres]]. This is called
the **J-homomorphism**.

=--

### Delooped: On classifying spaces and K-theory classes

+-- {: .num_remark #DeloopedJ}
###### Remark

Since the maps of def. \ref{OrthogonalActionOnSphereOnHomotopyGroups}
are [[∞-group]] [[homomorphisms]], there exists their [[delooping]]

$$
  B J \;\colon\; B O \longrightarrow B GL_1(\mathbb{S}) = B H
  \,.
$$

=--

+-- {: .num_remark }
###### Remark

Here $GL_1(\mathbb{S})$ is the [[∞-group of units]] of the [[sphere spectrum]].

=--

This map $B J$ is the [[universal characteristic class]] of stable [[vector bundles]] with values in [[spherical fibrations]]:

+-- {: .num_defn #SphereBundleOfVectorBundle}
###### Definition

For $V \to X$ a [[vector bundle]], write $S^V$ for its [[fiber]]-wise [[one-point compactification]]. This is a [[sphere bundle]]/[[spherical fibration]]. Write $\mathbb{S}^V$ for the $X$-[[parameterized spectrum]] which is fiberwise the [[suspension spectrum]] of $S^V$.

=--

It is immediate that:

+-- {: .num_prop #SphericalFibrationsOfVectorBundlesClassifiedViaJ}
###### Proposition
**([[spherical fibrations]] of [[vector bundles]] classified via J-homomorphism)**

For $V \to X$ a [[vector bundle]] classified by a map $X \to B O$,
the corresponding [[spherical fibration]] $\mathbb{S}^V$, def. \ref{SphereBundleOfVectorBundle}, is classified by $X \to B O \stackrel{B J}{\longrightarrow} B GL_1(\mathbb{S})$, def. \ref{DeloopedJ}.

=--

This construction descends to a map

$$
  KO^0(X) \longrightarrow Sph(X)
$$

from [[topological K-theory]] to [[spherical fibrations]]

(e.g. [Clausen 11, Section 2](#Clausen11), [Westerland 14, slide 5](#Westerland14), [MO discussion](http://mathoverflow.net/a/156369/381))

## Properties
 {#Properties}

### Image of the J-homomorphism
 {#Image}

#### Traditional formulation
 {#ImageOfJTradtionalFormulation}

##### Description of the image

The following characterization of the [[image]] of the J-homomorphism on [[homotopy groups]] derives from a statement that was first conjectured in ([Adams 66](#Adams66)) -- and since called the _[[Adams conjecture]]_ -- and then proven in ([Quillen 71](#Quillen71), [Sullivan 74](#Sullivan74)).

+-- {: .num_remark}
###### Remark

By the discussion at _[orthogonal group -- homotopy groups](orthogonal%20group#HomotopyGroups)_ we have that the [[homotopy groups]] of the [[stable orthogonal group]] are

| $n\;mod\; 8$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|-----|---|---|---|---|---|---|---|---|
| $\pi_n(O)$ | $\mathbb{Z}_2$ | $\mathbb{Z}_2$ |0 | $\mathbb{Z}$ | 0 | 0 | 0 | $\mathbb{Z}$ |

Because all groups appearing here and in the following are [[cyclic groups]], we instead write down the [[order of a group|order]]

| $n\;mod\; 8$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|-----|---|---|---|---|---|---|---|---|
| ${\vert\pi_n(O)\vert}$ | 2 | 2 | 1 | $\infty$ | 1 | 1 | 1 | $\infty$ |

=--

For the following statement it is convenient to restrict to J-homomorphism to the stable [[special orthogonal group]] $S O$, which removes the lowest degree homotopy group in the above

| $n\;mod\; 8$ | $n = 0$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|-----|---|---|---|---|---|---|---|---|---|
| $\pi_n(S O)$ | 0 | $\mathbb{Z}_2$ | $\mathbb{Z}_2$ | 0 | $\mathbb{Z}$ | 0 | 0 | 0 | $\mathbb{Z}$ |

| $n\;mod\; 8$ | $n = 0$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|-----|---|---|---|---|---|---|---|---|---|
| ${\vert\pi_n(S O)\vert}$ | 1 | 2 | 2 | 1 | $\infty$ | 1 | 1 | 1 | $\infty$ |

+-- {: .num_theorem #AdamsQuillenTheorem}
###### Theorem

The [[stable homotopy groups of spheres]] $\pi_n(\mathbb{S})$ are the [[direct sum]] of the ([[cyclic group|cyclic]]) [[image]]
$im(J|_{SO})$ of the J-homomorphism, def. \ref{JHom}, applied to the [[special orthogonal group]] and the [[kernel]] of the [[Adams e-invariant]].

Moreover,

* for $n = 0 \;mod \;8$ and $n = 1 \;mod \; 8$ and $n$ positive the J-homomorphism $\pi_n(J) \colon \pi_n(S O) \to \pi_n(\mathbb{S})$ is [[injection|injective]], hence its image is $\mathbb{Z}_2$,

* for $n = 3\; mod\; 8$ and $n = 7 \; mod \; 8$ hence for $n = 4 k -1$, the [[order of a group|order]] of the image is equal to the [[denominator]] of $B_{2k}/4k$ in its reduced form, where $B_{2k}$ is the [[Bernoulli number]]

* for all other cases the image is necessarily zero.

=--

This characterization of the image of $J$ is due to ([Adams 66](#Adams66), [Quillen 71](#Quillen71), [Sullivan 74](#Sullivan74)).
Specifically the identification of $J(\pi_{4n-1}(S O))$ is ([Adams 65a, theorem 3.7](#Adams65a) and the direct summand property is ([Adams 66, theorems 1.1-1.6.](#Adams66)).
That the image is a direct summand of the codomain is proven for instance in ([Switzer 75, end of chapter 19](#Switzer75)).

A modern version of the proof, using methods from [[chromatic homotopy theory]], is surveyed in some detail in ([Lorman 13](#Lorman13)).

The statement of the theorem is recalled for instance as ([Ravenel, chapter 1, theorem 1.1.13](#RavenelCh1)).
Another computation of the image of $J$ is in ([Ravenel, chapter 5, section 3](#RavenelChapter5)).

+-- {: .num_remark}
###### Remark

The order of $J(\pi_{4k-1} O)$ in theorem \ref{AdamsQuillenTheorem}
is for low $k$ given by the following table

| k | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|---|---|---|---|---|---|---|---|---|---|---|
| $\vert J(\pi_{4k-1}(O))\vert$ | 24 | 240 | 504 | 480 | 264 | 65,520 | 24 | 16,320 | 28,728 | 13,200 |

=--

See for instance ([Ravenel, Chapt. 1,  p. 5](#RavenelCh1)).

+-- {: .num_remark}
###### Remark

Therefore we have in low degree the following situation

[[!include image of J -- table]]

=--

The following tables show the [[p-primary components]] of the [[stable homotopy groups of spheres]] for low values, the image of J appears as the bottom row.

Here the horizontal index is the degree $n$ of the stable homotopy group $\pi_n$. The appearance of a string of $k$ connected dots vertically above index $n$ means that there is a [[direct sum|direct summand]] [[primary group]] of [[order of a group|order]] $p^k$. See example \ref{InterpretTable} below for illustration.

(The tables are taken from ([Hatcher](#Hatcher)), where in turn were they were generated based on ([Ravenel 86](#RavenelCh1)).

**at $p = 2$**

**at $p = 3$**

**at $p = 5$**

+-- {: .num_example #InterpretTable}
###### Example

The [[finite abelian group]] $\pi_3(\mathbb{S}) \simeq \mathbb{Z}_{24}$ decomposes into [[primary groups]] as $\simeq \mathbb{Z}_8 \oplus \mathbb{Z}_3$. Here $8 = 2^3$ corresponds to the three dots above $n = 3$ in the first table, and $3 = 3^1$ to the single dot over $n = 3$ in the second.

The [[finite abelian group]] $\pi_7(\mathbb{S}) \simeq \mathbb{Z}_{24}$ decomposes into [[primary groups]] as $\simeq \mathbb{Z}_{16} \oplus \mathbb{Z}_3 \oplus \mathbb{Z}_5$. Here $16 = 2^4$ corresponds to the four dots above $n = 7$ in the first table, and $3 = 3^1$ to the single dot over $n = 7$ in the second and $5 = 5^1$ to the single dot over $n = 7$ in the third table.

=--

##### Characterization via the Adams operations

(...)

We indicate how the [[Adams conjecture]]/Adams-Quillen-Sullivan theorem serves to identify the image of the J-homomorphism. We follow the modern account as reviewed in ([Lorman 13](#Lorman13)).

(...)

Write $\psi^k$ for the $k$th [[Adams operation]] on [[complex K-theory]].

Let $p$ be a [[prime]]. Consider $k$ coprime to $p$.

The [[Adams conjecture]] implies that completed at $p$, the
J-homomorphism factors through the [[homotopy fiber]] of $1 - \psi^k$.

proof:

We have a homotopy-commuting diagram

$$
  \array{
    B U_p &\stackrel{1 - \psi^k}{\longrightarrow}& B U_p
    \\
    \downarrow &\swArrow_\simeq& \downarrow
    \\
    \ast &\stackrel{0}{\longrightarrow}& B H_p
  }
  \,.
$$

The [[pasting]] composite with the [[homotopy pullback]] that witnesses the [[homotopy fiber]] of $1 - \psi^k$ induces via the [[universal property]] of the [[loop space object]] a canonical map $fib(1-\psi^k) \longrightarrow H_p$:

$$
  \array{
    fib(1-\psi^k) &\longrightarrow& \ast
    \\
    \downarrow &&  \downarrow
    \\
    B U_p &\stackrel{1 - \psi^k}{\longrightarrow}& B U_p
    \\
    \downarrow &\swArrow_\simeq& \downarrow
    \\
    \ast &\stackrel{0}{\longrightarrow}& B H_p
  }
  \;\;\;
  \simeq
  \;\;\;
  \array{
    fib(1-\psi^k) &\longrightarrow& \ast
    \\
    \downarrow &&  \downarrow
    \\
    H_p &\stackrel{}{\longrightarrow}& \ast
    \\
    \downarrow &\swArrow_\simeq& \downarrow
    \\
    \ast &\stackrel{0}{\longrightarrow}& B H_p
  }
  \,.
$$

(...)

##### The J-spectrum

The [[J-spectrum]] is a [[spectrum]] whose [[homotopy groups]] are close to being the image of the J-homomorphism.

