motivic slice tower

## Idea

In [[motivic homotopy theory]], the slice filtration is a filtration of $\mathrm{H}(S)$ and of $SH(S)$ which is analogous to the [[Postnikov tower in an (∞,1)-category|Postnikov filtration]] for [[(∞,1)-topoi]]. It generalizes the _coniveau filtration_ in [[algebraic K-theory]], the _fundamental filtration_ on [[Witt groups]], and the _weight filtration_ on [[mixed Tate motives]].

If $S$ is [[smooth scheme|smooth]] over a [[field]], the layers of the slice filtration of a motivic spectrum (called its _slices_) are modules over the [motivic Eilenberg–Mac Lane spectrum](#MotivicCohomology) $H(\mathbb{Z})$. At least if $S$ is a [[field]] of characteristic zero, this is the same thing as an integral [[motive]]. The [[spectral sequences]] associated to the slice filtration are analogous to the [[Atiyah-Hirzebruch spectral sequence]]s in that their first page consists of [[motivic cohomology]] groups.

## Related concepts

* [[Postnikov tower]], [[slice filtration]]
* [[motivic homotopy theory]]

## References

* [[Vladimir Voevodsky]], _Open problems in the motivic stable homotopy theory, I_, [web](http://www.math.uiuc.edu/K-theory/0392/).

* [[Vladimir Voevodsky]], _A possible new approach to the motivic spectral sequence for algebraic K-theory_, [web](http://www.math.uiuc.edu/K-theory/469/).

* [[Marc Levine]], _Motivic Postnikov towers_, talk notes, 2007, [pdf](https://www.uni-due.de/~bm0032/publ/Postnikov2.pdf).

In the [[unstable motivic homotopy category]]:

* [[Pablo Pelaez]], _The Unstable Slice Filtration_, [arXiv:1201.3657](http://arxiv.org/abs/1201.3657).

[[!redirects motivic slice filtration]]
[[!redirects motivic Postnikov tower]]
[[!redirects Voevodsky slice tower]]
[[!redirects Voevodsky slice filtration]]


motivic spectrum

#Contents#
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## Idea

An object in [[motivic homotopy theory]] [[stabilization|stabilized]] not with respect to the categorical sphere, but the geometric (algebraic) sphere.

See at _[motivic homotopy theory -- The stable motivic homotopy category](motivic+homotopy+theory#StableCategory)_.

## References

In the context of [[Snaith's theorem]]:

* [[David Gepner]], [[Victor Snaith]], _On the motivic spectra representing algebraic cobordism and algebraic K-theory_, Documenta Math.  2008 ([arXiv:0712.2817](http://arxiv.org/abs/0712.2817))
 {#GepnerSnaith08}

[[!redirects motivic spectra]]


motivic sphere

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## Idea

(...)

## Related concepts

* [[Tate sphere]]

* [[motivic homotopy theory]]

## References

* MO discussion [here](http://mathoverflow.net/q/115657/381)

On ([[stable cohomotopy|stable]]) [[motivic cohomology|motivic]] [[Cohomotopy]] of [[schemes]] (as  [[motivic homotopy theory|motivic homotopy classes]] of maps into [[motivic sphere|motivic]] [[Tate spheres]]):

* [[Aravind Asok]], [[Jean Fasel]], [[Mrinal Kanti Das]], *Euler class groups and motivic stable cohomotopy*, Journal of the EMS **24** 8 (2022) 2775–2822 [[arXiv:1601.05723](https://arxiv.org/abs/1601.05723), [doi:10.4171/jems/1156](https://doi.org/10.4171/jems/1156)]

* [[Samuel Lerbet]], *Motivic stable cohomotopy and unimodular rows*, Advances in Mathematics **436** 109415 (2024) [[arXiv:2206.11688](https://arxiv.org/abs/2206.11688), [doi:10.1016/j.aim.2023.109415](https://doi.org/10.1016/j.aim.2023.109415)]

[[!redirects motivic spheres]]


motivic Thom spectrum

#Contents#
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## Idea

The _motivic Thom spectrum_ $MGL$ is the [[motivic spectrum]] in the [[stable motivic homotopy category]] representing [[algebraic cobordism]].  It is the algebraic or [[motives|motivic]] analogue of the [[Thom spectrum]] in the [[stable homotopy category]] which represents [[complex cobordism]].

## Definition

In analogy with the [[complex cobordism]] spectrum $MU$, [Voevodsky](#Voevodsky98) defined the algebraic cobordism spectrum $MGL_S$ in the [[stable motivic homotopy category]] by the formula

$$ MGL_S = colim_{n\to\infty} \Omega^n_{\mathbf{P}^1} Th(V_n), $$

where $S$ is the base scheme, $Th(V_n)$ is the (infinite suspension of) the [[Thom space]] of the tautological vector bundle $V_n$ over the infinite Grassmannian of $n$-planes $Gr(n)$. More precisely, $Gr(n)$ is defined as the colimit of the smooth $S$-schemes $Gr(n,k)$ as $k\to\infty$, and $V_n$ is similarly the colimit of the tautological vector bundles over $Gr(n,k)$.
The notation $\Omega^n_{\mathbf{P}^1}$ denotes the $n$th $\mathbf{P}^1$-loop space.

## Properties

Like the [[motivic Eilenberg-Mac Lane spectrum]] $H\mathbf{Z}$ and the [[algebraic K-theory]] spectrum $KGL$, $MGL$ is an $E_\infty$-motivic [[ring spectrum]].

The spectrum $MGL$ was used by Voevodsky in his proof of the [[Bloch-Kato conjecture]]. It is also the starting point of [[chromatic motivic homotopy theory]].

## Related concepts

* [[algebraic cobordism]]
* [[complex cobordism]]
* [[Thom spectrum]]
* [[motivic homotopy theory]]
* [[Snaith's theorem]]
* [[motivic Eilenberg-Mac Lane spectrum]]
* [[algebraic K-theory]]

## References

* {#Voevodsky98} [[Vladimir Voevodsky]], _$\mathbf{A}^1$-Homotopy Theory_, Doc. Math., Extra Vol. ICM 1998(I), 417-442, [web](http://www.mathematik.uni-bielefeld.de/documenta/xvol-icm/00/Voevodsky.MAN.html).

* [[Ivan Panin]], K. Pimenov, [[Oliver Röndigs]], _A universality theorem for Voevodsky's algebraic cobordism spectrum_, Homology, Homotopy and Applications, 2008, 10(2), 211-226, [arXiv](http://arxiv.org/abs/0709.4116).

* [[Ivan Panin]], K. Pimenov, [[Oliver Röndigs]], _On the relation of Voevodsky's algebraic cobordism to Quillen's K-theory_, Inventiones mathematicae, 2009, 175(2), 435-451, [DOI](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.244.7301), [arXiv](http://arxiv.org/abs/0709.4124).

* [[Markus Spitzweck]], _Algebraic cobordism in mixed characteristic_, [arXiv](http://arxiv.org/abs/1404.2542).

* [[Marc Hoyois]], _From algebraic cobordism to motivic cohomology_, [pdf](http://math.mit.edu/~hoyois/papers/hopkinsmorel.pdf), [arXiv](http://arxiv.org/abs/1210.7182).

* {#GepnerSnaith08} [[David Gepner]], [[Victor Snaith]], _On the motivic spectra representing algebraic cobordism and algebraic K-theory_, Documenta Math.  2008 ([arXiv:0712.2817](http://arxiv.org/abs/0712.2817)).

A discussion on MathOverflow:

* _Interdependence between A^1-homotopy theory and algebraic cobordism_, [MO/36659](http://mathoverflow.net/questions/36659/interdependence-between-a1-homotopy-theory-and-algebraic-cobordism/36698#36698).

[[!redirects algebraic Thom spectrum]]
[[!redirects algebraic Thom complex]]
[[!redirects motivic cobordism spectrum]]
[[!redirects MGL]]


Motzkin number

[[!redirects Motzkin numbers]]

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## Idea

(...)

## Definition

(....)

## Properties

### Relation to standard Young tableaux

The $n$th Motzkin number equals the
the number $\left\vert sYT_n(3) \right\vert$ of [[standard Young tableaux]] with $n$ boxes and $\leq 3$ rows.

([Regev 81](#Regev81))

## Related concepts

* [[Catalan number]]

* [[chord diagram]]

## References

See also

* Wikipedia, *[Motzkin number](https://en.wikipedia.org/wiki/Motzkin_number)*

As counting [[standard Young tableaux]] with $\leq 3$ rows:

* {#Regev81} [[Amitai Regev]], *Asymptotic values for degrees associated with strips of young diagrams*, Advances in Mathematics Volume 41, Issue 2, August 1981, Pages 115-136 (doi:10.1016/0001-8708(81)90012-8)


Moufang loop

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## Idea

A Moufang loop is a set with a [[binary operation]] that is similar to that of a [[group]], but does not require [[associativity]]. Making this precise is a little fiddly.

## Definition

A **Moufang loop** is a set $Q$ with a binary operation $\cdot\colon Q\times Q \to Q$ (a [[magma]]) with two-sided unit $e$ such that left and right multiplication, $y\cdot -\colon Q \to Q$ and $-\cdot x\colon Q \to Q$ respectively, are [[isomorphisms]] (i.e. it is a unital [[quasigroup]] or _[[loop (algebra)|loop]]_) and such that the _Moufang identities_ hold:

* $(u v)(w u) = (u(v w))u = u((v w)u)$

* $((u v)u)w = u(v(u w))$

* $((u v)w)v = u(v(w v))$

One may consider the weaker analogous structure without unit $e$, the Moufang quasigroup, but the Moufang identities, by a result of [[Kenneth Kunen]] imply that the quasigroup is a loop.

## Properties

Every element in a Moufang loop has a multiplicative inverse; a priori there are only left and right inverses, but these coincide by the Moufang identities.

Since right and left multiplication give isomorphisms of the underlying set, one can 'divide' by any element of the Moufang loop, on the right or on the left (recall we are not assuming commutativity). Thus one can define a Moufang loop as a set together with a multiplication as above, together with right and left division operations $/, \backslash \colon Q\times Q \to Q$, again satisfying the Moufang identities. Thus Moufang loops are algebras for a [[Lawvere theory]], and thus can be defined [[internalization|internal]] to any category with [[finite products]].

Moufang loops are _[[power-associative algebra|power-associative]]_, in that any bracketing of a string consisting of copies of the same element multiply to a unique element. In fact, more is true, in that any two elements generate a genuine group; that is, Moufang loops are _[[alternative algebra|alternative]]_.

## Examples

* Any group is a Moufang loop.

* The non-zero [[octonions]] form a Moufang loop, as does the multiplicatively closed subset of octonions of norm 1 (which form the [[7-sphere]]). The invertible elements in any alternative ring or [[alternative algebra]] form a Moufang loop.

* The basis octonions, $1,e_1,\ldots,e_7$ and their additive inverses $-1,-e_1,\ldots,-e_7$ form a finite Moufang loop of order 16 (compare with the case of the quaternions, where the basis elements and their inverses form a group of order 8, the [[quaternion group]] $Q_8$).

* There is a finite Moufang loop of order $2^{13}$ which [[John Conway]] used to construct the [[Monster group]]. This Moufang loop is a central extension by $\mathbb{Z}/2$ of the [[binary Golay code]], an abelian group of order $2^{12}$, see ([Hsu](#Hsu)). This is a special case of a [[code loop]], which is a specific sort of Moufang loop.

## References

Related concepts in $n$Lab: [[quasigroup]], [[Bol loop]], [[composition algebra]], [[identities of Bol-Moufang type]]

* Wikipedia: [Moufang loop](http://en.wikipedia.org/wiki/Moufang_loop)

* [[eom]]: [Moufang loop](http://www.encyclopediaofmath.org/index.php?title=M/m065050)

* R. Moufang, _Zur Struktur von Alternativkörpern_, Math. Ann. __110__ (1935) 416–430 ([gdz](http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=GDZPPN002277301))

* Tim Hsu, _Explicit constructions of code loops as centrally twisted products_, Math. Proc. Cambridge Philos. Soc. __128__ (2000), no. 2, 223–232 ([journal](http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=37707&fulltextType=RA&fileId=S030500419900403X) or [preprint](http://www.math.sjsu.edu/~hsu/pdf/tp.pdf))
{#Hsu}

* [[Kenneth Kunen]], _Moufang quasigroups_, J. Algebra __183__:1 (1996) 231–234 [doi](http://dx.doi.org/10.1006/jabr.1996.0216)
* E. N. Kuz'min, , Algebra i Logika __10__ (1) (1971), pp. 3–22, in Russian [pdf](), English translation: _On a relation between Malcev algebras and analytic Moufang loops_, Algebra Logic __10__ (1972), pp. 1–14

[[!redirects Moufang loops]]
[[!redirects Moufang quasigroup]]


Mouzhe Xie

* [personal page](https://sites.google.com/view/xiemouzhe/home)

## Selected writings

On [[quantum sensing]] via [[spin resonance]] on the [[nitrogen-vacancy center in diamond]]:

* [[Mouzhe Xie]] et al., *Biocompatible surface functionalization architecture for a diamond quantum sensor*, PNAS **119** 8 (2022) e2114186119 [[arXiv:2108.04843](https://arxiv.org/abs/2108.04843), [doi:10.1073/pnas.2114186119](https://doi.org/10.1073/pnas.2114186119)]

Survey:

* [[Mouzhe Xie]], *Diamond-based quantum sensor for molecular analytics*, [talk at](Center+for+Quantum+and+Topological+Systems#XieApr2023) [[CQTS]] (Apr 2023) [slides: [[Xie-CQTS-Apr2023.pdf:file]], video: [YT](https://www.youtube.com/watch?v=f4qv-H5vkR0)]

category: people


movable pro-object

## Definition

Let $\Lambda$ be a [[directed set]], which we may also view as the associated [[order category]], a directed category. View a [[pro-object]] $X:\Lambda^{op}\to C$ as a family of objects $\{X_\lambda\}_{\lambda\in\Lambda}$ in $C$ and bonding morphisms $p_{\lambda\lambda'}:X_{\lambda'}\to X_\lambda$ whenever $\lambda'\gt\lambda$.