(...)

#### Formulation in chromatic homotopy theory
 {#ImageOfJInChromotopy}

In terms of [[chromatic homotopy theory]] the nature of the image of the J-homomorphism can be formulated more succinctly as follows.

Write $E(1)$ for the first [[Morava E-theory]] [[spectrum]] at given [[prime number]] $p$. Write
$L_{E(1)}\mathbb{S}$ for the [[Bousfield localization of spectra]] of the [[sphere spectrum]] at $E(1)$.

+-- {: .num_theorem }
###### Theorem

The [[homotopy groups]] of the $E(1)$-localized sphere spectrum are

$$
  \pi_n L_{E(1)} \mathbb{S}
  \simeq
  \left\{
    \array{
      \mathbb{Z} & if\; n = 0
       \\
       \mathbb{Q}_p/\mathbb{Z}_p & if\; n= -2
       \\
       \mathbb{Z}/p^{k+1}\mathbb{Z} & if\; n+1 = 2(p-1)p^k m \;with\; m \neq 0\;mod\;p
        \\
        0 & otherwise
    }
  \right.
  \,.
$$

=--

This appears as ([Lurie 10, theorem 6](#Lurie)) (note that there are two typos: first, the theorem is stated with $L_{K(1)} \mathbb{S}$ rather than $L_{E(1)} \mathbb{S}$, and second, it should be $n+1 = 2(p-1)p^k m$ but the $2$ is missing.)

+-- {: .num_defn }
###### Definition

Write $\mathbb{S}_p$ for the [[p-localization]] of the [[sphere spectrum]].
For $n \in \mathbb{Z}$, write $im(J)_n$ for the [[image]] of the $p$-localized J-homomorphism

$$
  J \;\colon\; \pi_n(O) \longrightarrow \pi_n(\mathbb{S}) \longrightarrow \pi_n(\mathbb{S}_{(p)})
  \,.
$$

=--

+-- {: .num_theorem }
###### Theorem

For $n \in \mathbb{N}$, the further [[Bousfield localization]] at [[Morava E-theory|Morava E(1)-theory]]  $\mathbb{S}_{(p)} \longrightarrow L_{E(1)}\mathbb{S}$ induces a [[isomorphism]]

$$
  im(J)_n \stackrel{\simeq}{\longrightarrow} \pi_n (L_{E(1)} \mathbb{S})
$$

between the image of the $J$-homomorphism and the $E(1)$-local [[stable homotopy groups of spheres]].

=--

In this form this appears as ([Lurie 10, theorem 7](#Lurie)). See also ([Behrens 13, section 1](#Behrens13)).

+-- {: .num_cor }
###### Corollary

The $E(1)$-[[Bousfield localization of spectra|localization map]] is surjective on non-negative homotopy groups:

$$
  \pi_n(\mathbb{S}_{(p)}) \longrightarrow \pi_n(L_{E(1)} \mathbb{S})
  \,.
$$

=--

For review see also ([Lorman 13](#Lorman13)). That $J$ factors through $L_{K(1)}\mathbb{S}$ is in ([Lorman 13, p. 4](#Lorman13))

+-- {: .num_remark }
###### Remark

Hence: the image of $J$ is essentially the first [[chromatic layer]] of the [[sphere spectrum]].

=--

## Related concepts

* [[e-invariant]]

* [[Thom spectra]], [[MO]]

* [[orientation in generalized cohomology]]

* [[J-homomorphism and chromatic homotopy]]

* [[scanning map equivalence]]

* [[twisted cohomotopy]]

\linebreak

## References

### General

The J-homomorphism was introduced in

* {#Whitehead42} [[George Whitehead]], _On the homotopy groups of spheres and rotation groups_, Annals of Mathematics. Second Series 43 (4): 634–640 (1942) ([jstor:1968956](http://www.jstor.org/stable/1968956)).

Textbook accounts:

* {#Kosinski93} [[Antoni Kosinski]], chapter IX 6.3 of: _Differential manifolds_, Academic Press 1993 ([pdf](http://www.maths.ed.ac.uk/~v1ranick/papers/kosinski.pdf), [ISBN:978-0-12-421850-5](https://www.sciencedirect.com/bookseries/pure-and-applied-mathematics/vol/138/suppl/C))

* [[Dai Tamaki]], [[Akira Kono]], Section 4.4 in: _Generalized Cohomology_, Translations of Mathematical Monographs, American Mathematical Society, 2006 ([ISBN: 978-0-8218-3514-2](https://bookstore.ams.org/mmono-230))

Lecture notes:

* [[Akhil Mathew]], _The Adams conjecture I_ ([web](http://amathew.wordpress.com/2013/01/23/the-adams-conjecture-i/))

* [[Akhil Mathew]], _Notes on the J-homomorphism_, 2012 ([pdf](http://math.uchicago.edu/~amathew/j.pdf), [[MathewJHomomorphism.pdf:file]])

* [[Michael Hopkins]] (notes by [[Akhil Mathew]]), Lecture 13 in: _Spectra and stable homotopy theory_, 2012 ([pdf](http://math.uchicago.edu/~amathew/256y.pdf), [[HopkinsMathewStableHomotopyTheory.pdf:file]])

* [[Arpon Raksit]], _Vector fields and the J-homomorphism_, 2014 ([pdf](http://stanford.edu/~arpon/math/files/vfields.pdf), [[Raksit_JHomomorphism.pdf:file]])

* {#Westerland14} [[Craig Westerland]], _Views on the J-homomorphism_, MSRI talk, 2014 ([recording](https://mathinstitutes.org/videos/videos/1464), [pdf](https://docs.google.com/viewer?url=https://www.msri.org/workshops/685/schedules/17895/documents/1978/assets/20102), [[WesterlandJHomomorphism.pdf:file]])

* [[Gereon Quick]], _The $e$-invariant and the $J$-homomorphism_, lecture notes in: _[Advanced algebraic topology](https://folk.ntnu.no/gereonq/Math231br.html)_, 2014 ([[QuickEInvariantAndJHomomorphism.pdf:file]])

Discussion in [[higher algebra]] in term of [[(∞,1)-module bundles]] is in

* [[Matthew Ando]], [[Andrew Blumberg]], [[David Gepner]], _Twists of K-theory and TMF_ ([arXiv:1002.3004](http://arxiv.org/abs/1002.3004))

and in higher [[chromatic homotopy theory]]:

* [[Craig Westerland]], _A higher chromatic analogue of the image of J_, Geometry & Topology 21 (2017) 1033–1093 ([doi:10.2140/gt.2017.21.1033](https://doi.org/10.2140/gt.2017.21.1033))

Discussion in [[p-adic geometry]]:

* {#Clausen11} Dustin Clausen, _$p$-adic J-homomorphisms and a product formula_ ([arXiv:1110.5851](https://arxiv.org/abs/1110.5851))

The [[complex geometry|complex]] J-homomorphism is discussed in

* [[Victor Snaith]], _The complex J-homomorphism_, Proc. London Math. Soc. (1977) s3-34 (2): 269-302 ([journal](http://plms.oxfordjournals.org/content/s3-34/2/269.full.pdf+html))

* [[Victor Snaith]], _Infinite loop maps and the complex $J$-homomorphism_, Bull. Amer. Math. Soc. Volume 82, Number 3 (1976), 508-510. ([Euclid](http://projecteuclid.org/euclid.bams/1183537922))

A [[p-adic number|p-adic]] J-homomorphism is described in

* Dustin Clausen, _p-adic J-homomorphisms and a product formula_ ([arXiv:1110.5851](http://arxiv.org/abs/1110.5851))

### The image of J

The analysis of the image of $J$ is due to

* [[John Adams]], _On the groups $J(X)$ I_, Topology 2 (3) (1963) ([pdf](http://math1.unice.fr/~cazanave/Gdt/ImJ/J-I.pdf))

* {#Adams65a} [[John Adams]], _On the groups $J(X)$ II_, Topology 3 (2) (1965) ([pdf](http://math1.unice.fr/~cazanave/Gdt/ImJ/J-II.pdf))

* {#Adams65b} [[John Adams]], _On the groups $J(X)$ III_, Topology 3 (3) (1965) ([pdf](http://math1.unice.fr/~cazanave/Gdt/ImJ/J-III.pdf))

* {#Adams66} [[John Adams]], _On the groups $J(X)$ IV_, Topology 5: 21,(1966)  _Correction_, Topology 7 (3): 331 (1968) ([pdf](http://math.unice.fr/~cazanave/Gdt/ImJ/J-IV.pdf))

* {#Quillen71} [[Daniel Quillen]], _The Adams conjecture_, Topology. an International Journal of Mathematics 10: 67–80 (1971) ([pdf](http://math1.unice.fr/~cazanave/Gdt/ImJ/Quillen.pdf))

* {#Sullivan74} [[Dennis Sullivan]], _Genetics of homotopy theory and the Adams conjecture_, Ann. of Math. 100 (1974), 1–79.

* {#Switzer75} [[Robert Switzer]], _Algebraic topology–homotopy and homology_, Springer-Verlag, New York, 1975.

The statement of the theorem about the characterization of the image is reviewed in

* {#RavenelCh1} [[Doug Ravenel]], _[[Complex cobordism and stable homotopy groups of spheres]]_, chapter 1, _An introduction to the homotopy groups of spheres_ ([pdf](http://www.math.rochester.edu/people/faculty/doug/mybooks/ravenel1.pdf))

see there also around theorem 3.4.16.

The details of the proof are surveyed in

* {#RavenelChapter5} [[Doug Ravenel]], _[[Complex cobordism and stable homotopy groups of spheres]]_, chapter 5, _The chromatic spectral sequence_ ([pdf](http://www.math.rochester.edu/people/faculty/doug/mybooks/ravenel5.pdf))

Tables showing the image of $J$ at low primes are in

* {#Hatcher} [[Allen Hatcher]], _Stable homotopy groups of spheres_ ([html](http://www.math.cornell.edu/~hatcher/stemfigs/stems.html))

Other reviews include

* [[Mark Mahowald]], _The Image of J in the EHP Sequence_, Annals of Mathematics   Second Series, Vol. 116, No. ([JSTOR](http://www.jstor.org/stable/2007048))

* [[Johannes Ebert]], _The Adams conjecture after Edgar Brown_, ([pdf](http://www.math.uni-muenster.de/u/jeber_02/talks/adams.pdf))

Discussion from the point of view of [[chromatic homotopy theory]] is in

* {#Lurie} [[Jacob Lurie]], _The image of $J$_, lecture 35 in _[[Chromatic Homotopy Theory]]_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture35.pdf))

* {#Behrens13} [[Mark Behrens]], section 1 of Introduction talk at _[Talbot 2013: Chromatic Homotopy Theory](http://math.mit.edu/conferences/talbot/index.php?year=2013&sub=talks)_ ([pdf](http://math.mit.edu/conferences/talbot/2013/1-Behrens-intro.pdf), [pdf](http://math.mit.edu/conferences/talbot/2013/Behrens-Introduction-chromatic-homotopy-thy-4-22-13.pdf))

* {#Knudsen13} [[Ben Knudsen]], _First chromatic layer of the sphere spectrum = homotopy of the $K(1)$-local sphere_, talk at _[2013 Pre-Talbot Seminar](http://math.northwestern.edu/~htanaka/pretalbot2013/index.php)_ ([pdf](http://math.northwestern.edu/~htanaka/pretalbot2013/notes/2013-04-04-Rob-Legg-K1-local-sphere.pdf))

* {#Lorman13} [[Vitaly Lorman]], _Chromatic homotopy theory at height 1 and the image of $J$_, talk at _[Talbot 2013: Chromatic Homotopy Theory](http://math.mit.edu/conferences/talbot/index.php?year=2013&sub=talks)_ ([pdf](https://math.mit.edu/events/talbot/2013/Image%20of%20J-1.pdf))

Generalization to [[equivariant cohomology]] ([[equivariant K-theory]]) is discussed in

* Z. Fiedorowicz, H. Hauschild, [[Peter May]], theorem 0.4 of _Equivariant algebraic K-theory_, _Equivariant algebraic K-theory_, Algebraic K-Theory. Springer, Berlin, Heidelberg, 1982. 23-80 ([pdf](http://math.uchicago.edu/~may/PAPERS/40.pdf))

* Henning Hauschild, [[Stefan Waner]], theorem 0.1 of _The equivariant Dold theorem mod $k$ and the Adams conjecture_, Illinois J. Math. Volume 27, Issue 1 (1983), 52-66. ([euclid:1256065410](https://projecteuclid.org/euclid.ijm/1256065410))

* Kuzuhisa Shimakawa, _Note on the equivariant $K$-theory spectrum_, Publ. RIMS, Kyoto Univ. **29** (1993), 449-453 ([pdf](http://www.ems-ph.org/journals/show_pdf.php?issn=0034-5318&vol=29&iss=3&rank=5), [doi](https://doi.org/10.2977/prims/1195167052))

* Christopher French, theorem 2.4 in _The equivariant $J$–homomorphism for finite groups at certain primes_, Algebr. Geom. Topol. Volume 9, Number 4 (2009), 1885-1949 ([euclid:1513797069](https://projecteuclid.org/euclid.agt/1513797069))

### Equivariant enhancement

The generalization to [[equivariant cohomology]] ([[equivariant K-theory]] and the [[equivariant Adams conjecture]]) is discussed in

* [[Tammo tom Dieck]], theorem 11.3.8 in _[[Transformation Groups and Representation Theory]]_ Lecture Notes in Mathematics 766 Springer 1979

* Z. Fiedorowicz, H. Hauschild, [[Peter May]], theorem 0.4 of _Equivariant algebraic K-theory_, _Equivariant algebraic K-theory_, Algebraic K-Theory. Springer, Berlin, Heidelberg, 1982. 23-80 ([pdf](http://math.uchicago.edu/~may/PAPERS/40.pdf))

* Henning Hauschild, [[Stefan Waner]], theorem 0.1 of _The equivariant Dold theorem mod $k$ and the Adams conjecture_, Illinois J. Math. Volume 27, Issue 1 (1983), 52-66. ([euclid:1256065410](https://projecteuclid.org/euclid.ijm/1256065410))

* Kuzuhisa Shimakawa, _Note on the equivariant $K$-theory spectrum_, Publ. RIMS, Kyoto Univ. **29** (1993), 449-453 ([pdf](http://www.ems-ph.org/journals/show_pdf.php?issn=0034-5318&vol=29&iss=3&rank=5), [doi](https://doi.org/10.2977/prims/1195167052))

* Christopher French, theorem 2.4 in _The equivariant $J$–homomorphism for finite groups at certain primes_, Algebr. Geom. Topol. Volume 9, Number 4 (2009), 1885-1949 ([euclid:1513797069](https://projecteuclid.org/euclid.agt/1513797069))

### Relation to $O$-action on general spectra

Similarly there is a canonical $O(n)$-[[∞-action]] on an [[n-fold loop space]], not just on the [[sphere spectrum]]. But the general case is closely related to the J-homomorphism. Discussion includes

* {#GaudensMenichi07} Gerald Gaudens, [[Luc Menichi]],  section 5 of _Batalin-Vilkovisky algebras and the $J$-homomorphism_, Topology and its Applications Volume 156, Issue 2, 1 December 2008, Pages 365–374 ([arXiv:0707.3103](http://arxiv.org/abs/0707.3103))

and in the context of the [[cobordism hypothesis]]:

* {#Lurie} [[Jacob Lurie]], example 2.4.15 of _[[On the Classification of Topological Field Theories]]_, Current Developments in Mathematics Volume 2008 (2009), 129-280 ([arXiv:0905.0465](http://arxiv.org/abs/0905.0465))

[[!redirects J-homomorphisms]]

[[!redirects image of J]]
[[!redirects image of J-homomorphism]]
[[!redirects image of the J-homomorphism]]

[[!redirects equivariant J-homomorphism]]
[[!redirects equivariant J-homomorphisms]]


J-homomorphism and chromatic homotopy

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Stable Homotopy theory
+--{: .hide}
[[!include stable homotopy theory - contents]]
=--
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

> This entry explains the [[J-homomorphism]], states how its image is the first ([[chromatic layer|chromatic]]) layer of the [[sphere spectrum]]; and then  motivated by this explains some basic notions of [[chromatic homotopy theory]], notably the origin of the general $E$-[[Adams spectral sequence]].

#Contents#
* table of contents
{:toc}

## **I)** The J-homomorphism

The _[[J-homomorphism]]_ is a map from the [[homotopy groups]] of the [[stable orthogonal group]] (which are completely understood) to the [[stable homotopy groups of spheres]] (which in their totality are hard to compute).

### On groups
 {#OnGroups}

+-- {: .num_defn #SphereAsCompactification}
###### Definition

For $n \in \mathbb{N}$ regard the $n$-[[sphere]] (as a [[topological space]]) as the [[one-point compactification]] of the [[Cartesian space]] $\mathbb{R}^n$

$$
  S^n \simeq (\mathbb{R}^n)^\ast
  \,.
$$

=--

+-- {: .num_remark #ActionOfOrthogonalOnSphere}
###### Remark

Since the process of [[one-point compactification]] is a [[functor]] on [[proper maps]], hence on [[homeomorphisms]], via def. \ref{SphereAsCompactification} the $n$-sphere inherits from the canonical [[action]] of the [[orthogonal group]] $O(n)$ on $\mathbb{R}^n$ an [[action]]

$$
  O(n) \times S^n \longrightarrow S^n
$$

(by [[continuous maps]]) which preserves the base point (the "point at infinity").

=--

For definiteness we distinguish in the following notationally between

1. the $n$-[[sphere]] $S^n \in Top$ regarded as a [[topological space]];

1. its [[homotopy type]] $\Pi(S^n) \in L_{whe} Top \simeq$ [[∞Grpd]] given by its [[fundamental ∞-groupoid]].

Similarly we write $\Pi(O(n))$ for the [[homotopy type]] of the [[orthogonal group]], regarded as a [[group object in an (∞,1)-category]] in [[∞Grpd]] (using that the [[shape modality]] $\Pi$ preserves [[finite products]]).

+-- {: .num_defn #AutoequivalencesOfnSphere}
###### Definition

For $n \in \mathbb{N}$ write $H(n)$ for the [[automorphism ∞-group]] of homotopy self-equivalences $S^n \longrightarrow S^n$, hence

$$
  H(n) \coloneqq Aut_{\infty Grpd^{\ast/}}(\Pi(S^n))
  \,.
$$

=--

+-- {: .num_remark }
###### Remark

The [[∞-group]] $H(n)$, def. \ref{AutoequivalencesOfnSphere}, constitutes the two [[connected components]] of the $n$-fold based [[loop space]] $\Omega^n S^n$ corresponding to  the [[homotopy groups]] $\pm 1 \in \pi_n(S^n)$.

=--

+-- {: .num_defn #OrthogonalActionOnSphereOnHomotopyGroups}
###### Definition

Via the presentation of [[∞Grpd]] by the [[cartesian closed model structure|cartesian closed]] [[model structure on topological spaces|model structure on compactly generated topological spaces]] (and using that $S^n$ and $O(n)$ and hence their product are [[compact topological spaces|compact]]) we have that for $n \in \mathbb{N}$ the [[continuous function|continuous]] [[action]] of $O(n)$ on $S^n$
of remark \ref{ActionOfOrthogonalOnSphere}, which by [[cartesian closed category|cartesian closure]] is equivalently a homomorphism of [[topological groups]] of the form

$$
  O(n) \longrightarrow Aut_{Top^{\ast/}}(S^n)
  \,,
$$

induces a homomorphism of [[∞-groups]] of the form

$$
  \Pi(O(n)) \longrightarrow Aut_{\infty Grpd^{\ast/}}(\Pi(S^n))
  \,.
$$

This in turn induces for each $i \in \mathbb{N}$ homomorphisms of [[homotopy groups]] of the form

$$
  \pi_i(O(n)) \longrightarrow \pi_i(\Omega^n S^n) \simeq \pi_{n+i}(S^n)
  \,.
$$

=--

+-- {: .num_remark #OrthogonalActionOnSphereOnHomotopyGroups}
###### Remark

By construction, the homomorphisms of remark \ref{OrthogonalActionOnSphereOnHomotopyGroups} are compatible with  [[suspension]] in that for all $n \in \mathbb{N}$ the [[diagrams]]

$$
  \array{
    O(n) &\longrightarrow& Aut_{Top^{\ast/}}(S^n)
    \\
    \downarrow && \downarrow
    \\
    O(n+1) &\longrightarrow& Aut_{Top^{\ast/}}(S^{n+1})
  }
$$

in $Grp(Top)$ commute, and hence so do the diagrams

$$
  \array{
    \Pi(O(n)) &\longrightarrow& Aut_{\infty Grpd^{\ast/}}(\Pi(S^n))
    \\
    \downarrow && \downarrow
    \\
    \Pi(O(n+1)) &\longrightarrow& Aut_{\infty Grpd^{\ast/}}(\Pi(S^{n+1}))
  }
$$

in $Grp(\infty Grpd)$, up to [[homotopy]].