A pro-object $X$ is movable if for every $\lambda\in\Lambda$ there is $\lambda'\gt\lambda$ ('the movability index of $\lambda$') such that for every $\lambda''\gt\lambda'$ there is a morphism $r : X_{\lambda'}\to X_{\lambda''}$ such that
$p_{\lambda\lambda''} \circ r = p_{\lambda\lambda'}$.

#### Relative version

Let $C$ be a category and $C_0\subset C$ a full subcategory. $(\Lambda, X, p)$ in pro-$C$ is movable with respect to $C_0$ (or $C_0$-movable) if every $\lambda\in\Lambda$ admits $\lambda'\gt\lambda$ such that for every $\lambda''\gt\lambda$ and every morphism $h:Y\to X_{\lambda'}$ in $C$ such that $Y\in C_0$ there is a morphism
$r:Y\to X_{\lambda''}$ in pro-$C$ such that
$$p_{\lambda\lambda''}\circ r = p_{\lambda\lambda'}\circ h.$$

## Movability of topological spaces

Term movability comes from topology where it denotes a property studied by Borsuk and which generalizes the class of topological spaces having the [[shape]] of an [[absolute neighborhood retract]]. This can be restated using expansions by pro-objects. There are versions for topological spaces and for pointed topological spaces. As pointed space is a special case of a pair of topological spaces it is natural to define  the movability of pairs by requiring that there exist an expansion of the pair by the pro-object in the category of pairs in the homotopy category of polyhedra which is movable as an abstract pro-object.

## Properties

Every [[retract]] in pro-$C$ of a movable pro-object is a movable pro-object.

Let $\mathcal{F}$ be a full subcategory of the category Grp of groups whose objects are the free groups. Then a pro-group is $\mathcal{F}$-movable iff the underlying pro-object in the category of [[pointed set]]s is movable. A pointed pro-set is movable iff it satisfies the [[Mittag-Leffler property]]. This may be used to show that every retract of a pro-object satisfying Mittag-Leffler property is also Mittag-Leffler.

## Literature

* Ch. II, Sec. 2 in [[S. Mardešić]], J. Segal, _Shape theory_, North Holland 1982


movable singularity

# Fixed and movable singularities
* table of contents
{: toc}

Given an [[ordinary differential equation]] (ODE) for an analytic function from a domain in the complex plane (or on a Riemann surface) to the complex numbers, a point is a __fixed singularity__ if it is a [[singular point]] of every solution of the ODE and a __movable singularity__ if the position of the singular point depends on the solution (or equivalently on the initial conditions).

For example, the [[Ricatti equation]] $u'' + u^2 = 0$ has a general solution $u(z) = 1/(z-z_0)$, where $z_0\in\mathbb{C}$ is an arbitrary constant. The position of the pole is at $z_0$, hence it depends on a particular solution.

An ODE satisfies the [[Painlevé property]] if all solutions are single-valued around every movable singularity.

## Related entries

Related entries include [[monodromy]], [[holomorphic function]], [[Riemann-Hilbert problem]], [[Painlevé transcendent]], [[Milnor fiber]], [[Stokes phenomenon]], [[semiclassical approximation]], [[Fuchsian equation]], [[hypergeometric function]], and [[Picard-Lefschetz theory]].

## References

* Martin D. Kruskal, Nalini Joshi, Rod Halburd, _Analytic and asymptotic methods for nonlinear singularity analysis: a review and extensions of tests for the Painlevé property_, 1996 [pdf](http://www.ucl.ac.uk/~ucahrha/Publications/Pond-97.pdf)

* Henryk Żołądek, _The monodromy group_, Monografie Matematyczne __67__, 588 pp. Birkhäuser 2006

* Vladimir V. Golubev, _Lekcii po analitičeskoi teorii differencial'nyh uravnenii_. 1ed 1941, 2nd ed. Moscow-Leningrad. 1950; German transl. _Vorlesungen ueber
Differentialgleichungen im Komplexen_, Deutsche Verlag Wissenschaft., 1958

As a possible generalization for PDEs,
one can consider _analytic movable singularity manifolds_ as in

* J. Weiss, M. Tabor, G. Carnevale, _The Painlevé  property
for partial differential equations_, J. Math. Phys. __24__, 522–526 (1983)

category: analysis

[[!redirects movable singular point]]
[[!redirects movable singular points]]
[[!redirects movable singularity]]
[[!redirects movable singularities]]

[[!redirects fixed singular point]]
[[!redirects fixed singular points]]
[[!redirects fixed singularity]]
[[!redirects fixed singularities]]


movie

#Contents#
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{:toc}

## Idea

One way of visualizing and/or presenting higher dimensional [[bordisms]]/[[tangles]]/[[knots]] is as a sequence of [[pre-images]] of a sequence of values of a suitable [[Morse function]]. If one thinks of the Morse function here as assigning a "[[time]]"-cordinate to the [[bordism]]/[[tangle]]/[[knot]] then each of these pre-images is like a frame of a movie ([Carter-Saito 93](#CarterSaito93)).

## References

The term was introduced, in the context of [[surface knots]] in:

* {#CarterSaito93} [[Scott Carter]], Masahico Saito, _Reidemeister moves for surface isotopies and their interpretations as moves to movies_,  Journal of Knot Theory and Its RamificationsVol. 02, No. 03, pp. 251-284 (1993) ([doi:10.1142/S0218216593000167](https://doi.org/10.1142/S0218216593000167))

Review:

* [[Carlo Collari]],  Chapter 2, Section 1.2 in: _The Functoriality of Khovanov Homology and the Monodromy of Knots_, 2013 ([pdf](https://core.ac.uk/download/pdf/19204099.pdf), [[CollariKhovanovHomology.pdf:file]])

[[!redirects movies]]

[[!redirects movie move]]
[[!redirects movie moves]]


Moyal deformation quantization

[[!redirects Moyal quantization]]

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## Idea

The _Moyal product_ is a [[formal deformation quantization]] of a linear [[Poisson manifold]], hence of a [[vector space]] $V$ equipped with a [[Poisson bivector]] $\pi \in V \wedge V$, regarded as a constant (translation invariant) [[tensor field|bivector field]].

Moyal quantization serves as an intermediate step in quantization of more general situations:

Given a [[symplectic manifold]], then Moyal quantization applies in each [[fiber]] of the [[tangent bundle]]. The resulting [[fiber bundle]] of Moyal algebras admits a [[flat connection]] (non-uniquely) compatible with the algebra structure. The [[covariantly constant sections]] of this Moyal-algebra bundle constitute a [[formal deformation quantization]] of the symplectic manifold, see at _[[Fedosov's deformation quantization]]_.

With a little care, the Moyal construction applies also to infinite-dimensional Poisson vector spaces such as appear in [[local field theory]].  Here the Moyal quantization yields [[formal deformation quantization]] of [[free field theories]] to [[perturbative quantum field theories]], the result are the _[[Wick algebras]]_ of free field theory ([Dito 90](#Dito90), [Dütsch-Fredenhagen 01](#DutschFredenhagen01)). Combining this this with [[Fedosov's deformation quantization]] as above yields [[interaction|interacting]] [[perturbative quantum field theories]] as constructed via [[causal perturbation theory]] ([Collini 16](#Collini16)), see at _[[locally covariant perturbative quantum field theory]]_ for more on this.

## Definition

The Moyal [[star product]] on [[smooth functions]] $C^\infty(V)$ is given on $f,g \in C^\infty(V)$ by

$$
  f \star g \coloneqq prod \circ \exp(\hbar \pi)(f , g)
  \,,
$$

where in the exponent we regard $\pi$ as an [[endomorphism]] on the [[tensor product]] $C^\infty(V) \otimes C^\infty(V)$ by [[differentiation]] in each argument, where the [[exponential]] denotes the corresponding [[formal power series]] of iterated applications of this endomorphism, and where $prod \colon C^\infty(V) \otimes C^\infty(V) \to C^\infty(V)$ is the usual pointwise product of functions.

This means that given a choice of [[basis]] $\{x^i\}_i$ of $V$ such that $\pi$ has components $\{\pi^{i j}\}_{i j}$ in this basis, the resulting [[formal power series]] in the formal parameter $\hbar$ ("[[Planck's constant]]") starts out as

$$
  (f \star g) =
  f \cdot g
  +
  \hbar
  \sum_{i,j} \pi^{i j} \frac{\partial f}{\partial x^i}\cdot \frac{\partial g}{\partial x^j}
  +
  \frac{1}{2}
  \hbar^2
  \sum_{i,j,k, l} \pi^{k l} \pi^{i j} \frac{\partial^2 f}{\partial x^k\partial x^i}\cdot \frac{\partial^2 g}{\partial x^l \partial x^j}
   +
  \cdots
$$

## Properties

### Integral representation

+-- {: .num_prop #IntegralRepresentationOfStarProduct}
###### Proposition
**([[integral]] representation of [[star product]])**

If the functions $f,g$ admit [[Fourier analysis]] (are [[functions with rapidly decreasing partial derivatives]]), then  their [[star product]] is equivalently given by the following [[integral]] expression:

$$
  \begin{aligned}
  \left(f \star_\omega g\right)(x)
   &=
  \frac{(det(\omega)^{2n})}{(2 \pi \hbar)^{2n} }
  \int
    e^{ - \tfrac{1}{i \hbar} \omega((x - \tilde y),(x-y))}
    f(y)
    g(\tilde y)
   \,
   d^{2 n} y
   \,
   d^{2 n} \tilde y
  \end{aligned}
$$

=--

([Baker 58](#Baker58), see at _[[star product]]_ [this prop](star+product#IntegralRepresentationOfStarProduct)).

### Via geometric quantization
 {#ViaGeometricQuantization}

The Moyal quantization of a Poisson vector space $(V,\pi)$ arises equivalently as the canonical [[geometric quantization of symplectic groupoids]] of the [[symplectic groupoid]] which is the [[Lie integration]] of the corresponding [[Poisson Lie algebroid]] ([Weinstein 91, p. 446](#Weinstein91), [Garcia-Bondia & Varilly 94, section V](#GBV), [Hawkins 06](#EH)).

See at _[[star product]]_ [this prop.](star+product#PolarizedSymplecticGroupoidConvolutionProductOfSymplecticVectorSpaceIsMoyalStarProduct) for the **proof**; and see at _[geometric quantization of symplectic groupoids -- Examples -- Moyal quantization](geometric+quantization+of+symplectic+groupoids#MoyalQuantizationofPoissonVectorSpace)_ for more.

## References
 {#References}

The Moyal product was introduced independently in

* [[Hilbrand Groenewold]], _On the Principles of elementary quantum mechanics_, Physica,12 (1946) pp. 405-460.

* [[José Moyal]], _Quantum mechanics as a statistical theory_. Mathematical Proceedings of the Cambridge Philosophical Society 45: 99 (1949)

The integral expression (prop. \ref{IntegralRepresentationOfStarProduct}) is apparently due to

* {#Baker58} George A. Baker, _Formulation of Quantum Mechanics Based on the Quasi-Probability Distribution Induced on Phase Space_, Jr. Phys. Rev. 109, 2198 – Published 15 March 1958 ([doi:10.1103/PhysRev.109.2198](https://doi.org/10.1103/PhysRev.109.2198))

General accounts include

* D. B. Fairlie, _Moyal Brackets, Star Products and the Generalised Wigner Function_ ([arXiv:hep-th/9806198](https://arxiv.org/abs/hep-th/9806198))

* Maciej Blaszak, Ziemowit Domanski, _Maciej Blaszak, Ziemowit Domanski_ ([arXiv:1009.0150](https://arxiv.org/abs/1009.0150))

The understanding of the Moyal product as the [[polarization|polarized]] [[groupoid convolution algebra]] of the corresponding [[symplectic groupoid]], hence as an example of [[geometric quantization of symplectic groupoids]] had been suggested without proof in

* {#Weinstein91} [[Alan Weinstein]], p. 446 in P. Donato et al. (eds.) _Symplectic Geometry and Mathematical Physics,  (Birkhäuser, Basel, 1991);

and was proven in detail in

* {#GBV} [[José Gracia-Bondia]], [[Joseph Varilly]], _From geometric quantization to Moyal quantization_, J. Math. Phys. 36 (1995) 2691-2701 ([arXiv:hep-th/9406170](http://arxiv.org/abs/hep-th/9406170))

In a broader context this was reconsidered in

* {#EH} [[Eli Hawkins]], example 6.2 of _A groupoid approach to quantization_,  J. Symplectic Geom. Volume 6, Number 1 (2008), 61-125. ([arXiv:math.SG/0612363](http://arxiv.org/abs/math.SG/0612363))

The observation that Moyal deformation quantization applied to the [[Peierls-Poisson  bracket]] yields the [[Wick algebra]] quantization of [[free field theories]] is due to

* {#Dito90} J. Dito, _Star-product approach to quantum field theory: The free scalar field_. Letters in Mathematical Physics, 20(2):125–134, 1990 ([spire](https://inspirehep.net/record/303898/))

and was amplified in the broader context of [[perturbative AQFT]] in

* {#DutschFredenhagen01} [[Michael Dütsch]], [[Klaus Fredenhagen]], _Perturbative algebraic field theory, and deformation quantization_, in [[Roberto Longo]] (ed.), _Mathematical Physics in Mathematics and Physics, Quantum and Operator Algebraic Aspects_, volume 30 of Fields Institute Communications, pages 151–160. American Mathematical Society, 2001

That moreover the corresponding [[Fedosov deformation quantization]] based on this free field theory star product yields the [[causal perturbation theory]] quantization of interacting field theories is due to

* {#Collini16} [[Giovanni Collini]], _Fedosov Quantization and Perturbative Quantum Field Theory_ ([arXiv:1603.09626](https://arxiv.org/abs/1603.09626))

[[!redirects Moyal product]]
[[!redirects Moyal products]]

[[!redirects Moyal star-product]]
[[!redirects Moyal star-products]]

[[!redirects Moyal star product]]
[[!redirects Moyal star products]]

[[!redirects Moyal *-product]]
[[!redirects Moyal *-products]]

[[!redirects Moyal star]]

[[!redirects Moyal deformation quantization]]


MP

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#### Cohomology
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## Idea

The [[periodic cohomology theory|2-periodic]] version of [[complex cobordism cohomology theory]] [[MU]] is usually denoted $MP$ (e.g. [Lurie lecture 18, example 5](#LurieLect18)).