=--

Therefore one can take the [[direct limit]] over $n$:

+-- {: .num_defn #JHom}
###### Definition

By remark \ref{OrthogonalActionOnSphereOnHomotopyGroups}
there is induced a homomorphism

$$
  J_i \;\colon\; \pi_\bullet(O) \longrightarrow \pi_\bullet(\mathbb{S})
$$

from the [[homotopy groups]] of the [[stable orthogonal group]]
to the [[stable homotopy groups of spheres]]. This is called
the **[[J-homomorphism]]**.

=--

### On classifying spaces

+-- {: .num_remark #DeloopedJ}
###### Remark

Since the maps of def. \ref{OrthogonalActionOnSphereOnHomotopyGroups}
are [[∞-group]] [[homomorphisms]], there exists their [[delooping]]

$$
  B J \;\colon\; B O \longrightarrow B GL_1(\mathbb{S}) = B H
  \,.
$$

=--

+-- {: .num_remark }
###### Remark

Here $GL_1(\mathbb{S})$ is the [[∞-group of units]] of the [[sphere spectrum]].

=--

This map $B J$ is the [[universal characteristic class]] of stable [[vector bundles]] with values in [[spherical fibrations]]:

+-- {: .num_defn #SphereBundleOfVectorBundle}
###### Definition

For $V \to X$ a [[vector bundle]], write $S^V$ for its [[fiber]]-wise [[one-point compactification]]. This is a [[sphere bundle]], a [[spherical fibration]]. Write $\mathbb{S}^V$ for the $X$-[[parameterized spectrum]] which is fiberwise the [[suspension spectrum]] of $S^V$.

=--

It is immediate that:

+-- {: .num_prop}
###### Proposition

For $V \to X$ a [[vector bundle]] classified by a map $X \to B O$,
the corresponding [[spherical fibration]] $\mathbb{S}^V$, def. \ref{SphereBundleOfVectorBundle}, is classified by $X \to B O \stackrel{B J}{\longrightarrow} B GL_1(\mathbb{S})$, def. \ref{DeloopedJ}.

=--

## **II)** The image of the J-homomorphism

Since the [[J-homomorphism]] maps from something well-understood to something hard to understand, it is of interst to characterize its [[image]], "the [[image of J]]".

### Explicit description

The following characterization of the [[image]] of the J-homomorphism on [[homotopy groups]] derives from a statement that was first conjectured in ([Adams 66](Adams+spectrals+sequence#Adams66)) -- and since called the _[[Adams conjecture]]_ -- and then proven in ([Quillen 71](Adams+spectrals+sequence#Quillen71), [Sullivan 74](Adams+spectrals+sequence#Sullivan74)).

+-- {: .num_remark}
###### Remark

By the discussion at _[orthogonal group -- homotopy groups](orthogonal%20group#HomotopyGroups)_ we have that the [[homotopy groups]] of the [[stable orthogonal group]] are

| $n\;mod\; 8$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|-----|---|---|---|---|---|---|---|---|
| $\pi_n(O)$ | $\mathbb{Z}_2$ | $\mathbb{Z}_2$ |0 | $\mathbb{Z}$ | 0 | 0 | 0 | $\mathbb{Z}$ |

Because all groups appearing here and in the following are [[cyclic groups]], we instead write down the [[order of a group|order]]

| $n\;mod\; 8$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|-----|---|---|---|---|---|---|---|---|
| ${\vert\pi_n(O)\vert}$ | 2 | 2 | 1 | $\infty$ | 1 | 1 | 1 | $\infty$ |

=--

For the following statement it is convenient to restrict to J-homomorphism to the stable [[special orthogonal group]] $S O$, which removes the lowest degree homotopy group in the above

| $n\;mod\; 8$ | $n = 0$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|-----|---|---|---|---|---|---|---|---|---|
| $\pi_n(S O)$ | 0 | $\mathbb{Z}_2$ | $\mathbb{Z}_2$ |0 | $\mathbb{Z}$ | 0 | 0 | 0 | $\mathbb{Z}$ |

| $n\;mod\; 8$ | $n = 0$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|-----|---|---|---|---|---|---|---|---|---|
| ${\vert\pi_n(S O)\vert}$ | 1 | 2 | 2 | 1 | $\infty$ | 1 | 1 | 1 | $\infty$ |

+-- {: .num_theorem #AdamsQuillenTheorem}
###### Theorem

The [[stable homotopy groups of spheres]] $\pi_n(\mathbb{S})$ are the [[direct sum]] of the ([[cyclic group|cyclic]]) [[image]]
$im(J|_{SO})$ of the J-homomorphism, def. \ref{JHom}, applied to the [[special orthogonal group]] and the [[kernel]] of the [[Adams e-invariant]].

Moreover,

* for $n = 0 \;mod \; 8$ and $n = 1 \;mod \; 8$ and $n$ positive the J-homomorphism $\pi_n(J) \colon \pi_n(S O) \to \pi_n(\mathbb{S})$ is [[injection|injective]], hence its image is $\mathbb{Z}_2$,

* for $n = 3\; mod\; 8$ and $n = 7 \; mod \; 8$ hence for $n = 4 k -1$, the [[order of a group|order]] of the image is equal to the [[denominator]] of $B_{2k}/4k$ in its reduced form, where $B_{2k}$ is the [[Bernoulli number]]

* for all other cases the image is necessarily zero.

=--

The characterization of this image is due to ([Adams 66](Adams+spectrals+sequence#Adams66), [Quillen 71](Adams+spectrals+sequence#Quillen71), [Sullivan 74](Adams+spectrals+sequence#Sullivan74)).
Specifically the identification of $J(\pi_{4n-1}(S O))$ is ([Adams 65a, theorem 3.7](Adams+spectrals+sequence#Adams65a) and the direct summand property is ([Adams 66, theorems 1.1-1.6.](Adams+spectrals+sequence#Adams66)).
That the image is a direct summand of the codomain is proven for instance in ([Switzer 75, end of chapter 19](Adams+spectrals+sequence#Switzer75)).

A modern version of the proof, using methods from [[chromatic homotopy theory]], is surveyed in some detail in ([Lorman 13](#Lorman13)).

The statement of the theorem is recalled for instance as ([Ravenel, chapter 1, theorem 1.1.13](#Ravenel)).
Another computation of the image of $J$ is in ([Ravenel, chapter 5, section 3](#Ravenel)).

+-- {: .num_remark}
###### Remark

The order of $J(\pi_{4k-1} O)$ in theorem \ref{AdamsQuillenTheorem}
is for low $k$ given by the following table

| k | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|---|---|---|---|---|---|---|---|---|---|---|
| $\vert J(\pi_{4k-1}(O))\vert$ | 24 | 240 | 504 | 480 | 264 | 65,520 | 24 | 16,320 | 28,728 | 13,200 |

=--

See for instance ([Ravenel, Chapt. 1,  p. 5](#Ravenel)).

+-- {: .num_remark}
###### Remark

Therefore we have in low degree the following situation

[[!include image of J -- table]]

=--

+-- {: .num_example #TableOfPPrincipalComponentsInHomotopyGroups}
###### Example

The following tables show the [[p-primary components]] of the [[stable homotopy groups of spheres]] for low values, the image of J appears as the bottom row.

Here the horizontal index is the degree $n$ of the stable homotopy group $\pi_n$. The appearance of a string of $k$ connected dots vertically above index $n$ means that there is a [[direct sum|direct summand]] [[primary group]] of [[order of a group|order]] $p^k$. See example \ref{InterpretTable} below for illustration. (These tables are taken from ([Hatcher](homotopy+groups+of+spheres#Hatcher)), where in turn they were generated based on ([Ravenel 86](#Ravenel))).

**$p = 2$-primary component** (e.g. [Ravenel 86, theorem 3.2.11, figure 4.4.46](#Ravenel))

**$p = 3$-primary component**

**$p = 5$-primary component**

=--

We illustrate how to read these tables:

+-- {: .num_example #InterpretTable}
###### Example

The [[finite abelian group]] $\pi_3(\mathbb{S}) \simeq \mathbb{Z}_{24}$ decomposes into [[primary groups]] as $\simeq \mathbb{Z}_8 \oplus \mathbb{Z}_3$. Here $8 = 2^3$ corresponds to the three dots above $n = 3$ in the first table, and $3 = 3^1$ to the single dot over $n = 3$ in the second.

The [[finite abelian group]] $\pi_7(\mathbb{S}) \simeq \mathbb{Z}_{240}$ decomposes into [[primary groups]] as $\simeq \mathbb{Z}_{16} \oplus \mathbb{Z}_3 \oplus \mathbb{Z}_5$. Here $16 = 2^4$ corresponds to the four dots above $n = 7$ in the first table, and $3 = 3^1$ to the single dot over $n = 7$ in the second and $5 = 5^1$ to the single dot over $n = 7$ in the third table.

The [[finite abelian group]] $\pi_11(\mathbb{S}) \simeq \mathbb{Z}_{504}$ has [[primary group]]-decomposition $\cdots \simeq \mathbb{Z}_{2^3} \oplus \mathbb{Z}_{3^2} \oplus \mathbb{Z}_7$ and so this corresponds to the three connected dots over $n = 11$ in the first table and the two connected dots over $n = 11$ in the second (and there will be one dot over $n = 11$ in the fourth table for $p = 7$ not shown here).

The groups $\pi_1(\mathbb{S}) \simeq \pi_2(\mathbb{S}) \simeq \pi_6(\mathbb{S}) \simeq \pi_{10}(\mathbb{S}) \simeq \mathbb{Z}_2$ correspond to the single dots over $n = 1,2,6,10$ in the first table, respectively.

The group $\pi_8(\mathbb{S}) \simeq \mathbb{Z}_2 \oplus \mathbb{Z}_2$ corresponds to the two _unconnected_ dots over $n = 8$ in the first table.