## Properties

### As a Thom spectrum

$MP$ may be realized as the [[Thom  spectrum]] of the [[universal vector bundle|universal]] [[virtual vector bundle|virtual]] [[complex vector bundle|complex bundle]] of arbitrary [rank](rank#rank_of_a_vector_bundle) over the space $B U \times \mathbb{Z}$ ([Lurie lecture 18, Example 5](#LurieLect18)).

### Landweber exact functor therem

There is an analogue of the [[Landweber exact functor theorem]] for [[even cohomology theory|even]] 2-[[periodic cohomology theories]], with [[MU]] replaced by [[MP]] ([Lurie lecture 18, prop. 11](#LurieLect18)).

## Related concepts

[[!include flavours of cobordism cohomology theories -- table]]

## References

* {#LurieLect18} [[Jacob Lurie]], in _[[Chromatic Homotopy Theory]]_, Lecture series 2010 , Lecture 18 _Even periodic cohomology theories_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture18.pdf))

[[!redirects periodic complex cobordism cohomology theory]]
[[!redirects periodic complex cobordism]]
[[!redirects periodic complex bordism]]

[[!redirects periodic complex cobordism spectrum]]

[[!redirects MP theory]]
[[!redirects MP-theory]]


MR

Can refer to either:

* [[Mathematical Reviews]]

* [[MR cohomology theory]]

category: disambiguation


MR cohomology theory

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## Idea

What is called _$MR$ cohomology theory_ or _Real cobordism_ ([Landweber 68](#Landweber68), [Landweber 69](#Landweber69)) is the $\mathbb{Z}_2$-[[equivariant cohomology theory]] version of [[complex cobordism]] $MU$.

There is an evident [[action]] of $\mathbb{Z}_2$ on [[formal group laws]] given by negation in the formal group (the [[inversion involution]]), and this lifts to an [[involution|involutive]] [[automorphism]] $MU \stackrel{\simeq}{\to} MU$ of the [[spectrum]] [[MU]]. This induces an $\mathbb{Z}_2$-[[equivariant spectrum]] $M\mathbb{R}$, and real cobordism is the [[cohomology theory]] that it [[Brown representability theorem|represents]].  This is directly analogous to how [[complex K-theory]] [[KU]] gives $\mathbb{Z}_2$-equivariant [[KR-theory]], both are examples of _[[real-oriented cohomology theories]]_.

A modern review in in ([Kriz 01, section 2](#Kriz01)).

## Properties

### Relation to real manifolds

While $\pi_\bullet(M \mathbb{R})$ is _not_ the [[cobordism ring]] of [[real manifolds]], still every [[real manifold]] does give a class in $M \mathbb{R}$ ([Hu 99](#Hu99),  [Kriz 01, p. 13](#Kriz01)).

### $E_\infty$-structure

$M \mathbb{R}$ is naturally an [[E-∞ ring]] spectrum. (reviewed as [Kriz 01, prop. 3.1](#Kriz01))

### Universal real orientation

In direct analogy with the situation for [[complex cobordism theory]] in [[complex oriented cohomology theory]], $M \mathbb{R}$ is the univeral [[real oriented cohomology theory]]:

Equivalence classes of [[real oriented cohomology theory|real orientations]] of a $\mathbb{Z}/2\mathbb{Z}$-[[equivariant spectrum|equivariant]] [[E-∞ ring]] $E$ are in [[bijection]] to equivalence classes of [[E-∞ ring]] homomorphisms

$$
  M \mathbb{R}\longrightarrow E
  \,.
$$

([Kriz 01, theorem 2.25](#Kriz01))

## Related concepts

* [[equivariant cobordism cohomology theory]]

* [[real-oriented cohomology theory]]

  * [[KR-theory]]

  * [[BPR-theory]]

\linebreak

[[!include flavours of cobordism cohomology theories -- table]]

## References
 {#References}

The definition of Real cobordism cohomology goes back to:

* {#Landweber67} [[Peter Landweber]], *Fixed point free conjugations on complex manifolds*, Annals of Mathematics **86** (2) (1967) 491-502 [[jstor:](https://www.jstor.org/stable/1970612)]

* {#Landweber68} [[Peter Landweber]], _Conjugations on complex manifolds and equivariant homotopy of $MU$_, Bulletin of the American Mathematical Society **74** (1968) 271-274 [[Euclid](https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society/volume-74/issue-2/Conjugations-on-complex-manifolds-and-equivariant-homotopy-of-MU/bams/1183529527.full)]

and in the broader context of [[real-oriented cohomology theories]]:

* {#Araki78} [[Shôrô Araki]], *$\tau$-Cohomology Theories*, Japanese Journal of Mathematics **4** 2 (1978) 363-416 [[doi:10.4099/math1924.4.363](https://doi.org/10.4099/math1924.4.363)]

* {#Araki79} [[Shôrô Araki]], *Forgetful spectral sequences*, Osaka Journal of Mathematics **16** 1 (1979) 173-199 [[Euclid](https://projecteuclid.org/journals/osaka-journal-of-mathematics/volume-16/issue-1/Forgetful-spectral-sequences/ojm/1200771837.full)]

* [[Shôrô Araki]], *Orientations in $\tau$-cohomology theories*, Japan Journal of Mathematics (N.S.) **5** 2 (1979) 403-430 [[doi:10.4099/math1924.5.403](https://doi.org/10.4099/math1924.5.403)]

The [[Adams spectral sequence]] for Real cobordism:

* [[Po Hu]], *The cobordism of Real manifolds and calculations with the Real Adams-Novikov spectral sequence*, (1998) [[hdl:2027.42/130996](http://hdl.handle.net/2027.42/130996), [pdf](https://deepblue.lib.umich.edu/bitstream/handle/2027.42/130996/9825253.pdf?sequence=2&isAllowed=n)]

* {#Kriz01} [[Po Hu]], [[Igor Kriz]], _Real-oriented homotopy theory and an analogue of the Adams-Novikov spectral sequence_, Topology 40 (2001) 317-399 [doi:10.1016/S0040-9383(99)00065-8, [pdf](http://www.math.rochester.edu/people/faculty/doug/otherpapers/hukriz.pdf)]

Further discussion:

* {#Hu99} [[Po Hu]], *The cobordism of Real manifolds*, Fundamenta Mathematicae (1999) Volume: 161, Issue: 1-2, page 119-136 [[dml:212395](https://eudml.org/doc/212395), [pdf](http://matwbn.icm.edu.pl/ksiazki/fm/fm161/fm16115.pdf)]

* [[Po Hu]], [[Igor Kriz]], *Some Remarks on Real and Algebraic Cobordism*, K-Theory **22** (2001) 335–366 [[pdf](https://deepblue.lib.umich.edu/bitstream/handle/2027.42/43150/10977_2004_Article_321295.pdf?sequence=1), [doi:10.1023/A:1011196901303](http://dx.doi.org/10.1023/A:1011196901303)]

* [[Po Hu]], [[Igor Kriz]], *Topological Hermitian Cobordism*, Journal of Homotopy and Related Structures, **11** (2016) 173–197 [[arXiv:1110.5608](https://arxiv.org/abs/1110.5608), [doi:10.1007/s40062-014-0100-9](https://doi.org/10.1007/s40062-014-0100-9)]

[[!redirects MR-theory]]

[[!redirects Real cobordism cohomology]]


Mrinal Kanti Das

* [institute page](https://publications.iitpkd.ac.in/researcher/mrinal-kanti-das)

* [ResearchGate page](https://www.researchgate.net/profile/Mrinal-Das-9)

## Selected writings

On ([[stable cohomotopy|stable]]) [[motivic cohomology|motivic]] [[Cohomotopy]] of [[schemes]] (as  [[motivic homotopy theory|motivic homotopy classes]] of maps into [[motivic sphere|motivic]] [[Tate spheres]]):

* [[Aravind Asok]], [[Jean Fasel]], [[Mrinal Kanti Das]], *Euler class groups and motivic stable cohomotopy*, Journal of the EMS **24** 8 (2022) 2775–2822 [[arXiv:1601.05723](https://arxiv.org/abs/1601.05723), [doi:10.4171/jems/1156](https://doi.org/10.4171/jems/1156)]

category: people


MSet

[[!redirects M Set]]
If $M$ is a [[monoid]], then $M Set$ or $M‑Set$ is the [[category]] of $M$-sets.  That is, an [[object]] of this category is a [[set]] $X$ equipped with an [[action]] of $M$ on $X$, and a [[morphism]] is an [[equivariant function]].

Don't confuse this with the [[MultiSet|category of multisets]].

* See also [[Understanding M-Set]]

category: category

[[!redirects M-Set]]
[[!redirects M‑Set]]


MSO

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## Idea

(...)

## Related concepts

[[!include flavours of cobordism cohomology theories -- table]]

## References

* [[Michael Atiyah]], _Bordism and Cobordism_, Mathematical Proceedings of the Cambridge Philosophical Society, Volume 57, Issue 2, April 1961, pp. 200 - 208 ([doi:10.1017/S0305004100035064](https://doi.org/10.1017/S0305004100035064), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/atiyahb.pdf))

* {#Novikov62} [[Sergei Novikov]], _Homotopy properties of Thom complexes_, Mat. Sbornik 57 (1962), no. 4, 407–442, 407–442 ([pdf](http://www.mi-ras.ru/~snovikov/6.pdf), [[NovikovThomComplexes.pdf:file]])

* {#Stong68} [[Robert Stong]], Chapter IX of: _Notes on Cobordism theory_, Princeton University Press, 1968 ([toc pdf](http://pi.math.virginia.edu/StongConf/Stongbookcontents.pdf), [ISBN:9780691649016](http://press.princeton.edu/titles/6465.html), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/stongcob.pdf))

* [[Manifold Atlas]], _[Oriented bordism](http://www.map.mpim-bonn.mpg.de/Oriented_bordism)_

[[!redirects oriented bordism]]
[[!redirects oriented cobordism]]


MSp

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## Idea

By $MSp$ one denotes the [[Thom spectrum]] for stable [[Sp(n)]]-[[G-structure|structure]], [[Brown representability theorem|representing]] what is called "symplectic" or "quaternionic" [[cobordism cohomology theory]]. The coefficient ring is, accordingly, the "quaternionic"/"symplectic" [[bordism ring]], usually denoted $\Omega^{Sp}_\bullet$.

## Properties

### Relation to $MU$

The canonical [[topological group]]-[[subgroup|inclusions]]

$$
  1
  \;\subset\;
  Sp(k)
  \;\subset\;
  SU(2k)
  \;\subset\;
  U(2k)
$$

([[trivial group]] into [[quaternionic unitary group]] into [[special unitary group]] into [[unitary group]]) induce [[ring spectrum]]-[[homomorphism]] of [[Thom spectra]]

$$
  M Fr
   \;\longrightarrow\;
  M Sp
   \;\longrightarrow\;
  M SU
   \;\longrightarrow\;
  M \mathrm{U}
$$

(from [[MFr]] to [[MSp]] to [[MSU]] to [[MU]])

and hence corresponding [[multiplicative cohomology theory]]-[[homomorphisms]] of [[cobordism cohomology theories]].