Similarly the group $\pi_9(\mathbb{S}) \simeq \mathbb{Z}_2 \oplus \mathbb{Z}_2 \oplus \mathbb{Z}_2$ corresponds to the three unconnected dots above $n = 9$ in the first table.

=--

### Chromatic formulation

The above tables, example \ref{TableOfPPrincipalComponentsInHomotopyGroups}, suggest that the [[image of the J-homomorphism]] is
in some sense the "lowest order layer" of the [[stable homotopy groups of spheres]]. This is made precise by the following characterization of the image in [[stable homotopy theory]]. We bluntly state this here and give all the relevant definitions [below](#ELocalStableHomotopyTheory).

$\,$

Write $E(1)$ for the first [[Morava E-theory]] [[spectrum]] at given [[prime number]] $p$. Write
$L_{E(1)}\mathbb{S}$ for the [[Bousfield localization of spectra]] of the [[sphere spectrum]] at $E(1)$.

+-- {: .num_theorem }
###### Theorem

The [[homotopy groups]] of the $E(1)$-localized sphere spectrum are

$$
  \pi_n L_{E(1)} \mathbb{S}
  \simeq
  \left\{
    \array{
      \mathbb{Z} & if\; n = 0
       \\
       \mathbb{Q}_p/\mathbb{Z}_p & if\; n= -2
       \\
       \mathbb{Z}_{p^{k+1}} & if\; n+1 = (p-1)p^k m \;with\; m \neq 0\;mod\;p
        \\
        0 & otherwise
    }
  \right.
  \,.
$$

=--

This appears as ([Lurie 10, theorem 6](#LurieLecture))

+-- {: .num_defn }
###### Definition

Write $\mathbb{S}_p$ for the [[p-localization]] of the [[sphere spectrum]].
For $n \in \mathbb{Z}$, write $im(J)_n$ for the [[image]] of the $p$-localized J-homomorphism

$$
  J \;\colon\; \pi_n(O) \longrightarrow \pi_n(\mathbb{S}) \longrightarrow \pi_n(\mathbb{S}_{(p)})
  \,.
$$

=--

+-- {: .num_theorem }
###### Theorem

For $n \in \mathbb{N}$, the further [[Bousfield localization]] at [[Morava E-theory|Morava E(1)-theory]]  $\mathbb{S}_{(p)} \longrightarrow L_{E(1)}\mathbb{S}$ induces a [[isomorphism]]

$$
  im(J)_n \stackrel{\simeq}{\longrightarrow} \pi_n (L_{E(1)} \mathbb{S})
$$

between the image of the $J$-homomorphism and the $E(1)$-local [[stable homotopy groups of spheres]].

=--

In this form this appears as ([Lurie 10, theorem 7](#LurieLecture)). See also ([Behrens 13, section 1](#Behrens13)).

+-- {: .num_cor }
###### Corollary

The $E(1)$-[[Bousfield localization of spectra|localization map]] is surjective on non-negative homotopy groups:

$$
  \pi_n(\mathbb{S}_{(p)}) \longrightarrow \pi_n(L_{E(1)} \mathbb{S})
  \,.
$$

=--

For review see also ([Lorman 13](#Lorman13)). That $J$ factors through $L_{K(1)}\mathbb{S}$ is in ([Lorman 13, p. 4](#Lorman13))

+-- {: .num_remark }
###### Remark

Hence: the image of $J$ is essentially the first [[chromatic layer]] of the [[sphere spectrum]].

=--

## **III)** $E$-Local stable homotopy theory
 {#ELocalStableHomotopyTheory}

To say what all this means, we recall now [[Bousfield localization of spectra]] and then indicate the tower of localizations at the [[Morava E-theory]] spectra,  the "[[chromatic filtration]]".

### Bousfield localization of spectra

+-- {: .num_defn #AcyclicAndLocal}
###### Definition

Let $E \in Spec$ be a [[spectrum]].

Say that another spectrum
$X \in Spec$ is an **$E$-acyclic spectrum** if the [[smash product]]
is [[zero object|zero]], $E \wedge X \simeq 0$.

Say that $X$ is an **$E$-local spectrum** if every [[morphism]]
$Y \longrightarrow X$ out of an $E$-acyclic spectrum $Y$ is homotopic to the [[zero morphism]].

Say that a morphism $f \colon X \to Y$ is an **$E$-equivalence** if it becomes an [[equivalence]] after [[smash product]] with $E$.

=--

(e.g. [Lurie, Lecture 20, example 4](#LurieLecture))

+-- {: .num_prop #LocalizationCofiber}
###### Proposition

For $E$ a [[spectrum]], every other spectrum sits in an essentially unique [[homotopy cofiber sequence]]

$$
  G_E(X) \to X \to L_E(X)
  \,,
$$

where $G_E(X)$ is $E$-acyclic, and $L_E(X)$ is $E$-local, def. \ref{AcyclicAndLocal}.

Here $X \to L_E (X)$ is characterized by two properties

1. $L_E(X)$ is $E$-local;

1. $X \to L_E(X)$ is an $E$-equivalence

according to def. \ref{AcyclicAndLocal}.

=--

(e.g. [Lurie, Lecture 20, example 4](#LurieLecture))

+-- {: .num_defn}
###### Definition

Given $E \in Spec$,
the [[natural transformation|natural morphisms]] $X \longrightarrow L_E X$
in prop. \ref{LocalizationCofiber} exhibit the [[localization of an (infinity,1)-category]] called **Bousfield localization** at $E$.

=--

+-- {: .num_example #EModulesAreELocal}
###### Example

For $E$ an [[E-∞ ring]], every [[∞-module]] $X$ over $E$ is $E$-local, def. \ref{AcyclicAndLocal}.

=--

(e.g. [Lurie, Lecture 20, example 5](#LurieLecture))

+-- {: .num_example}
###### Example

For $E$ an [[E-∞ algebra]] over an [[E-∞ ring]] $S$ and for $X$ an $S$-[[∞-module]], consider the dual [[Cech nerve]] [[cosimplicial object]]

$$
  E^{\wedge_S^{\bullet+1}}\wedge_S X \;\colon\; \Delta \longrightarrow Spectra
  \,.
$$

By example \ref{EModulesAreELocal} each term is $E$-local, so that the map  to the  [[totalization]]

$$
  X \longrightarrow \underset{\leftarrow}{\lim} E^{\wedge_S^{\bullet+1}} \wedge_S X
$$

factors through the $E$-localization of $X$

$$
  X \longrightarrow
  L_E X
  \longrightarrow
  \underset{\leftarrow}{\lim} E^{\wedge_S^{\bullet+1}} \wedge_S X
  \,.
$$

Under suitable condition the second map here is indeed an [[equivalence]], in which case the [[totalization]] of the dual [[Cech nerve]] exhibits the $E$-localization. This happens for instance in the discussion of the [[Adams spectral sequence]], see the examples given there.

=--

+-- {: .num_example}
###### Example

For $p \in \mathbb{N}$ a [[prime number]], let

$$
  E \coloneqq H \mathbb{Z}/p\mathbb{Z}
$$

be the corresponding [[Eilenberg-MacLane spectrum]]. Then a spectrum which corresponds to a [[chain complex]] under the [stable Dold-Kan corespondence](module+spectrum#StableDoldKanCorrespondence) is $E$-local, def. \ref{AcyclicAndLocal}, if that chain complex has [[chain homology]] groups being $\mathbb{Z}[p^{-1}]$-modules.

The $E$-localization of a spectrum in this case is called _[[p-localization]]_.

=--

(e.g. [Lurie, Lecture 20, example 8](#LurieLecture))

### Chromatic layers

Let

* $k$ be a [[perfect field]] of [[characteristic]] $p$;

* $f$ be a [[formal group]] of [[height of a formal group|height]] $n$ over $k$.

+-- {: .num_defn}
###### Definition

Write $W(k)$ for the [[ring of Witt vectors]]. Write

$$
  R \coloneqq W(k)[ [ v_1, \cdots, v_{n-1} ] ]
$$

for the ring of [[formal power series]] over this ring, in $n-1$ [[variables]]; called the _[[Lubin-Tate ring]]_.

=--

+-- {: .num_theorem}
###### Theorem

The [[Lubin-Tate formal group]] $\overline{f}$ is the [[universal property|universal]] deformation of $f$ in that for every [[infinitesimal object|infinitesimal thickening]] $A$ of $k$, $\overline{f}$ induces a [[bijection]]

$$
  Hom_{/k}(R,A) \stackrel{\simeq}{\longrightarrow} Def(A)
$$

between the $k$-[[associative algebra|algebra]]-[[homomorphisms]] from $R$ into $A$ and the [[deformations]] of $A$.

=--

(e.g. [Lurie 10, lect 21, theorem 5](#LurieLecture))

By the discussion there, this is [[Landweber exactness|Landweber exact]], hence defines a [[cohomology theory]]. Therefore by the [[Landweber exact functor theorem]] there is an [[even periodic cohomology theory]] $E(n)^\bullet$ [[Brown representability theorem|represented by]] a [[spectrum]] $E(n)$ with the property that its [[homotopy groups]] are

$$
  \pi_\bullet(E(n))
  \simeq
  W(k)[ [v_1, \cdots, v_{n-1} ] ] [ \beta^{\pm 1} ]
$$

for $\beta$ of degree 2. This is called alternatively $n$th _[[Morava E-theory]]_, or _[[Lubin-Tate theory]]_ or _[[Johnson-Wilson theory]]_.

(e.g. [Lurie, lect 22](#LurieLecture))

For each [[prime]] $p \in \mathbb{N}$ and for each [[natural number]]  $n \in \mathbb{N}$ there is a [[Bousfield localization of spectra]]

$$
  L_n \coloneqq L_{E(n)}
  \,,
$$

where $E(n)$ is the $n$th [[Morava E-theory]] (for the given [[prime]] $p$), called the $n$th _[[chromatic localization]]_. These arrange into the _[[chromatic tower]]_ which for each spectrum $X$ is of the form

$$
  X \to \cdots \to L_n X \to L_{n-1} X \to \cdots \to L_0 X
  \,.
$$

The [[homotopy fibers]] of each stage of the tower

$$
  M_n(X) \coloneqq fib(L_{E(n)}X \longrightarrow L_{E(n-1)}(X))
$$

is called the $n$th _[[monochromatic layer]]_ of $X$.