(e.g. [Conner-Floyd 66, p. 27 (34 of 120)](#ConnerFloyd66))

## Related entries

* [[quaternionic oriented cohomology theory]]

[[!include flavours of cobordism cohomology theories -- table]]

## References

* {#Novikov62} [[Sergei Novikov]], _Homotopy properties of Thom complexes_, Mat. Sbornik 57 (1962), no. 4, 407–442, 407–442 ([pdf](http://www.mi-ras.ru/~snovikov/6.pdf), [[NovikovThomComplexes.pdf:file]])

* [[Arunas Liulevicius]], _Notes on Homotopy of Thom Spectra_, American Journal of Mathematics Vol. 86, No. 1 (1964), pp. 1-16 ([jstor:2373032](https://www.jstor.org/stable/2373032))

* {#ConnerFloyd66} [[Pierre Conner]], [[Edwin Floyd]], Section 5 of: _[[The Relation of Cobordism to K-Theories]]_, Lecture Notes in Mathematics __28__ Springer 1966 ([doi:10.1007/BFb0071091](https://link.springer.com/book/10.1007/BFb0071091), [MR216511](http://www.ams.org/mathscinet-getitem?mr=216511))

* [[Robert Stong]], _Some Remarks on Symplectic Cobordism_, Annals of Mathematics Second Series, Vol. 86, No. 3 (Nov., 1967), pp. 425-433 ([doi:10.2307/1970608](https://doi.org/10.2307/1970608))

* D. M. Segal, _On the symplectic cobordism ring_, Commentarii Mathematici Helvetici 45, 159–169 (1970) ([doi:10.1007/BF02567323](https://doi.org/10.1007/BF02567323))

* [[Nigel Ray]], _The symplectic bordism ring_, Volume 71, Issue 2 March 1972, pp. 271-282 ([doi:10.1017/S0305004100050519](https://doi.org/10.1017/S0305004100050519))

* [[Stanley Kochman]], _The symplectic cobordism ring I_, Memoirs of the American Mathematical Society 1980, Volume 24, Number 271 ([doi:10.1090/memo/0271](http://dx.doi.org/10.1090/memo/0271))

* [[Stanley Kochman]], _The symplectic cobordism ring II_, Memoirs of the American Mathematical Society 1982
Volume 40, Number 271 ([doi:10.1090/memo/0271](http://dx.doi.org/10.1090/memo/0271))

* [[Andrew Baker]], _Some chromatic phenomena in the homotopy of $MSp$_, in: N. Ray, G. Walker (eds.), _Adams Memorial Symposium on Algebraic Topology, Vol. 2_ editors, Cambridge University Press (1992), 263–80 ([pdf](http://www.maths.gla.ac.uk/~ajb/dvi-ps/bpmsp.pdf), [[BakerMSp.pdf:file]])

* {#GorbunovRay92} [[Vassily Gorbounov]], [[Nigel Ray]], _Orientations of $Spin$ Bundles and Symplectic Cobordism_, Publ. RIMS, Kyoto Univ. 28 (1992), 39-55 ([[GorbunovRaySpinBundles.pdf:file]], [doi: 10.2977/prims/1195168855](https://www.ems-ph.org/journals/show_abstract.php?issn=0034-5318&vol=28&iss=1&rank=4))

* [[Vassily Gorbounov]], [[Nigel Ray]], [[Paul Turner]], _On the Hopf Ring for a Symplectic Oriented Spectrum_, American Journal of Mathematics
Vol. 117, No. 4 (Aug., 1995), pp. 1063-1088 ([doi:10.2307/2374960](https://doi.org/10.2307/2374960))

* {#Laughton08} [[Craig Laughton]], _Quasitoric manifolds and cobordism theory_, Manchester 2008 ([pdf](http://www.ma.man.ac.uk/~nige/craiglphd.pdf), [[LaughtonQuasitoric.pdf:file]])

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

[[!redirects quaternionic cobordism]]
[[!redirects quaternionic cobordism theory]]

[[!redirects quaternionic bordism ring]]
[[!redirects quaternionic bordism rings]]

[[!redirects quaternionic cobordism ring]]
[[!redirects quaternionic cobordism rings]♧

[[!redirects symplectic bordism theory]]
[[!redirects symplectic bordism homology theory]]

[[!redirects symplectic bordism ring]]
[[!redirects symplectic cobordism ring]]


MSpin

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## Idea

The universal [[Thom spectrum]] for [[spin structure]] is denoted $M Spin$.

## Related concepts

* [[Atiyah-Bott-Shapiro orientation]]

\linebreak

[[!include flavours of cobordism cohomology theories -- table]]


MSpin^c

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## Idea

The [[cobordism theory]] for manifolds equipped with (stable) [[Spin^c structure]].

## Properties

### Atiyah-Bott-Shapiro orientation

(...)

see at _[[Atiyah-Bott-Shapiro orientation]]_

(...)

### Relation to complex K-theory

$M Spin^c$ is related to [[KU]] in a variant of the [[Conner-Floyd isomorphism]], via the [[Atiyah-Bott-Shapiro orientation]] ([Hopkins-Hovey 92, Thm. 1](#HopkinsHovey92))

## Related concepts

[[!include flavours of cobordism cohomology theories -- table]]

## References

* {#HopkinsHovey92} [[Michael Hopkins]], [[Mark Hovey]], _Spin cobordism determines real K-theory_, Mathematische Zeitschrift volume 210, pages 181–196 (1992) ([doi:10.1007/BF02571790](https://doi.org/10.1007/BF02571790), [[HopkinsHoveyCobordismK.pdf:file]])


MSSM

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=--
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{:toc}

## Idea

In [[particle physics]] the letters _MSSM_ are an abbreviation for "minimally supersymmetric standard model": an extension of the [[standard model of particle physics]] to a [[model (in particle phyiscs)]] with a minimum amount of extra [[global symmetry|global]] [[supersymmetry]].

## Examples

* [[G2-MSSM]]

## Related concepts

* [[super QCD]]

* [[R-parity]]

* [[intersecting D-brane model]]

* [[gauge coupling unification]]

* [[supersymmetry and Calabi-Yau manifolds]]

## References

### General

The original articles are

* S. Dimopoulos and H. Georgi, Phys. Lett. B 117, 287
(1982).

* [[Steven Weinberg]], Phys. Rev. D 26, 287 (1982)

Theoretical predictions

* Stefano Frixione, Benjamin Fuks, Valentin Hirschi, Kentarou Mawatari, Hua-Sheng Shao, Marthijn P. A. Sunder, Marco Zaro, _Automated simulations beyond the Standard Model: supersymmetry_ ([arXiv:1907.04898](https://arxiv.org/abs/1907.04898))

Review:

* S. F. King, S. Moretti, R. Nevzorov, _A Review of the Exceptional Supersymmetric Standard Model_ ([arXiv:2002.02788](https://arxiv.org/abs/2002.02788))

* [[Michael Dine]], _Supersymmetry and string theory: beyond the standard model_, Cambridge University Press (2006, 2007) [[ISBN:9781009290920](https://www.cambridge.org/ae/academic/subjects/physics/particle-physics-and-nuclear-physics/supersymmetry-and-string-theory-beyond-standard-model-2nd-edition?format=HB), [pdf](http://www.nucleares.unam.mx/~alberto/apuntes/dine.pdf)]

### Relation to flavour anomalies

Suggestion that the [[MSSM]] with [[R-parity]] violation could explain the [[flavour anomalies]]:

* {#WangYangYuan19} Dong-Yang Wang, Ya-Dong Yang, Xing-Bo Yuan, _$b \to c \tau \bar \nu$ decays in supersymmetry with R-parity violation_ ([arXiv:1905.08784](https://arxiv.org/abs/1905.08784))

* Quan-Yi Hu, Lin-Lin Huang,
_Explaining $b\to s \ell^+ \ell^-$ data by sneutrinos in the R-parity violating MSSM_ ([arXiv:1912.03676](https://arxiv.org/abs/1912.03676))

* Quan-Yi Hu, Ya-Dong Yang, Min-Di Zheng, _Revisiting the B-physics anomalies in R-parity violating MSSM_ ([arXiv:2002.09875](https://arxiv.org/abs/2002.09875))

* Wolfgang Altmannshofer, P. S. Bhupal Dev, Amarjit Soni, Yicong Sui, _Addressing $R_{D^\ast}$, $R_{K^\ast}$, muon $g-2$ and ANITA anomalies in a minimal R-parity violating supersymmetric framework_ ([arXiv:2002.12910](https://arxiv.org/abs/2002.12910))

### In heterotic string theory
 {#ReferencesInHeteroticStringTheory}

The origin of all [[string phenomenology]] is the [[top-down model building|top-down approach]] in the [[heterotic string]] due to ([Candelas-Horowitz-Strominger-Witten 85](heterotic+string#CHSW85)).

A brief review of motivations for [[GUT]] models in [[heterotic string theory]] is in

* {#Witten02} [[Edward Witten]], _Quest For Unification_, Heinrich Hertz lecture at [SUSY 2002](http://www.desy.de/susy02/) at DESY, Hamburg ([arXiv:hep-ph/0207124](http://arxiv.org/abs/hep-ph/0207124))

The following articles establish the existences of exact realization of the [[gauge group]] and [[matter]]-content of the [[MSSM]] in [[heterotic string theory]] (not yet checking [[Yukawa couplings]]):

* [[Volker Braun]], [[Yang-Hui He]], [[Burt Ovrut]], [[Tony Pantev]], _A Heterotic Standard Model_, Phys. Lett. B618 : 252-258 2005 ([arXiv:hep-th/0501070](http://arxiv.org/abs/hep-th/0501070))

* [[Volker Braun]], [[Yang-Hui He]], [[Burt Ovrut]], [[Tony Pantev]], _The Exact MSSM Spectrum from String Theory_, JHEP 0605:043,2006 ([arXiv:hep-th/0512177](http://arxiv.org/abs/hep-th/0512177))

* [[Vincent Bouchard]], [[Ron Donagi]], _An SU(5) Heterotic Standard Model_, Phys. Lett. B633:783-791,2006 ([arXiv:hep-th/0512149](http://arxiv.org/abs/hep-th/0512149))

A computer search through the "[[landscape of string theory vacua|landscape]]" of [[Calabi-Yau varieties]] showed several hundreds more such exact heterotic standard models (about one billionth of all CYs searched, and most of them arising as $SU(5)$-[[GUTs]])

* [[Lara Anderson]], [[Yang-Hui He]], [[Andre Lukas]], _Heterotic Compactification, An Algorithmic Approach_, JHEP 0707:049, 2007 ([arXiv:hep-th/0702210](https://arxiv.org/abs/hep-th/0702210))

* [[Lara Anderson]], James Gray, [[Andre Lukas]], [[Eran Palti]], _Two Hundred Heterotic Standard Models on Smooth Calabi-Yau Threefolds_ ([arXiv:1106.4804](http://arxiv.org/abs/1106.4804))

* [[Lara Anderson]], James Gray, [[Andre Lukas]], [[Eran Palti]], _Heterotic Line Bundle Standard Models_ JHEP06(2012)113 ([arXiv:1202.1757](https://arxiv.org/abs/1202.1757))

* [[Lara Anderson]], Andrei Constantin, James Gray, [[Andre Lukas]], [[Eran Palti]], _A Comprehensive Scan for Heterotic SU(5) GUT models_, JHEP01(2014)047 ([arXiv:1307.4787](https://arxiv.org/abs/1307.4787))

* [[Yang-Hui He]], Seung-Joo Lee, [[Andre Lukas]], Chuang Sun, _Heterotic Model Building: 16 Special Manifolds_ ([arXiv:1309.0223](http://arxiv.org/abs/1309.0223))

* {#CHE18} Andrei Constantin, [[Yang-Hui He]], [[Andre Lukas]], _Counting String Theory Standard Models_ ([arXiv:1810.00444](https://arxiv.org/abs/1810.00444))

The resulting database of compactifications is here:

* [[Lara Anderson]], James Gray, [[Andre Lukas]], [[Eran Palti]], _Heterotic standard model database_ ([web](http://www-thphys.physics.ox.ac.uk/projects/CalabiYau/linebundlemodels/index.html.))

Review includes

* [[Lara Anderson]], _New aspects of heterotic geometry and phenomenology_, talk at [Strings2012](http://wwwth.mpp.mpg.de/conf/strings2012/), Munich 2012 ([pdf](http://wwwth.mpp.mpg.de/members/strings/strings2012/strings_files/program/Talks/Wednesday/Anderson.pdf))

* {#He18} [[Yang-Hui He]], _Deep-learning the landscape_, talk at _[String and M-Theory: The new geometry of the 21st century](https://ims.nus.edu.sg/events/2018/wstring/wk.php)_ ([pdf slides](https://ims.nus.edu.sg/events/2018/wstring/files/yang.pdf), [video recording](https://www.youtube.com/watch?v=x3ThgBgkPlE))

Computation of [[metrics]] on these Calabi-Yau compactifications (eventually needed for computing their induced [[Yukawa couplings]]) is started in

* [[Volker Braun]], Tamaz Brelidze, [[Michael Douglas]], [[Burt Ovrut]], _Calabi-Yau Metrics for Quotients and Complete Intersections_, JHEP 0805:080, 2008 ([arXiv:0712.3563](https://arxiv.org/abs/0712.3563))

This "heterotic standard model" has a "hidden sector" copy of the actual standard model, more details of which are discussed here:

* [[Volker Braun]], [[Yang-Hui He]], [[Burt Ovrut]], _Supersymmetric Hidden Sectors for Heterotic Standard Models_ ([arXiv:1301.6767](http://arxiv.org/abs/1301.6767))

The issue of [[moduli stabilization]] in these kinds of models is discussed in

* Michele Cicoli, Senarath de Alwis, Alexander Westphal, _Heterotic Moduli Stabilization_ ([arXiv:1304.1809](http://arxiv.org/abs/1304.1809))

* [[Lara Anderson]], James Gray, [[Andre Lukas]], [[Burt Ovrut]], _Vacuum Varieties, Holomorphic Bundles and Complex Structure Stabilization in Heterotic Theories_ ([arXiv:1304.2704](http://arxiv.org/abs/1304.2704))

Principles singling out heterotic models with three generations of fundamental particles are discussed in:

* [[Philip Candelas]], [[Xenia de la Ossa]], [[Yang-Hui He]], Balazs Szendroi, _Triadophilia: A Special Corner in the Landscape_, Adv.Theor.Math.Phys.12:2,2008 ([arXiv:0706.3134](http://arxiv.org/abs/0706.3134))

See also

* Hajime Otsuka, _$SO(32)$ heterotic line bundle models_, ([arXiv:1801.03684](https://arxiv.org/abs/1801.03684))

### In F-theory

A [[geometric engineering]] of the MSSM in [[F-theory]] (see at [[string phenomenology]]) is claimed in

* {#CLLO18} [[Mirjam Cvetič]], Ling Lin, Muyang Liu, Paul-Konstantin Oehlmann, _An F-theory Realization of the Chiral MSSM with $\mathbb{Z}_2$-Parity_ ([arXiv:1807.01320](https://arxiv.org/abs/1807.01320))

A large number of realizations of the exact field content of the MSSM in [[F-theory]] is claimed to be realized in

* {#CveticHalversonLinLiuTian19} [[Mirjam Cvetic]], [[James Halverson]], Ling Lin, Muyang Liu, Jiahua Tian, _A Quadrillion Standard Models from F-theory_ ([arXiv:1903.00009](https://arxiv.org/abs/1903.00009))

### Experimental status

* Jin Min Yang, Pengxuan Zhu, Rui Zhu, *A brief survey of low energy supersymmetry under current experiments* ([arXiv:2211.06686](https://arxiv.org/abs/2211.06686))

[[!redirects minimally supersymmetric standard model]]


MSU

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cobordism theory
+--{: .hide}
[[!include cobordism theory -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The [[cobordism cohomology theory]] for [[special unitary group]]-[[G-structure|structure]].