[[!include chromatic tower examples - table]]

## **IV)** Adams spectral sequence for $E$-local homotopy groups

Summing up the above, we need a means to compute [[homotopy groups]] of $E$-[[Bousfield localization of spectra|localized]] [[spectra]].
In ([Lurie, Higher Algebra, section 1.2.2](#LurieHigherAlgebra)) is given a general [[spectral sequence of a filtered stable homotopy type]] which computes homotopy groups of [[spectra]], and in ([Lurie 10, lectures 8 and 9](#LurieLecture)) is discussed that the [[totalization]] of the [[coskeleton]] filtration on the dual [[Cech nerve]] of an [[E-∞ algebra]] yields the $E$-localization. Taken together this is just what we need... and this is the general _$E$-[[Adams spectral sequence]]_. We follow the nice exposition in ([Wilson 13](#Wilson13)).

First we recall

* [Spectral sequences computing homotopy groups of filtered objects](#SpectralSequencesForHomotopyGroupsOfFilteredObjects)

for the general case of filtered objects in suitable [[stable (∞,1)-categories]]. Then we consider the specialization of that to the

* [Canonical cosimplicial resolutions of E-∞ algebras](#CanonicalCosimplicialResolutionOfEInfinityAlgebras).

In conclusion this yields for each suitable [[E-∞ algebra]] $E$ over $S$  and $S$-[[∞-module]] $X$ a [[spectral sequence]] converging to the [[homotopy groups]] of the $E$-[[Bousfield localization of spectra|localization]] of $X$, and this is

* [The E-Adams-Novikov spectral sequence](#TheEAdamsSpectralSequence).

### Spectral sequences for homotopy groups of filtered spectra
 {#SpectralSequencesForHomotopyGroupsOfFilteredObjects}

We discuss the [[spectral sequence of a filtered stable homotopy type]].

Let throughout $\mathcal{C}$ be a [[stable (∞,1)-category]], $\mathcal{A}$ an [[abelian category]], and $\pi \;\colon\; \mathcal{C}\longrightarrow \mathcal{A}$ a [[homological functor]] on $\mathcal{C}$, i.e., a functor that transforms every [[cofiber sequence]]

$$ X\to Y\to Z\to \Sigma X $$

in $\mathcal{C}$ into a long exact sequence

$$ \dots \to \pi(X)\to \pi(Y)\to \pi(Z)\to \pi(\Sigma X) \to \dots $$

in $\mathcal{A}$. We write $\pi_n=\pi\circ \Sigma^{-n}$.

+-- {: .num_example}
###### Example

* $\mathcal{C}$ is arbitrary, $\mathcal{A}$ is the category of [[abelian groups]] and $\pi$ is taking the 0th [[homotopy group]] $\pi_0 \mathcal{C}(S,-)$ of the [[mapping spectrum]] out of some [[object]] $S\in\mathcal{C}$

* $\mathcal{C}$ is equipped with a [[t-structure]], $\mathcal{A}$ is the [[heart of a stable (∞,1)-category|heart]] of the t-structure, and $\pi$ is the canonical functor.

* $\mathcal{C} = D(\mathcal{A})$ is the [[derived category]] of the abelian category $\mathcal{A}$ and $\pi=H_0$ is the degree-0  [[chain homology]] functor.

* Any of the above with $\mathcal{C}$ and $\mathcal{A}$ replaced by their [[opposite categories]].

=--

+-- {: .num_defn #GeneralizedFilteredObject}
###### Definition

A _[[filtered object in an (∞,1)-category]]_ in $\mathcal{C}$ is
simply a [[sequential diagram]] $X \colon (\mathbb{Z}, \lt) \to \mathcal{C}$

$$
  \cdots
  X_{n-1} \to X_n \to X_{n+1} \to \cdots
  \,.
$$

=--

This appears as ([[Higher Algebra|Higher Algebra, def. 1.2.2.9]]).

We will take the view that the object being filtered is the [[homotopy limit]]

$$
  X \coloneqq \underset{\leftarrow}{\lim}_n X_n.
$$

We could also consider the sequential diagram as a filtering of its [[homotopy colimit]], but this is really an equivalent point of view since we can replace $\mathcal{C}$ by $\mathcal{C}^{op}$.

+-- {: .num_defn #ChainComplexInStableInfinityCategory}
###### Definition

Let $I$ be a [[linearly ordered set]]. An $I$-chain complex in a [[stable (∞,1)-category]] $\mathcal{C}$ is an [[(∞,1)-functor]]

$$
  F \;\colon\; I^{\Delta[1]} \longrightarrow \mathcal{C}
$$

such that

1. for each $n \in I$, $F(n,n) \simeq 0$ is the [[zero object]];

1. for all $i \leq j \leq k$ the induced [[diagram]]

   $$
     \array{
        F(i,j) &\longrightarrow& F(i,k)
        \\
        \downarrow && \downarrow
        \\
        F(j,j) &\longrightarrow& F(j,k)
     }
   $$

   is a [[homotopy pullback]] square.

=--

This is [[Higher Algebra|Higher Algebra, def. 1.2.2.2]].

+-- {: .num_remark }
###### Remark

Given a $\mathbb{Z}$-chain complex $F$ in $\mathcal{C}$ as in def. \ref{ChainComplexInStableInfinityCategory},
setting

$$
  C_n \coloneqq \Sigma^{-n} F(n,n+1)
$$

and defining a [[differential]] induced from the [[connecting homomorphisms]] of the defining [[homotopy fiber sequences]]

$$
  F(n-1,n) \to F(n-1, n+1) \to F(n,n+1)
$$

yields an ordinary [[chain complex]] $C_\bullet$ in the [[homotopy category of an (∞,1)-category|homotopy category]].

=--

([[Higher Algebra|Higher Algebra, remark 1.2.2.3]])

+-- {: .num_prop #ChainComplexesFromFilteredObjects}
###### Proposition

Consider the inclusion of [[posets]]

$$
  (\mathbb{Z}, \leq)
  \to
  (\mathbb{Z}\cup \{\infty\}, \leq)^{\Delta[1]}
$$

given by

$$
  n \mapsto (n,\infty)
  \,.
$$

The induced [[(∞,1)-functor]]

$$
  Func((\mathbb{Z}\cup \{\infty\}, \leq)^{\Delta[1]} , \mathcal{C})
  \longrightarrow
  Func((\mathbb{Z}, \leq), \mathcal{C})
$$

restricts to an [[equivalence of (∞,1)-categories|equivalence]] between the (∞,1)-category of $\mathbb{Z}\cup \{\infty\}$-chain complexes in $\mathcal{C}$ (def. \ref{ChainComplexInStableInfinityCategory}) and that of generalized filtered objects in $\mathcal{C}$ (def. \ref{GeneralizedFilteredObject}).

=--

This is [[Higher Algebra|Higher Algebra, lemma 1.2.2.4]]. The inverse functor can be described informally as follows: given a filtered object $X_\bullet$, the associated chain complex $X(\bullet,\bullet)$ is given by

$$ X(n, n+r) = \operatorname{fib}(X_n\to X_{n+r}). $$

+-- {: .num_defn }
###### Definition

Let $X_\bullet$ be a filtered object in the sense of def. \ref{GeneralizedFilteredObject}. Write $X(\bullet,\bullet)$ for the corresponding chain complex, according to prop. \ref{ChainComplexesFromFilteredObjects}.

Then for all $i \leq j \leq k$ there is a [[long exact sequence of homotopy groups]] in $\mathcal{A}$ of the form

$$
  \cdots \to
  \pi_n X(i,j)
  \to
  \pi_n X(i,k)
  \to
  \pi_n X(j,k)
  \to
  \pi_{n-1}X(i,j)
  \to \cdots
  \,.
$$

Define then for $p,q \in \mathbb{Z}$ and $r \geq 1$ the object $E^r_{p,q}$ by the canonical [epi-mono factorization](abelian+category#FactorizationOfMorphisms)

$$
    \pi_{p} X(q-r+1,q+1)
    \twoheadrightarrow
    E^r_{p,q}
    \hookrightarrow
    \pi_{p} X(q, q+r)
$$

in the abelian category $\mathcal{A}$, and define the [[differential]]

$$
  d^r \;\colon\; E_{p,q}^r \to E_{p-1, q-r}^r
$$

to be the restriction of the [[connecting homomorphism]]

$$
 \pi_{p} X(q,q+r) \to \pi_{p-1} X(q-r, q)
$$

from the above long exact sequence (with $i=q-r$, $j=q$, and $k=q+r$).

=--

([[Higher Algebra|Higher Algebra, construction 1.2.2.6]])

+-- {: .num_prop}
###### Proposition

$d^r\circ d^r = 0$ and there are natural (in $X_\bullet$) isomorphisms

$$
E^{r+1}\cong \operatorname{ker}(d^r)/\operatorname{im}(d^r).
$$

Thus, $\{E^r_{*,*}\}_{r\geq 1}$ is a bigraded [[spectral sequence]]  in the [[abelian category]] $\mathcal{A}$, functorial in the filtered object $X_\bullet$, with

$$ E^1_{p,q} = \pi_p \operatorname{fib}(X_q\to X_{q+1}), \qquad d^r: E^r_{p,q}\to E^r_{p-1,q-r}. $$

=--

([[Higher Algebra|Higher Algebra, prop. 1.2.2.7]])

If [[sequential limits]] and [[sequential colimits]] exist in $\mathcal{A}$, we can form the limiting term $E^\infty_{*,*}$ of this spectral sequence.

On the other hand, the [[graded object]] $\pi_\bullet (X)$ admits a [[filtered object|filtration]] by

$$ F_q \pi_p (X) = \operatorname{ker}(\pi_p (X)\to \pi_p(X_q)) $$

and we would like to compare $E^\infty_{*,*}$ with the [[associated graded]] of this filtration. We say that

+-- {: .num_defn #WeakAndStrongConvergence}
###### Definition

The spectral sequence **converges weakly** if there is a canonical isomorphism

$$ E^\infty_{p,q} \cong F_q\pi_p(X)/ F_{q-1}\pi_p(X) $$

for every $p,q\in\mathbb{Z}$.

We say that the spectral sequence **converges strongly** if it converges weakly and if, in addition, the filtration $F_\bullet\pi_p(X)$ is complete on both sides.