## Properties

We write $\Omega^{SU}_\bullet$ for the [[bordism ring]] for [[stable tangent bundle|stable]] [[special unitary group|SU]]-[[G-structure|structure]].

### Relation to $MU$

The canonical [[topological group]]-[[subgroup|inclusions]]

$$
  1
  \;\subset\;
  Sp(k)
  \;\subset\;
  SU(2k)
  \;\subset\;
  U(2k)
$$

([[trivial group]] into [[quaternionic unitary group]] into [[special unitary group]] into [[unitary group]]) induce [[ring spectrum]]-[[homomorphism]] of [[Thom spectra]]

$$
  M Fr
   \;\longrightarrow\;
  M Sp
   \;\longrightarrow\;
  M SU
   \;\longrightarrow\;
  M \mathrm{U}
$$

(from [[MFr]] to [[MSp]] to [[MSU]] to [[MU]])

and hence corresponding [[multiplicative cohomology theory]]-[[homomorphisms]] of [[cobordism cohomology theories]], so in particular [[ring homomorphisms]] of [[bordism rings]]

\[
  \label{HomomorphismsOfCobordismRings}
  \Omega^{fr}_{\bullet}
   \longrightarrow
  \Omega^{Sp}_{\bullet}
   \longrightarrow
  \Omega^{SU}_{\bullet}
   \longrightarrow
  \Omega^{U}_{\bullet}
\]

(e.g. [Conner-Floyd 66, p. 27 (34 of 120)](#ConnerFloyd66))

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

The [[kernel]] of the [[forgetful functor|forgetful]] morphism (eq:HomomorphismsOfCobordismRings)

$$
  \Omega^{SU}_\bullet
  \longrightarrow
  \Omega^{\mathrm{U}}_\bullet
$$

from the [[SU-bordism ring]] to the [[complex bordism ring]], is pure [[torsion subgroup|torsion]].

=--

([CLP 19, Thm. 5.8a](#CLP19))

+-- {: .num_prop #TorsionInSUBordismConcentratedInDegrees1And2Mod8}
###### Proposition

The [[torsion subgroup]] of the [[SU-bordism ring]] is concentrated in degrees $8k+1$ and $8k+2$, for $k \in \mathbb{N}$.

=--

([CLP 19, Thm. 5.11a](#CLP19))

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

Every [[torsion element]] in the [[SU-bordism ring]] $\Omega^{SU}_\bullet$ has [[order of an element|order]] 2.

=--

([CLP 19, Thm. 5.8b](#CLP19))

+-- {: .num_prop #SUBordismRingAwayFromTwo}
###### Proposition
**([[SU-bordism ring]] [[localization of a ring|away from 2]] is [[polynomial algebra]])**

The [[SU-bordism ring]] with [[localization of a ring|2 inverted]] is the [[polynomial algebra]] over $\mathbb{Z}\big[\tfrac{1}{2}\big]$ on one generator in every even degree $\geq 4$:

$$
  \Omega^{SU}\big[\tfrac{1}{2}\big]
  \;\simeq\;
  \mathbb{Z}
  \big[
    \tfrac{1}{2}
  \big]
  \big[
    \{
      y_{2i+4}
    \}_{i \in \mathbb{N}}
  \big]
  \,.
$$

=--

(due to [Novikov 62](#Novikov62), review in [LLP 17, Thm. 1.2](#LLP17))

### Relation to Calabi-Yau manifolds
 {#RelationToCalabiYauManifolds}

We discuss the classes of [[Calabi-Yau manifolds]] in the [[SU-bordism ring]]. For more see at _[[Calabi-Yau manifolds in SU-bordism theory]]_.

+-- {: .num_prop #K3SurfaceSpansSUBordismRingInDegree4}
###### Proposition
**([[K3-surface]] spans [[SU-bordism ring]] in degree 4)**

The degree-4 generator $y_4 \in \Omega^{SU}_4$ in the [[SU-bordism ring]] (Prop. \ref{SUBordismRingAwayFromTwo}) is represented by minus the class of any (non-[[torus]]) [[K3-surface]]:

\[
  \label{K3GeneratesNonTorsionSUBordimsRing}
  \Omega^{SU}_4
    \;\simeq\;
  \mathbb{Z}\big\langle -[K3] \big\rangle
  \,.
\]

=--

([LLP 17, Example 3.1](#LLP17), [CLP 19, Theorem 13.5a](#CLP19))

+-- {: .num_cor #K3SurfaceInUBordismRing}
###### Corollary
**([[K3-surface]] represents non-trivial element in [[U-bordism ring]])**

The image in the [[MU]]-[[cobordism ring]] of the class of the [[K3-surface]] $[K3] \in \Omega^{SU}_4$ (eq:K3GeneratesNonTorsionSUBordimsRing) under the canonical morphism $\Omega^{SU}_4 \to \Omega^{\mathrm{U}}_4$ (eq:HomomorphismsOfCobordismRings) is non-trivial.

In fact, the canonical morphism is an [[injection]] in this degree

$$
  \array{
    \Omega^{SU}_4
      &\hookrightarrow&
    \Omega^{\mathrm{U}}_4
    \\
    [K3] &\mapsto& [K3]
    \,.
  }
$$

=--

(This is vaguely indicated in [Novikov 86, p. 216 (218 of 321)](#Novikov86).)

\begin{proof}
  By Prop. \ref{KernelOfMapFromSUBordismToUBordismIsTorsion} the
  kernel of the map to $\Omega^{\mathrm{U}}_4$ is torsion, but by Prop. \ref{K3SurfaceSpansSUBordismRingInDegree4} $[K3]$ represents a non-torsion element. Since it is in fact a non-torsion generator, the kernel vanishes (as also implied by Prop. \ref{TorsionInSUBordismConcentratedInDegrees1And2Mod8}).

\end{proof}

+-- {: .num_prop }
###### Proposition
**([[Calabi-Yau manifolds]] generate the [[SU-bordism ring]] [[localization of a ring|away from 2]])**

The [[SU-bordism ring]] [[localization of a ring|away from 2]] is multiplicatively generated by [[Calabi-Yau manifolds]].

=--

([LLP 17, Theorem 2.4](#LLP17))

+-- {: .num_prop }
###### Proposition
**([[Calabi-Yau manifolds]] in complex dim $\leq 4$ span the [[SU-bordism ring]] in $deg \leq 8$ [[localization of a ring|away from 2]])**

There are [[Calabi-Yau manifolds]] of complex dimension $3$ and $4$ whose whose SU-bordism classes equal the generators $\pm y_6$ and $\pm y_8$ in Prop. \ref{SUBordismRingAwayFromTwo}.

Together with the [[K3 surface]] representing $- y_4$ (Prop. \ref{K3SurfaceSpansSUBordismRingInDegree4}), this means that [[CYs]] [[linear space|span]] $\Omega^{SU}_{\leq 8}\big[ \tfrac{1}{2}\big]$.

=--

([CLP 19, Theorem 13.5](#CLP19))

## Related concepts

* [[Calabi-Yau manifold]]

* [[generalized Calabi-Yau manifold]]

\linebreak

[[!include flavours of cobordism cohomology theories -- table]]

## References

* {#ConnerFloyd66} [[Pierre Conner]], [[Edwin Floyd]], Section 5 of: _[[The Relation of Cobordism to K-Theories]]_, Lecture Notes in Mathematics __28__ Springer 1966 ([doi:10.1007/BFb0071091](https://link.springer.com/book/10.1007/BFb0071091), [MR216511](http://www.ams.org/mathscinet-getitem?mr=216511))

On the SU-bordism ring structure away from 2:

* {#Novikov62} [[Sergei Novikov]], _Homotopy properties of Thom complexes_, Mat. Sbornik 57 (1962), no. 4, 407–442, 407–442 ([pdf](http://www.mi-ras.ru/~snovikov/6.pdf), [[NovikovThomComplexes.pdf:file]])

* {#Stong68} [[Robert Stong]], Chapter X of: _Notes on Cobordism theory_, Princeton University Press, 1968 ([toc pdf](http://pi.math.virginia.edu/StongConf/Stongbookcontents.pdf), [ISBN:9780691649016](http://press.princeton.edu/titles/6465.html), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/stongcob.pdf))

Survey:

* {#Novikov86} [[Sergei Novikov]], p. 218 in: _Topology I -- General survey_, in: Encyclopedia of Mathematical Sciences Vol. 12, Springer 1986 ([doi:10.1007/978-3-662-10579-5](https://link.springer.com/book/10.1007/978-3-662-10579-5), [pdf](https://web.math.rochester.edu/people/faculty/doug/otherpapers/novikovsurv.pdf))

On its [[torsion subgroups]]:

* [[Pierre Conner]], [[Edwin Floyd]], _Torsion in $SU$-bordism_, Memoirs of the American Mathematical Society 1966 ([ISBN:978-1-4704-0007-1](https://bookstore.ams.org/memo-1-60))

Relation to [[Calabi-Yau manifolds]]:

* {#LLP17} [[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))

On the (failure of) the [[Conner-Floyd isomorphism]] for $MSU \to $ [[KO]]:

* [[Serge Ochanine]], _Modules de SU-bordisme. Applications_, Bulletin de la Société Mathématique de France, Tome 115 (1987) , pp. 257-289 ([doi:10.24033/bsmf.2078]( https://doi.org/10.24033/bsmf.2078))

Survey:

* {#CLP19} 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))

* [[Taras Panov]], _A geometric view on $SU$-bordism_, talk at Moscow State University 2020 ([webpage](https://www.math.princeton.edu/events/geometric-view-su-bordism-2020-09-17t170000), [pdf](http://higeom.math.msu.su/people/taras/talks/2019SPb-Panov.pdf), [[PanovSU-Bordism.pdf:file]])

[[!redirects SU-bordism]]
[[!redirects SU-bordism theory]]

[[!redirects SU-cobordism]]
[[!redirects SU-cobordism theory]]

[[!redirects special unitary bordism ring]]
[[!redirects special unitary bordism rings]]

[[!redirects SU-bordism ring]]
[[!redirects SU-bordism rings]]

[[!redirects special unitary cobordism ring]]
[[!redirects special unitary cobordism rings]]

[[!redirects SU-cobordism ring]]
[[!redirects SU-cobordism rings]]


MSUFr

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cobordism theory
+--{: .hide}
[[!include cobordism theory -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In joint generalization of the [[cobordism cohomology theories]] [[MSU]] and [[MFr]] of [[closed manifold|closed]] $SU$-manifolds and of $Fr$-manifolds, respectively, an _$(SU,fr)$-manifold_ ([Conner-Floyd 66, Section 16, p. 103 onwards](#ConnerFloyd66)) is a [[compact topological space|compact]] [[manifold with boundary]] equipped with [[special unitary group]]-[[tangential structure]] on its [[stable tangent bundle]] and equipped with a [[trivial vector bundle|trivialization]] (stable [[framed manifold|framing]]) of that over the [[boundary]].

The corresponding [[bordism classes]] form a [[bordism ring]] denoted $\Omega^{SU,fr}_\bullet$.

## Properties

### Representing spectrum

In generalization to how $\Omega^{SU}_{2k}$ is represented by [[homotopy classes]] of [[maps]] into the [[Thom spectrum]] [[MSU]], so $\Omega^{SU,fr}_{2k}$ is represented by maps into the [[quotient spaces]] $MSU_{2k}/S^{2k}$ (for $S^{2k} = Th(\mathbb{C}^{k}) \to Th( \mathbb{C}^k \times_{U(k)} E U(k) ) = MU_{2k}$ the canonical inclusion):

\[
  \label{InTermsOfHomotopyGroupsOfQuotientedThomSpace}
  \Omega^{(SU,fr)}_\bullet
  \;=\;
  \pi_{\bullet + 2k}
  \big(
    MSU_{2k}/S^{2k}
  \big)
  \,,
  \;\;\;\;\;
  \text{for any}
  \;
  2k \geq \bullet + 2
  \,.
\]

([Conner-Floyd 66, p. 103](#ConnerFloyd66))

### Relation to $MSU$ and $MFr$

In every eighth degree, the [[bordism rings]] for [[MSU]], $MSUFr$ and [[MFr]] sit in a [[short exact sequence]] of the form ([Conner-Floyd 66, p. 104](#ConnerFloyd66)):

\[
  \label{ShortExactSequenceOfUFrBordismRings}
  0
  \to
  \Omega^{SU}_{8\bullet+4}
  \overset{i}{
    \longrightarrow
  }
  \Omega^{SU,fr}_{8\bullet+4}
  \overset{\partial}{
    \longrightarrow
  }
  \Omega^{fr}_{8\bullet + 3}
  \to
  0
  \,,
\]

where $i$ is the evident inclusion, while $\partial$ is restriction to the [[boundary]].

In particular, this means that $\partial$ is [[surjective function|surjective]], hence that every $Fr$-manifold of dimension $8k + 3$ is the boundary of a $(U,fr)$-manifold.

### Relation to Todd classes and the $e_{\mathbb{R}}$-invariant

In refinement of how the complex [[e-invariant is the Todd class of cobounding (U,fr)-manifolds]] we have for [[special unitary group]]-structure instead of [[unitary group]]-structure and in dimensions $8\bullet + 4$:

Since on $(8 \bullet + 4)$-dimensional $SU$-manifolds the [[Todd class]] is divisible by 2 [Conner-Floyd 66, Prop. 16.4](#ConnerFloyd66), we have ([Conner-Floyd 66, p. 104](#ConnerFloyd66)) the following [[short exact sequence]] of [[MSUFr]]-[[bordism rings]]:

\[
  \label{HalfToddClassesOnShortExactSequenceOfSUFrBordismRings}
  \array{
  0
  \to
  &
  \Omega^{SU}_{8\bullet+4}
  &
  \overset{i}{\longrightarrow}
  &
  \Omega^{SU,fr}_{8\bullet+4}
  &
  \overset{\partial}{
    \longrightarrow
  }
  &
  \Omega^{fr}_{8\bullet + 3}
  &
  \simeq
  &
  \pi^s_{8\bullet+3}
  \\
  &
  \big\downarrow{}^{\tfrac{1}{2}\mathrlap{Td}}
  &&
  \big\downarrow{}^{\tfrac{1}{2}\mathrlap{Td}}
  &&
  \big\downarrow{}^{}
  &&
  \big\downarrow{}^{e_{\mathbb{R}}}
  \\
  0
  \to
  &
  \mathbb{Z}
  &\overset{\;\;\;\;\;}{\hookrightarrow}&
  \mathbb{Q}
  &\overset{\;\;\;\;}{\longrightarrow}&
  \mathbb{Q}/\mathbb{Z}
  &=&
  \mathbb{Q}/\mathbb{Z}
  }
  \,.
\]

This produces $e_{\mathbb{R}}$, the [[Adams e-invariant]] with respect to [[KO]]-theory instead of [[KU]] ([Adams 66, p. 39](e-invariant#Adams66)), which, in degrees $8k + 3$, is indeed half of the e-invariant $e_{\mathbb{C}}$ for $KU$ (by [Adams 66, Prop. 7.14](e-invariant#Adams66)).

In fact, for $k = 0$ we have:

+-- {: .num_prop #AdamseRInvariantDetectsThirdStableHomotopyGroupOfSpheres}
###### Proposition
**([Adams 66, Example 7.17  and p. 46](e-invariant#Adams66))**

In degree 3, the [[KO]]-theoretic [[e-invariant]] $e_{\mathbb{R}}$ takes the value $\left[\tfrac{1}{24}\right] \in \mathbb{Q}/\mathbb{Z}$ on the [[quaternionic Hopf fibration]] $S^7 \overset{h_{\mathbb{H}}}{\longrightarrow} S^4$  and hence reflects the full [[third stable homotopy group of spheres]]:

$$
  \array{
    \pi^s_3
      &
      \underoverset{
        \simeq
      }{
        e_{\mathbb{R}}
      }{
        \;\;\longrightarrow\;\;
      }
      &
    \mathbb{Z}/24
    &
    \subset
    &
    \mathbb{Q}/\mathbb{Z}
    \\
    [h_{\mathbb{H}}]
    &&\mapsto&&
    \left[\tfrac{1}{24}\right]
  }
$$

while $e_{\mathbb{C}}$ sees only "half" of it (by [Adams 66, Prop. 7.14](e-invariant#Adams66)).

=--

## Related concepts

[[!include flavours of cobordism cohomology theories -- table]]

## References

The concept of $(SU,fr)$-bordism theory and its relation to the [[e-invariant]] originates with:

* {#ConnerFloyd66} [[Pierre Conner]], [[Edwin Floyd]], Section 16 from p. 103 onwards, in: _[[The Relation of Cobordism to K-Theories]]_, Lecture Notes in Mathematics __28__ Springer 1966 ([doi:10.1007/BFb0071091](https://link.springer.com/book/10.1007/BFb0071091), [MR216511](http://www.ams.org/mathscinet-getitem?mr=216511))


MU

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cobordism theory
+--{: .hide}
[[!include cobordism theory -- contents]]
=--
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
=--
=--

#Contents##
* table of contents
{:toc}

## Idea

$MU$ is the universal [[Thom spectrum]] for [[complex vector bundles]]. It is the spectrum [[Brown representability theorem|representing]] _complex [[cobordism cohomology theory]]_. It is the complex analog of [[MO]].

[[MR cohomology theory]], or _real cobordism_, ([Landweber 68](#Landweber68), [Landweber 69](#Landweber69)) is the $\mathbb{Z}_2$-[[equivariant cohomology theory]] version of $MU$ [[complex cobordism cohomology theory]].

## The $M U$ spectrum

The spectrum denoted $M U$
is, as a [[sequential spectrum]], in degree $2 n$ given by the [[Thom space]] of the underlying real vector bundle of the complex [[universal vector bundle]]:  the [[vector bundle]] that is  [[associated bundle|associated]] by the defining [[representation]] of the [[unitary group]] $U(n)$ on $\mathbb{C}^n$ to the $U(n)$-[[universal principal bundle]]:

$$
  M U(2n) = Thom
  \left(
    standard\;associated\;bundle\;to\;universal\;bundle
    \array{
      E U(n) \\ \downarrow \\ B U(n)
    }
  \right)
$$

A priori this yields a  [sequential S2-spectrum](Introduction+to+Stable+homotopy+theory+--+1#SequentialTSpectra), which is then turned into a sequential $S^1$-spectrum by taking the component spaces in odd degree to be the [[smash product]] of the [[circle]] $S^1$ with those in even degree.

This represents a [[complex oriented cohomology theory]] and indeed the universal one among these, see at _[[universal complex orientation on MU]]_.

The _periodic_ complex cobordism theory is given by adding up all the even degree powers of this theory:

$$
  M P = \vee_{n \in \mathbb{Z}} \Sigma^{2 n} M U
$$

The [[cohomology ring]] $M P({*})$ is the [[Lazard ring]] which is the universal coefficient ring for [[formal group laws]], see at _[[Milnor-Quillen theorem on MU]]_ .

The [[periodic cohomology theory|periodic]] version is sometimes written $PMU$.

## Properties

### Homotopy groups: Cobordism and Lazard ring
  {#HomotopyGroups}

The [[graded ring]] given by evaluating complex cobordism theory on the point is both the complex [[cobordism ring]] as well as the [[Lazard ring]] classifying [[formal group laws]].

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

Evaluation of $MU$ on the point yields the complex [[cobordism ring]], whose underlying group is

$$
  \pi_\ast MU \simeq MU_\ast(pt) \simeq \mathbb{Z}[x_1, x_2, \cdots]
  \,,
$$

where the generator $x_i$ is in degree $2 i$.

=--

This is due to ([Milnor 60](#Milnor60), [Novikov 60](#Novikov60), [Novikov 62](#Novikov62)). A review is in ([Ravenel theorem 1.2.18](#Ravenel), [Ravenel, ch. 3, theorem 3.1.5](#Ravenel)).

The [[formal group law]] associated with $MU$ as with any [[complex oriented cohomology theory]] is classified by a [[ring]] [[homomorphism]] $L \longrightarrow \pi_\bullet(MU)$ out of the [[Lazard ring]].

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

This canonical [[homomorphism]] is an [[isomorphism]]

$$
  L \stackrel{\simeq}{\longrightarrow} \pi_\bullet(MU)
$$

between the [[Lazard ring]] and the $MU$-[[cohomology ring]], hence by theorem \ref{RelationToCobordismRing} with the complex [[cobordism ring]].

=--

This is [[Quillen's theorem on MU]].
(e.g [Lurie 10, lect. 7, theorem 1](#LurieLect7))

### Universal complex orientation on $M U$
 {#UniversalComplexOrientation}

There is a canonical [[complex oriented cohomology theory|complex orientation]] on $MU$ obtained from the map

$$
  \omega : \mathbb{C}P^\infty \stackrel{\simeq}{\to}
  M U(1)
  \;\;\;\;
  M U(\mathbb{C}P^\infty)
$$

For $E$ a [[homotopy-commutative ring spectrum]] there is a [[bijection]] between [[complex oriented cohomology theory|complex orientation]] of $E$ and [[ring spectrum]] maps of the form

$$
  MU \longrightarrow E
  \,.
$$

(e.g [Lurie 10, lect. 6, theorem 8](#LurieLect6), [Ravenel, chapter 4, lemma 4.1.13](#Ravenel))

See also at _[[complex orientation and MU]]_.

### $MU$-homology of a manifold: bordisms in $X$

For $X$ a [[manifold]] or a [[topological space]], the $MU$-homology [[group]] $MU_\ast(X)$ of its underlying [[homotopy type]] is the group of equivalence classes of maps $\Sigma \to X$ from manifolds $\Sigma$ with [[complex structure]] on the stable [[normal bundle]], modulo suitable complex [[cobordisms]].

See [Ravenel chapter 1, section 2](#Ravenel).

For more information, see the article [[bordism homology theory]],
which treats the oriented case; the case of (stable almost) complex structure is similar.

### $MU$-cohomology of a manifold: cobordisms in $X$

$MU$-cohomology groups of a manifold $M$ can be expressed
in terms of bordisms given by proper [complex-oriented maps](cobordism+cohomology+theory#ComplexOrientedMaps) into $M$.

For more information, see the article _[[cobordism cohomology theory]]_.

### $MU$-homology of $MU$: Hopf algebroid structure on dual Steenrod algebra

Moreover, the dual $MU$-[[Steenrod algebra]] $MU_\bullet(MU)$
forms a [[commutative Hopf algebroid]] over the [[Lazard ring]].
This is the content of the _[[Landweber-Novikov theorem]]_.

### Nilpotence theorem

* [[nilpotence theorem]]

### Snaith's theorem

[[Snaith's theorem]] asserts that the [[periodic complex cobordism spectrum]] is the [[∞-group ∞-ring]] of the [[classifying space]] for stable [[complex vector bundles]] (the classifying space for [[topological K-theory]]) localized at the [[Bott element]] $\beta$:

$$
  PMU \simeq \mathbb{S}[B U][\beta^{-1}]
  \,.
$$

### $p$-Localization and Brown-Peterson spectrum

The [[p-localization]] of $MU$ decomposes into the
[[Brown-Peterson spectra]].

## Related concepts

* [[differential cobordism cohomology]]

[[!include flavours of cobordism cohomology theories -- table]]

## References

### General

* {#Novikov62} [[Sergei Novikov]], _Homotopy properties of Thom complexes_, Mat. Sbornik 57 (1962), no. 4, 407–442, 407–442 ([pdf](http://www.mi-ras.ru/~snovikov/6.pdf), [[NovikovThomComplexes.pdf:file]])

* {#ConnerFloyd66} [[Pierre Conner]], [[Edwin Floyd]], Section 12 of: _[[The Relation of Cobordism to K-Theories]]_, Lecture Notes in Mathematics __28__ Springer 1966 ([doi:10.1007/BFb0071091](https://link.springer.com/book/10.1007/BFb0071091), [MR216511](http://www.ams.org/mathscinet-getitem?mr=216511))

* {#Stong68} [[Robert Stong]], Chapter VII of: _Notes on Cobordism theory_, Princeton University Press, 1968 ([toc pdf](http://pi.math.virginia.edu/StongConf/Stongbookcontents.pdf), [ISBN:9780691649016](http://press.princeton.edu/titles/6465.html), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/stongcob.pdf))

* {#ConnerSmith69} [[Pierre Conner]], [[Larry Smith]], _On the complex bordism of finite complexes_, Publications Mathématiques de l'IHÉS, Tome 37 (1969) , pp. 117-221 ([numdam:PMIHES_1969__37__117_0](http://www.numdam.org/item/?id=PMIHES_1969__37__117_0))

* [[Larry Smith]], _On Realizing Complex Bordism Modules: Applications to the Stable Homotopy of Spheres_, American Journal of Mathematics Vol. 92, No. 4 (Oct., 1970), pp. 793-856 ([doi:10.2307/2373397](https://doi.org/10.2307/2373397))

* {#Landweber70} [[Peter Landweber]], _On the complex bordism and cobordism of infinite complexes_, Bull. Amer. Math. Soc. Volume 76, Number 3 (1970) ([Euclid](http://projecteuclid.org/euclid.bams/1183531832))

* {#Quillen71} [[Daniel Quillen]], *Elementary Proofs of Some Results of Cobordism Theory Using Steenrod Operations*, Advances in Mathematics **7** (1971) 29–56 [doi:10.1016/0001-8708(71)90041-7]