=--

+-- {: .num_remark}
###### Remark

The meaning of the word *canonical* in def. \ref{WeakAndStrongConvergence} is somewhat subtle since, in general, there is no map from one side to the other. However, there always exists a canonical *[[relation]]* between the two, and we ask that this relation be an isomorphism (see [Hilton-Stammbach, VIII.7](#HiltonStammbach)).

=--

+-- {: .num_prop #FiltrationSpectralSequence}
###### Proposition

Let $\mathcal{C}$ be a [[stable (∞,1)-category]] and let $\pi:\mathcal{C}\to\mathcal{A}$ be a homological functor where $\mathcal{A}$ is an [[abelian category]] which admits [[sequential limits]]. Let $X_\bullet$ be a filtered object in $\mathcal{C}$ such that $\underset{\leftarrow}{\lim} X_\bullet$ exists. Suppose further that:

1. For every $n$, the diagram $r\mapsto \operatorname{fib}(X_{n-r}\to X_n)$ has a limit in $\mathcal{C}$ and that limit is preserved by $\pi$.
2. For every $n$, $\pi_n(X_r)=0$ for $r\gg 0$.

Then the [[spectral sequence]] $\{E^r_{*,*}\}_{r\geq 1}$ in $\mathcal{A}$ converges strongly (def. \ref{WeakAndStrongConvergence}). We write:

$$
  E_{p,q}^1
  =
  \pi_{p} \operatorname{fib}(X_q\to X_{q+1})
  \Rightarrow
  \pi_{p} (\underset{\leftarrow}{\lim} X_\bullet)
$$

=--

There is also a dual statement in which limits are replaced by colimits, but it is in fact a special case of the proposition with $\pi$ replaced by $\pi^{op}$. A proof of this proposition (in dual form) is given in ([[Higher Algebra|Higher Algebra, prop. 1.2.2.14]]). Review is in ([Wilson 13, theorem 1.2.1](#Wilson13)).

For the traditional statement in the [[category of chain complexes]] see at _[[spectral sequence of a filtered complex]]_.

Plenty of types of [[spectral sequences]] turn out to be special cases of this general construction.

[[!include Lurie spectral sequences -- table]]

### Canonical cosimplicial resolution of $E_\infty$-algebras
 {#CanonicalCosimplicialResolutionOfEInfinityAlgebras}

We discuss now the special case of coskeletally filtered
totalizations coming from the canonical cosimplicial objects
induced from [[E-∞ algebras]] (dual [[Cech nerves]]/[[Sweedler corings]]/[[Amitsur complexes]]).

In this form this appears as ([Lurie 10, theorem 2](#LurieLecture)). A review is in ([Wilson 13, 1.3](#Wilson13)). For the analog of this in the traditional formulation see ([Ravenel, ch. 3, prop. 3.1.2](#Ravenel)).

+-- {: .num_defn #FiltrationOfTotalizationByTotalizationOfCoskeleta}
###### Definition

Given an [[cosimplicial object in an (∞,1)-category]]  with [[(∞,1)-colimits]]

$$
  Y \;\colon\; \Delta  \longrightarrow \mathcal{C}
$$

its [[totalization]] $Tot Y \simeq \underset{\leftarrow}{\lim}_n Y_n$
is [[filtered object in an (infinity,1)-category|filtered]], def. \ref{GeneralizedFilteredObject}, by the
totalizations of its [[coskeleta]]

$$
  Tot Y
  \to
  \cdots
  \to
  Tot (cosk_2 Y)
  \to
  Tot (cosk_1 Y)
  \to
  Tot (cosk_0 Y)
  \to 0
  \,.
$$

=--

+-- {: .num_defn #SpectralSequenceOfSimplicialStableHomotopyType}
###### Definition

The [[spectral sequence of a filtered stable homotopy type|filtration spectral sequence]], prop. \ref{FiltrationSpectralSequence},
applied to the filtration of a [[totalization]] by [[coskeleton|coskeleta]] as in def. \ref{FiltrationOfTotalizationByTotalizationOfCoskeleta}, we call the _[[spectral sequence of a simplicial stable homotopy type]]_.

=--

([[Higher Algebra|Higher Algebra, prop. 1.2.4.5]])

+-- {: .num_prop #E2PageByMooreComplex}
###### Proposition

The [[spectral sequence of a simplicial stable homotopy type]] has as first page/$E_1$-term the [[cohomology groups]] of the [[Moore complex]] associated with the [[cosimplicial objects]] of [[homotopy groups]]

$$
  E_2^{p,q}
  =
  H^p(\pi_q(Tot (cosk_\bullet(Y))))
  \Rightarrow
  \pi_{p-q} Tot(Y)
  \,.
$$

=--

By the discussion at _[[∞-Dold-Kan correspondence]]_ and _[[spectral sequence of a filtered stable homotopy type]]_. This appears as ([[Higher Algebra|Higher Algebra, remark 1.2.4.4]]). Review is around  ([Wilson 13, theorem 1.2.4](#Wilson13)).

+-- {: .num_defn}
###### Definition

Let $S$ be an [[E-∞ ring]] and let $E$ be an [[E-∞ algebra]] over $S$, hence an [[E-∞ ring]] equipped with a [[homomorphism]]

$$
  S \longrightarrow E
  \,.
$$

The _canonical [[cosimplicial object]]_
associated to this (the $\infty$-[[Cech nerve]]/[[Sweedler coring]]/[[Amitsur complex]]) is that given by the iterated [[smash product]]/[[tensor product]] over $S$:

$$
  E^{\wedge^{\bullet+1}_S} \;\colon\; \Delta \to \mathcal{C}
  \,.
$$

More generally, for $X$ an $S$-[[∞-module]], the canonical [[cosimplicial object]] is

$$
  E^{\wedge^{\bullet+1}_S}\wedge_S X \;\colon\; \Delta \to \mathcal{C}
  \,.
$$

=--

+-- {: .num_prop #FlatnessCondition}
###### Proposition

If $E$ is such that the self-[[generalized homology]]
$E_\bullet(E) \coloneqq \pi_\bullet(E \wedge_S E)$ (the dual $E$-[[Steenrod operations]]) is such that as a [[module]] over $E_\bullet \coloneqq \pi_\bullet(E)$ it is a [[flat module]], then there is a [[natural equivalence]]

$$
  \pi_\bullet
  \left(
    E^{\wedge^{n+1}_S}
    \wedge_S
    X
  \right)
  \simeq
  E_\bullet(E^{\wedge^n_S})
  \otimes_{E_\bullet}
  E_\bullet(X)
  \,.
$$

=--

Reviewed for instance as ([Wilson 13, prop. 1.3.1](#Wilson13)).

+-- {: .num_remark}
###### Remark

This makes $(E_\bullet, E_\bullet(E))$ be
the [[commutative Hopf algebroid]]
formed by the  $E$-[[Steenrod algebra]]. See there for more on this.

=--

+-- {: .num_example}
###### Example

The condition in prop. \ref{FlatnessCondition} is
satisfied for

* $E = H \mathbb{F}_p$ an [[Eilenberg-MacLane spectrum]] with $mod\;p$ [[coefficients]];

* $E = B P$ the [[Brown-Peterson spectrum]];

* $E = MU$ the [[complex cobordism cohomology theory|complec cobordism spectrum]].

It is NOT satisfied for

* $E = H \mathbb{Z}$ the [[Eilenberg-MacLane spectrum]] for [[integer|integers]] [[coefficients]];

* $E = M S U$.

=--

+-- {: .num_remark #ExtGroupsByMooreComplex}
###### Remark

Under good conditions (...), $\pi_\bullet$ of the canonical [[cosimplicial object]] provides a [[resolution]] of [[comodule]] [[tensor product]] and hence computes the [[Ext]]-groups over the [[commutative Hopf algebroid]]:

$$
  H^p(\pi_q(Tot(cosk_\bullet(E^{\wedge^{\bullet+1}_S } \wedge_S X))))
  \simeq
  Ext^p_{E_\bullet(E)}(\Sigma^q E_\bullet, E_\bullet(X))
  \,.
$$

(...)

=--

+-- {: .num_remark #CanonicalMapFromELocalizationToTotalization}
###### Remark

There is a canonical map

$$
  L_E X \stackrel{}{\longrightarrow} \underset{\leftarrow}{\lim}_n (E^{\wedge^{n+1}_S}\wedge_S X)
$$

from the $E$-[[Bousfield localization of spectra]] of $X$ into the [[totalization]].

=--

([Lurie 10, lecture 30, prop. 1](LurieLecture))

We consider now conditions for this morphism to be an [[equivalence]].

+-- {: .num_defn #CoreOfARing}
###### Definition

For $R$ a [[ring]], its _core_ $c R$ is the [[equalizer]] in

$$
  c R
    \longrightarrow
  R
    \stackrel{\longrightarrow}{\longrightarrow}
  R \otimes R
  \,.
$$

=--

+-- {: .num_prop #SufficientConditionsForTotalizationToBeELocalization}
###### Proposition

Let $E$ be a [[connective spectrum|connective]] [[E-∞ ring]] such that the core or $\pi_0(E)$, def. \ref{CoreOfARing} is either of

* the [[localization of a ring|localization]] of the [[integers]] at a set $J$ of [[primes]], $c \pi_0(E) \simeq \matbb{Z}[J^{-1}]$;

* $\mathbb{Z}_n$ for $n \geq 2$.

Then the map in remark \ref{CanonicalMapFromELocalizationToTotalization} is an equivalence

$$
  L_E X \stackrel{\simeq}{\longrightarrow}
  \underset{\leftarrow}{\lim}_n (E^{\wedge^{n+1}_S}\wedge_S X)
  \,.
$$

=--

([Bousfield 79](Bousfield+localization+of+spectra#Bousfield79), [Lurie 10, lecture 30, prop. 3](LurieLecture), [Lurie 10, lecture 31, ](LurieLecture)).