* [[John Frank Adams]], *[[Stable homotopy and generalized homology]]*, Chicago Lectures in Mathematics, The University of Chicago Press (1974) [[ucp:bo21302708](https://www.press.uchicago.edu/ucp/books/book/chicago/S/bo21302708.html), [pdf](https://www.uio.no/studier/emner/matnat/math/MAT9580/v17/documents/adams-shgh.pdf)]

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

* [[Stanley Kochman]], Section 4.4 of: _[[Bordism, Stable Homotopy and Adams Spectral Sequences]]_, Fields Institute Monographs, American Mathematical Society, 1996 ([cds:2264210](https://cdsweb.cern.ch/record/2264210))

* {#EKMM97} [[Anthony Elmendorf]], [[Igor Kriz]], [[Michael Mandell]], [[Peter May]], section VIII of _[[Rings, modules and algebras in stable homotopy theory]]_ 1997 ([pdf](http://www.math.uchicago.edu/~may/BOOKS/EKMM.pdf))

* {#Hopkins99} [[Mike Hopkins]], section 4 of _Complex oriented cohomology theories and the language of stacks_, course notes 1999 ([pdf](http://www.math.rochester.edu/u/faculty/doug/otherpapers/coctalos.pdf))

* [[Dai Tamaki]], [[Akira Kono]], Section 3.7 and Chapter 6 in: _Generalized Cohomology_, Translations of Mathematical Monographs, American Mathematical Society, 2006 ([ISBN: 978-0-8218-3514-2](https://bookstore.ams.org/mmono-230))

* [[Jacob Lurie]], _[[Chromatic Homotopy Theory]]_ Lecture series ([lecture notes](http://www.math.harvard.edu/~lurie/252x.html)), Lecture 5 _Complex bordism_
([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture5.pdf))

* {#LurieLect6} [[Jacob Lurie]], _[[Chromatic Homotopy Theory]]_ Lecture series ([lecture notes](http://www.math.harvard.edu/~lurie/252x.html)) Lecture 6 _MU and complex orientations_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture6.pdf))

* {#LurieLect7} [[Jacob Lurie]], _[[Chromatic Homotopy Theory]]_ Lecture series ([lecture notes](http://www.math.harvard.edu/~lurie/252x.html)),  Lecture 7 _The homology of MU_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture7.pdf))

* [[Manifold Atlas]], _[Complex bordism](http://www.map.mpim-bonn.mpg.de/Complex_bordism)_

* [[Neil Strickland]], _Products on $MU$-modules_ ([pdf](http://hopf.math.purdue.edu/Strickland/mult.pdf))

* [[Jesse McKeown]], _Complex Cobordism vs. Representing Formal Group Laws_ ([arXiv:1605.09252](http://arxiv.org/abs/1605.09252))

For general discussion of equivariant complex oriented cohomology see at _[equivariant cohomology -- References -- Complex oriented cohomology](equivariant+cohomology#InComplexOrientedGeneralizedCohomologyTheory)_

On the [[Chern-Dold character]] on complex cobordism:

* {#Buchstaber70} [[Victor Buchstaber]], _The Chern–Dold character in cobordisms. I_,

  Russian original: Mat. Sb. (N.S.), 1970 Volume 83(125), Number 4(12), Pages 575–595 ([mathnet:3530](http://m.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=sm&paperid=3530&option_lang=eng))

  English translation: Mathematics of the USSR-Sbornik, Volume 12, Number 4, AMS 1970 ([doi:10.1070/SM1970v012n04ABEH000939](https://iopscience.iop.org/article/10.1070/SM1970v012n04ABEH000939))

* [[Victor Buchstaber]], A. P. Veselov, _Chern-Dold character in complex cobordisms and abelian varieties_ ([arXiv:2007.05782](https://arxiv.org/abs/2007.05782))

### Differential and Hodge-filtered cobordism cohomology

On [[differential cobordism cohomology]] (enhancement of cobordism cohomology to [[differential cohomology]]):

* [[Ulrich Bunke]], [[Thomas Schick]], Ingo Schroeder, Moritz Wiethaup, *Landweber exact formal group laws and smooth cohomology theories*, Algebr. Geom. Topol. **9** (2009) 1751-1790 [[arXiv:0711.1134](https://arxiv.org/abs/0711.1134), [doi:10.2140/agt.2009.9.1751](https://doi.org/10.2140/agt.2009.9.1751)]

The notion of [[Hodge-filtered differential cohomology|Hodge filtered differential]] complex cobordism theory:

* {#HopkinsQuick14} [[Michael J. Hopkins]], [[Gereon Quick]], §5 in: *Hodge filtered complex bordism*, Journal of Topology **8** 1 (2014) 147-183 [[arXiv:1212.2173](https://arxiv.org/abs/1212.2173), [doi:10.1112/jtopol/jtu021](https://doi.org/10.1112/jtopol/jtu021)]

Introduction and survey:

* [[Gereon Quick]], *Geometric Hodge filtered complex cobordism*, [talk at](Center+for+Quantum+and+Topological+Systems#QuickMar2023) *[[CQTS]]* (March 2023) [video:[YT](https://www.youtube.com/watch?v=pMu0gT5kIBo)]

A geometric cocycle model:

* [[Knut B. Haus]], [[Gereon Quick]], *Geometric Hodge filtered complex cobordism* [[arXiv:2210.13259](https://arxiv.org/abs/2210.13259)]

Refinement of the [[Abel-Jacobi map]] to [[Hodge filtration|Hodge filtered]] [[differential cobordism cohomology theory|differential]] [[MU]]-[[cobordism cohomology theory]]:

* [[Gereon Quick]], *An Abel-Jacobi invariant for cobordant cycles*, Documenta Mathematica **21** (2016) 1645–1668 [[arXiv:1503.08449](https://arxiv.org/abs/1503.08449)]

On [[Umkehr maps]] in this context:

* [[Knut Bjarte Haus]], [[Gereon Quick]], *Geometric pushforward in Hodge filtered complex cobordism and secondary invariants* [[arXiv:2303.15899](https://arxiv.org/abs/2303.15899)]

[[!redirects Hodge-filtered differential cohomology theory]]

### Relation to CFT

A relation to [[2d CFT]] over [[Spec(Z)]] was suggested in

* Toshiyuki Katsura, Yuji Shimizu, [[Kenji Ueno]], _Complex cobordism ring and conformal field theory over $\mathbb{Z}$_, Mathematische Annalen
March 1991, Volume 291, Issue 1, pp 551-571 ([journal](http://link.springer.com/article/10.1007%2FBF01445226))

### Relation to divisors
 {#RelationToDivisors}

Relation of [[complex cobordism cohomology]] with [[divisors]], [[algebraic cycles]] and [[Chow groups]]:

* [[Burt Totaro]], _Torsion algebraic cycles and complex cobordism_, J. Amer. Math. Soc. 10 (1997), 467-493 ([doi:10.1090/S0894-0347-97-00232-4](https://doi.org/10.1090/S0894-0347-97-00232-4))

* [MO discussion](https://mathoverflow.net/a/272131/381)

[[!redirects MU-theory]]

[[!redirects complex cobordism spectrum]]

[[!redirects complex bordism ring]]
[[!redirects complex bordism rings]]

[[!redirects complex cobordism ring]]
[[!redirects complex cobordism rings]]

[[!redirects complex cobordism cohomology theory]]
[[!redirects complex cobordism]]

[[!redirects complex bordism]]

[[!redirects complex bordism ring]]
[[!redirects complex bordism rings]]

[[!redirects complex cobordism ring]]
[[!redirects complex cobordism rings]]

[[!redirects complex cobordism cohomology]]
[[!redirects complex cobordism cohomology theory]]

[[!redirects complex cobordism theory]]

[[!redirects U-bordism]]
[[!redirects U-bordisms]]

[[!redirects U-bordism ring]]
[[!redirects U-bordism rings]]

[[!redirects U-bordism theory]]
[[!redirects U-bordism theories]]


Mudumbai Narasimhan

* [Wikipedia entry](http://en.wikipedia.org/wiki/M._S._Narasimhan)

* Interview 2016 [recording](https://youtu.be/Y3n8mwko44g)

## Selected writings

On [[universal connections]] (as used in [[Chern-Weil theory]]):

* [[Mudumbai Narasimhan]] and [[Sundararaman Ramanan]], _Existence of Universal Connections_, American Journal of Mathematics Vol. 83, No. 3 (Jul., 1961), pp. 563-572 ([jstor:2372896 ](https://www.jstor.org/stable/2372896 ))

* [[Mudumbai Narasimhan]] and [[Sundararaman Ramanan]], _Existence of Universal Connections II_, American Journal of Mathematics Vol. 85, No. 2 (Apr., 1963), pp. 223-231 ([jstor:2373211](https://www.jstor.org/stable/2373211))

## Related entries

* [[Narasimhan-Seshadri theorem]]

* [[Harder-Narasimhan theorem]]

category: people

[[!redirects Narasimhan]]


MUFr

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cobordism theory
+--{: .hide}
[[!include cobordism theory -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In joint generalization of the [[cobordism cohomology theories]] [[MU]] and [[MFr]] of [[closed manifold|closed]] $U$-manifolds and of $Fr$-manifolds, respectively, a _$(U,fr)$-manifold_ ([Conner-Floyd 66, Section 16](#ConnerFloyd66), [Conner-Smith 69, Sections 6, 13](#ConnerSmith69)) is a [[compact topological space|compact]] [[manifold with boundary]] equipped with [[unitary group]]-[[tangential structure]] on its [[stable tangent bundle]] and equipped with a [[trivial vector bundle|trivialization]] (stable [[framed manifold|framing]]) of that over the [[boundary]].

The corresponding [[bordism classes]] form a [[bordism ring]] denoted $\Omega^{\mathrm{U},fr}_\bullet$.

## Properties

### Representing spectrum
 {#RepresentingSpectrum}

In generalization to how the complex [[cobordism ring]] $\Omega^U_{2k}$ is represented by [[homotopy classes]] of [[maps]] into the [[Thom spectrum]] [[MU]], so $\Omega^{\mathrm{U},fr}_{2k}$ is represented by maps into the [[quotient spaces]] $MU_{2k}/S^{2k}$ (for $S^{2k} = Th(\mathbb{C}^{k}) \to Th( \mathbb{C}^k \times_{\mathrm{U}(k)} E \mathrm{U}(k) ) = M \mathrm{U}_{2k}$ the canonical inclusion):

\[
  \label{InTermsOfHomotopyGroupsOfQuotientedThomSpace}
  \Omega^{(\mathrm{U},fr)}_\bullet
  \;=\;
  \pi_{\bullet + 2k}
  \big(
    MU_{2k}/S^{2k}
  \big)
  \,,
  \;\;\;\;\;
  \text{for any}
  \;
  2k \geq \bullet + 2
  \,.
\]

([Conner-Floyd 66, p. 97](#ConnerFloyd66))

Hence the representing spectrum $M(\mathrm{U},fr)$ is the [[homotopy cofiber]] of the [[ring spectrum]] unit $1^{M\mathrm{U}} \;\colon\; \mathbb{S} \longrightarrow M \mathrm{U}$ out of the [[sphere spectrum]] ([Conner-Smith 69, p. 156 (41 of 106)](#ConnerSmith69), [Smith 71](#Smith71)) which deserves to be denoted

$$
  M(\mathrm{U},fr)
  \;\coloneqq\;
  M \mathrm{U} / \mathbb{S}
  \,,
$$

but which in notation common around the [[Adams spectral sequence]] would be
"$\Sigma \overline {M \mathrm{U}}$" (as in [Adams 74, theorem 15.1 page 319](Adams+spectral+sequence#Adams74)) or just "$\overline{ M \mathrm{U} }$" (e.g. [Hopkins 99, Cor. 5.3](Adams+spectral+sequence#Hopkins99)):

\[
  \label{AsUnitCofiber}
  \array{
    \mathbb{S}
    &
      \overset{
        1^{M\mathrm{U}}
      }{
        \longrightarrow
      }
    &
    M \mathrm{U}
    \\
    \big\downarrow
    &
    {}^{{}_{(po)}}
    &
    \big\downarrow
    \\
    \ast
    &\longrightarrow&
    M \mathrm{U}/ \mathbb{S}
  }
\]

So in terms of [[stable homotopy groups]] of this spectrum we have the $(\mathrm{U},fr)$-[[cobordism ring]]

\[
  \label{UFrCobordismRing}
  \Omega^{\mathrm{U},fr}_{\bullet}
  \;\coloneqq\;
  \pi_{\bullet}
  \big(
    M\mathrm{U}/\mathbb{S}
  \big)
\]

### Boundary morphism to $MFr$
  {#BoundaryMorphism}

The realization (eq:AsUnitCofiber) makes it manifest
(this is left implicit in [Conner-Floyd 66, p. 99](#ConnerFloyd66))
that there is a [[cohomology operation]] to [[MFr]] of the form

\[
  \label{BoundaryCohomologyOperation}
  \array{
    M(\mathrm{U},fr)
    \;=
    &
    M \mathrm{U}/\mathbb{S}
    &
    \overset{
      \;\;\;
      \partial
      \;\;\;
    }{\longrightarrow}
    &
    \Sigma
    \mathbb{S}
    & =\;
    \Sigma Mfr
    \\
    \pi_{2d+2}\big( M(\mathrm{U},fr) \big)
    &&
    \longrightarrow
    &&
    \pi_{2d+1}\big( Mfr \big)
  }
  \,.
\]

Namely, $\partial$ is the second next step in the long [[homotopy cofiber]]-sequence starting with $1^{M \mathrm{U}}$. In terms of the [[pasting law]]:

\[
  \label{BoundaryOperationViaPastingLaw}
  \array{
    \mathbb{S}
    &
      \overset{
        1^{M\mathrm{U}}
      }{
        \longrightarrow
      }
    &
    M \mathrm{U}
    &
    \longrightarrow
    &
    \ast
    \\
    \big\downarrow
    &
    {}^{{}_{(po)}}
    &
    \big\downarrow
    &
    {}^{{}_{(po)}}
    &
    \big\downarrow
    \\
    \ast
    &
    \longrightarrow
    &
    M \mathrm{U}/ \mathbb{S}
    &
      \underset{
        \partial
      }{
        \longrightarrow
      }
    &
    \Sigma \mathbb{S}
  }
\]

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

The implicit idea in [Conner-Floyd 66, p. 99](#ConnerFloyd66) must be to see $\partial$ (eq:BoundaryCohomologyOperation) in terms of forming actual [[boundaries]] of representative [[manifolds with boundaries]] under a version of the [[Pontrjagin-Thom construction]]. This is certainly plausible, but not proven either, as they "forego the tedious details" on [p. 97](#ConnerFloyd66). NB: for the purpose on p. 99 they might just _define_ $\partial$ in this geometric way, but then the claim wrapping around p. 100-101 needs proof.  This claim however is immediate from the abstract homotopy theory, namely it is just the continuation yet one step further along the long cofiber sequence -- this is Prop. \ref{AShortExactSequenceOfUFrBordismRings} below.