### The $E$-Adams-Novikov spectral sequence
 {#TheEAdamsSpectralSequence}

Summing this up yields the general $E$-Adams(-Novikov) spectral sequence

+-- {: .num_cor}
###### Corollary

Let $E$ a [[connective spectrum|connective]] [[E-∞ ring]]
that satisfies the conditions of prop. \ref{SufficientConditionsForTotalizationToBeELocalization}.
Then by prop. \ref{FiltrationSpectralSequence} and prop. \ref{SufficientConditionsForTotalizationToBeELocalization} there is a strongly convergent multiplicative
[[spectral sequence]]

$$
  E^{p,q}_\bullet \Rightarrow \pi_{q-p} L_{c \pi_0 E} X
$$

converging to the [[homotopy groups]] of the  $ c \pi_0(E)$-[[Bousfield localization of spectra|localization]] of $X$. If moreover the dual $E$-[[Steenrod algebra]] $E_\bullet(E)$ is [[flat module|flat]] as a [[module]] over $E_\bullet$, then, by prop. \ref{E2PageByMooreComplex} and remark \ref{ExtGroupsByMooreComplex}, the $E_1$-term of this spectral sequence is given by the [[Ext]]-groups over the $E$-[[Steenrod algebra|Steenrod]] [[commutative Hopf algebroid|Hopf algebroid]].

$$
  E^{p,q}_\bullet
  =
  Ext^p_{E_\bullet(E)}(\Sigma^q E_\bullet, E_\bullet X)
  \,.
$$

=--

## References

An introduction to the chromatic perspective on the homotopy groups of spheres and the image of $J$ is in:

* [[Mark Mahowald]], [[Doug Ravenel]], _Towards  a Global Understanding of the Homotopy Groups of Spheres_, 1998 ([pdf](http://www.math.rochester.edu/people/faculty/doug/mypapers/global.pdf))

The bulk of the basic constructions is in

* [[Jacob Lurie]], _[[Chromatic Homotopy Theory]]_, lecture notes (2010)
 {#LurieLecture}

* [[Jacob Lurie]], _[[Higher Algebra]]_
 {#LurieHigherAlgebra}

Recent surveys of the modern picture are in

* [Talbot 2013: Chromatic Homotopy Theory](http://math.mit.edu/events/talbot/), _[2013 Pre-Talbot Seminar Chromatic homotopy theoy](http://math.northwestern.edu/~htanaka/pretalbot2013/index.php?pageID=links)_

and of relevance for the above discussion are particularly the following contributions there

* [[Mark Behrens]], section 1 of Introduction talk at _[Talbot 2013: Chromatic Homotopy Theory](http://math.mit.edu/events/talbot/index.php?year=2013&sub=talks)_ ([pdf](http://math.mit.edu/events/talbot/2013/1-Behrens-intro.pdf), [pdf](http://math.mit.edu/events/talbot/2013/Behrens-Introduction-chromatic-homotopy-thy-4-22-13.pdf))
 {#Behrens13}

* [[Dylan Wilson]] _Spectral Sequences from Sequences of Spectra: Towards the
Spectrum of the Category of Spectra_ lecture at _[2013 Pre-Talbot Seminar](http://math.northwestern.edu/~htanaka/pretalbot2013/index.php)_ ([pdf](http://math.northwestern.edu/~htanaka/pretalbot2013/notes/2013-03-21-Dylan-Wilson-ANSS.pdf))
 {#Wilson13}

* Ben Knudsen, _First chromatic layer of the sphere spectrum = homotopy of the $K(1)$-local sphere_, talk at _[2013 Pre-Talbot Seminar](http://math.northwestern.edu/~htanaka/pretalbot2013/index.php)_ ([pdf](http://math.northwestern.edu/~htanaka/pretalbot2013/notes/2013-04-04-Rob-Legg-K1-local-sphere.pdf))
 {#Knudsen13}

* [[Vitaly Lorman]], _Chromatic homotopy theory at height 1 and the
image of $J$_, talk at _[Talbot 2013: Chromatic Homotopy Theory](https://math.mit.edu/events/talbot/index.php?year=2013&sub=talks)_ ([pdf](https://math.mit.edu/events/talbot/2013/Image%20of%20J-1.pdf))
 {#Lorman13}

Loads of details for computations in the Adams spectral sequence are in

* [[Doug Ravenel]], _[[Complex cobordism and stable homotopy groups of spheres]]_, 1986
 {#Ravenel}

The construction of the image of the J-homomorphism when twisted by [[Dirichlet characters]] is in

* Ningchuan Zhang, _Analogs of Dirichlet L-functions in chromatic homotopy theory_, ([arXiv:1910.14582](https://arxiv.org/abs/1910.14582))


j-invariant

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Elliptic cohomology
+-- {: .hide}
[[!include elliptic cohomology -- contents]]
=--
#### Complex geometry
+--{: .hide}
[[!include complex geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

What is called the _$j$-invariant_ is an invariant of [[cubic curves]] and hence of [[elliptic curves]], partly characterizing them.

Over the [[complex numbers]] the _$j$-invariant_ is a [[modular function]] on the [[upper half plane]] which serves to characterize most of the properties of the [[moduli stack of elliptic curves]] in this case.

## Definition

### Over a general ring

With the Weierstrass parameterization discussed at _[elliptic curve -- Over general Rings -- As solution to the Weierstrass equation](elliptic%20curve#OverGeneralRingAsSolutionsToEquations)_, the $j$-invariant is the combination

$$
  j \coloneqq \frac{c_4^3}{\Delta}
  \,.
$$

In the case that 2 and 3 are invertible in the base ring, then this is equivalent to

(...)

### Over the complex numbers

Over the [[complex numbers]],  with $G_{2k}$ the [[Eisenstein series]] and with

$$
  g_2 \coloneqq 60 G_4
$$

$$
  g_3 \coloneqq 140 G_6
$$

and the [[discriminant]]

$$
  \Delta \coloneqq g_2^3 - 27 g_3^2
$$

then the _$j$-invariant_ is

$$
  j = 1728 \frac{g_2^3}{\Delta}
  \,.
$$

(Notice that $1728 = 12^3$.)

(e.g. [Miranda 88, def. I.2.1](#Miranda88)).

Notice the two special values

$$
  (j = 0 ) \Leftrightarrow (g_2 = 0)
$$

$$
  (j= 1728) \Leftrightarrow (g_3 = 0)
$$

### General

See at _[elliptic curve -- Definition for general rings -- j-invariant](elliptic+curve#WeierstrassCubicDiscriminantAndJInvariant)_.

## Properties

### Characterization of complex elliptic curves

Over the complex numbers, there are two [[elliptic curves]]
with special values

* the curve with $j = 0$, hence $g_2  = 0$, which is the one given by quotienting out an equilateral lattice; this has [[automorphism group]] $\mathbb{Z}/6\mathbb{Z}$;

* the curve with $j = 1728$, which corresponds to dividing out the lattice $(1,i)\mathbb{Z}$, this has automorphism group $\mathbb{Z}/4\mathbb{Z}$.

All other curves have automorphism $\mathbb{Z}/2\mathbb{Z}$, given by [[inversion involution]].

The case $j \to \infty$, hence $\Delta = 0 $ but $g_2 \neq 0$, corresponds to the [[nodal curve]] which is added in the [[Deligne-Mumford compactification]] of the [[moduli stack of elliptic curves]].

### As a branched cover of the complex plane
 {#AsABranchedCoverOfTheComplexPlane}

Over the [[complex numbers]], the $j$-invariant is a map

$$
  j \;\colon \; \mathfrak{h} \longrightarrow \mathbb{C}
$$

from the [[upper half plane]] to the [[complex numbers]]. This is a [[branched cover]], with two branching points being $0,\;1728 \in \mathbb{C}$.

The induced unramified covering

$$
  j \;\colon\; (\mathfrak{h}-j^{-1}(\{0,1728\})) \longrightarrow (\mathbb{C}-\{0,1728\})
$$

is a [[modular group]]($PSL_2(\mathbb{Z})$)-[[principal bundle]] and hence classified by a map

$$
  \mathbb{C} - \{0,1728\} \longrightarrow B PSL_2(\mathbb{C})
$$

from the [[plane]] with two points removed or equivalently

$$
  \pi_1(\mathbb{C}-\{0,1728\}) \longrightarrow PSL_2(\mathbb{Z})
  \,.
$$

(e.g. [Miranda 88, section VI.3, p.65](#Miranda88))

## Related concepts

* [[moduli space of elliptic curves]]

* [[elliptic genus]]

* [[Moonshine]], [[Monster vertex algebra]]

## References

* Wikipedia _[j-invariant](http://en.wikipedia.org/wiki/J-invariant)_

* {#Miranda88} [[Rick Miranda]], _The basic theory of elliptic surfaces_, lecture notes 1988 ([pdf](http://www.math.colostate.edu/~miranda/BTES-Miranda.pdf))

[[!redirects j-invariants]]


J-orthogonal matrix

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Linear algebra
+-- {: .hide}
[[!include homotopy - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

Given a [[square matrix]] $J$ (representing a [[bilinear form]]) over a [[commutative ring]] $k$, a _$J$-orthogonal matrix_ is any square matrix $A$ of the same size over $k$ such that

$$
  A^T J A = J
$$

where $A^T$ is the [[transpose matrix]] of $A$.

Most often one considers a case when $J$ is diagonal. For $J = 1$ one recovers the notion of an [[orthogonal matrix]].

All J-orthogonal matrices over the [[ground field]] $k$ form an [[algebraic group]] denoted $SO_J(n, k)$ or $SO(n,k; J)$ or alike; in real or complex case a Lie group. The matrices in the Lie algebra are J-skew-symmetric, $A^T J + J A = 0$.

## Related concepts

* [[orthogonal matrix]]

## References

* [[M M Postnikov]], _Lectures on geometry, Semester V, Lie groups and Lie algebras_

* Nicholas J. Higham, _J-orthogonal matrices: properties and generation_,
[doi](https://doi.org/10.1137/S0036144502414930)


J-spectrum

#Contents#
* table of contents
{:toc}

## Idea

The _J-spectrum_ is a [[spectrum]] whose [[homotopy groups]] are close to being the [[image of the J-homomorphism]].

(e.g [Mahowald-Ravenel, section 4](#MahowaldRavenel))

## References

* [[Mark Mahowald]], [[Doug Ravenel]], _Towards  a Global Understanding of the Homotopy Groups of Spheres_ ([pdf](http://www.math.rochester.edu/people/faculty/doug/mypapers/global.pdf))
 {#MahowaldRavenel}

[[!redirects J-spectra]]


J. Arthur Seebach

* [Wikipedia entry](https://en.wikipedia.org/wiki/J._Arthur_Seebach_Jr.)

## Selected writings

On [[topology]]:

* [[Lynn Arthur Steen]], [[J. Arthur Seebach]], _[[Counterexamples in Topology]]_, Springer 1970/1978 ([doi:10.1007/978-1-4612-6290-9](https://link.springer.com/book/10.1007/978-1-4612-6290-9))

category: people