=--

\linebreak

### Relation to $MU$ and $MFr$
 {#RelationToMUAndFr}

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

In [[positive number|positive]] degree, the underlying [[abelian groups]] of the [[bordism rings]] for [[MU]], [[MFr]] and $MUFr$ (eq:UFrCobordismRing) sit in [[short exact sequences]] of this form:

\[
  \label{ShortExactSequenceOfUFrBordismRings}
  0
  \to
  \Omega^{\mathrm{U}}_{2n + 2}
  \overset{i}{\longrightarrow}
  \Omega^{\mathrm{U},fr}_{2n + 2}
  \overset{\partial}{
    \longrightarrow
  }
  \Omega^{fr}_{2n + 1}
  \to
  0
  \,,
  \phantom{AAAA}
  n \in \mathbb{N}
  \,,
\]

where $i$ is the evident inclusion, while $\partial$ is the [[boundary]] homomorphism from [above](#BoundaryMorphism).

=--

This is stated without comment in [Conner-Floyd 66, p. 99](#ConnerFloyd66).  The beginning of an argument appears inside the proof of [CF66, Thm. 16.2  (p. 100)](#ConnerFloyd66), attributed there to [[Peter Landweber]] (see Remark \ref{BoundaryOperationFromFrBordToUFrBordIsSurjective} below). The idea for how to complete the argument is a little more explicit in [Stong 68, p. 102](#Stong68).

The following is the complete and quick proof using the formulation (eq:BoundaryOperationViaPastingLaw) via abstract homotopy [above](#BoundaryMorphism):

+-- {: .proof}
###### Proof

We have the [[long exact sequence of homotopy groups]] ([[long exact sequence in generalized cohomology]] on [[spheres]])  obtained from the [[cofiber sequence]] $\mathbb{S} \overset{1^{M\mathrm{U}}}{\longrightarrow} M \mathrm{U} \to M \mathrm{U}/\mathbb{S} \overset{\partial}{\to} \Sigma \mathbb{S}$ (eq:BoundaryOperationViaPastingLaw), the relevant part of which looks as follows:

\[
  \label{SToMULongExactSequenceOfHomotopyGroups}
  \array{
    \overset{
      \mathclap{
        \color{darkblue}
        pure \; torsion
      }
    }{
      \overbrace{
        \pi_{2d+2}
        \big(
          \mathbb{S}
        \big)
      }
    }
    &
    \overset{
      1^{M\mathrm{U}}
    }{
      \longrightarrow
    }
    &
    \overset{
      \mathclap{
        \color{darkblue}
        free \; abelian
      }
    }{
      \overbrace{
      \pi_{2d+2}
      \big(
        M\mathrm{U}
      \big)
      }
    }
    &
    \overset{
    }{\longrightarrow}
    &
    \pi_{2d+2}
    \big(
      M\mathrm{U}/\mathbb{S}
    \big)
    &
    \overset{
      \partial
    }{\longrightarrow}
    &
    \pi_{2d+1}\big(\mathbb{S}\big)
    &\longrightarrow&
    \overset{
      \color{darkblue}
      trivial
    }{
      \overbrace{
        \pi_{2d+1}\big(M\mathrm{U}\big)
      }
    }
    \\
    \big\downarrow{}^{\mathrlap{=}}
    &&
    \big\downarrow{}^{\mathrlap{=}}
    &&
    \big\downarrow{}^{\mathrlap{=}}
    &&
    \big\downarrow{}^{\mathrlap{=}}
    &&
    \big\downarrow{}^{\mathrlap{=}}
    \\
    \Omega^{fr}_{2d+2}
    &
    \underset{
      \color{green}
      0
    }{
      \longrightarrow
    }
    &
    \Omega^{\mathrm{U}}_{2d+2}
    &
    \underset{
      i
    }{\longrightarrow}
    &
    \Omega^{(\mathrm{U},fr)}_{2d+2}
    &
    \underset{
      \partial
    }{\longrightarrow}
    &
    \Omega^{fr}_{2d + 1}
    &
    \underset{
      \color{green}
      0
    }{\longrightarrow}
    &
    0
  }
\]

Observing now that the [[stable homotopy groups]] of $M\mathrm{U}$ are [[free abelian groups]] concentrated in [[even number|even]] degrees (by [this theorem](MU#RelationToCobordismRing) at _[[MU]]_) it follows that:

1. the rightmost morphism shown in (eq:SToMULongExactSequenceOfHomotopyGroups) is the [[zero morphism]] since its [[codomain]] is [[zero object|zero]];

1. the leftmost morphism shown in (eq:SToMULongExactSequenceOfHomotopyGroups) is the [[zero morphism]], since the [[stable homotopy groups of spheres]] are all pure [[torsion groups]] in [[positive number|positive]] degrees (by the [[Serre finiteness theorem]]), and the only morphism from a torsion group to a [[free abelian group]] is the zero morphism.

=--

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

Here is an explicit construction of a [[lift]] through the boundary map in (eq:ShortExactSequenceOfUFrBordismRings), depending on a choice of trivialization in $\pi_{2d-1}\big( M\mathrm{U}\big)$:

(The following is essentially a streamlined account of the construction in the first half of the proof of [Conner-Floyd 66, Thm. 16.2](#ConnerFloyd66); for more and a more abstract perspective see at _[[d-invariant]]_ the section _[Trivializations of the d-invariant](d-invariant#TrivializationsOfThedInvariant)_):

Let

$$
  \big[
    \Sigma^\infty
    S^{2(n+d)-1}
    \overset{
       \Sigma^{\infty}
       {\color{green} c }
    }{\longrightarrow}
    \Sigma^\infty S^{2n}
  \big]
  \;\in\;
  \pi^s_{2d-1}
  \;\simeq\;
  \Omega^{fr}_{2d-1}
$$

be a given class in [[stable Cohomotopy]], hence in the [[MFr]]-[[cobordism ring]], under the [[Pontryagin-Thom isomorphism|PT isomorphism]]. (We write this as the [[stabilization]] of a class ${\color{green} c}$ in unstable [[Cohomotopy]] just for emphasis that we can.)

Consider then following [[homotopy coherent diagram|homotopy]] [[pasting diagram]]:

\begin{imagefromfile}
    "file_name": "LiftingFromUBordismToUFrBordism.jpg",
    "width": 600,
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 10
    },
    "caption": "From [SS21](https://ncatlab.org/schreiber/show/Equivariant+Cohomotopy+and+Oriented+Cohomology+Theory)"
\end{imagefromfile}

Here all squares are [[homotopy pushout]]-squares, they arise as follows (beware that we say "square" for any _single_ cell and "rectangle" for the pasting composite of any adjacent _pair_ of cells):

* The top square witnesses the [[homotopy cofiber]] $C_c$, defined thereby;

* the left rectangle is a homotopy pushout by definition of [[suspension]],

* hence the bottom left square is so by the [[pasting law]].

* Again via the first point, a dashed morphism exists as shown, witnessing the fact that the pullback of the class $\Sigma^{2n}(1^{M\mathrm{U}})$ to an odd-dimensional square vanishes (as does every $MU$-class);

* with this, the bottom left rectangle exists, and it is a homotopy pushout by definition of $M \mathrm{U}/\mathbb{S}$ (eq:BoundaryOperationViaPastingLaw);

* hence the morphism $\color{magenta} M^{2 d}$ exists, and the resulting bottom middle square is a homotopy pushout, again by the [[pasting law]].

* Now the bottom right rectangle is defined to be a homotopy pushout, and thus looks as shown by the [[pasting law]];

* therefore the bottom right square exists, and it is a homotopy pushout by the [[pasting law]].

But now the commuting three morphism in the very bottom and right part of the diagram shows that ${\color{magenta}M^{2d}} \in \pi_{2d}\big( M(U,fr) \big)$ is a lift of ${\color{green} c} \in \pi_{2d-1}(M Fr)$ through $\partial$.

=--

### Relation to Todd classes and the e-invariant

+-- {: .num_prop #EInvariantIsToddClassOnCoboundingUFrManifold}
###### Proposition
**([[e-invariant is Todd class of cobounding (U,fr)-manifold]])**

Evaluation of the [[Todd class]] on $(U,fr)$-manifolds yields [[rational numbers]] which are [[integers]] on actual $U$-manifolds. It follows with the [[short exact sequence]] (eq:ShortExactSequenceOfUFrBordismRings) that assigning to $Fr$-manifolds the Todd class of any of their cobounding $(U,fr)$-manifolds yields a well-defined element in [[Q/Z]].

Under the [[Pontrjagin-Thom isomorphism]] between the [[framed bordism ring]] and the [[stable homotopy group of spheres]] $\pi^s_\bullet$, this assignment coincides with the [[Adams e-invariant]] in [its Q/Z-incarnation](Adams+e-invariant#TheEInvariantAsAnElementOfQModZ):

\[
  \label{ToddClassesOnShortExactSequenceOfUFrBordismRings}
  \array{
  0
  \to
  &
  \Omega^{\mathrm{U}}_{\bullet+1}
  &
  \overset{i}{\longrightarrow}
  &
  \Omega^{\mathrm{U},fr}_{\bullet+1}
  &
  \overset{\partial}{
    \longrightarrow
  }
  &
  \Omega^{fr}_\bullet
  &
  \simeq
  &
  \pi^s_\bullet
  \\
  &
  \big\downarrow{}^{\mathrlap{Td}}
  &&
  \big\downarrow{}^{\mathrlap{Td}}
  &&
  \big\downarrow{}^{}
  &&
  \big\downarrow{}^{e}
  \\
  0
  \to
  &
  \mathbb{Z}
  &\overset{\;\;\;\;\;}{\hookrightarrow}&
  \mathbb{Q}
  &\overset{\;\;\;\;}{\longrightarrow}&
  \mathbb{Q}/\mathbb{Z}
  &=&
  \mathbb{Q}/\mathbb{Z}
  }
  \,,
\]

=--

([Conner-Floyd 66, Theorem 16.2](#ConnerFloyd66))

The first step in the proof of (eq:ToddClassesOnShortExactSequenceOfUFrBordismRings) is the observation ([Conner-Floyd 66, p. 100-101](#ConnerFloyd66)) that the representing map (eq:InTermsOfHomotopyGroupsOfQuotientedThomSpace) for a $(U,fr)$-manifold $M^{2k}$ cobounding a $Fr$-manifold represented by a map $f$ is given by the following [[homotopy coherent diagram|homotopy]] [[pasting diagram]] (see also at _[[Hopf invariant]] -- [In generalized cohomology](Hopf+invariant#InGeneralizedCohomology)_):

\begin{imagefromfile}
        "file_name": "CoboundingUFrManifoldDiagrammatically.jpg",
        "web": "nlab",
        "width": 470,
        "unit": "px",
        "margin": {
            "top": -20,
            "right": 0,
            "bottom": 20,
            "left": 20,
            "unit": "px"
        },
        "alt": "homotopy pasting diagram exhibiting cobounding UFr-manifolds",
        "caption": "from [SS21](https://ncatlab.org/schreiber/show/Equivariant+Cohomotopy+and+Oriented+Cohomology+Theory)"
\end{imagefromfile}

## Related concepts

[[!include flavours of cobordism cohomology theories -- table]]

## References

The concept of $(U,fr)$-bordism theory and its relation to the [[e-invariant]] originates with:

* {#ConnerFloyd66} [[Pierre Conner]], [[Edwin Floyd]], Section 16 of: _[[The Relation of Cobordism to K-Theories]]_, Lecture Notes in Mathematics __28__ Springer 1966 ([doi:10.1007/BFb0071091](https://link.springer.com/book/10.1007/BFb0071091), [MR216511](http://www.ams.org/mathscinet-getitem?mr=216511))

* {#ConnerSmith69} [[Pierre Conner]], [[Larry Smith]], Section 6 of: _On the complex bordism of finite complexes_, Publications Mathématiques de l'IHÉS, Tome 37 (1969) , pp. 117-221 ([numdam:PMIHES_1969__37__117_0](http://www.numdam.org/item/?id=PMIHES_1969__37__117_0))

Analogous discussion for [[MO]]-bordism with [[MSO]]-boundaries:

* G. E. Mitchell, _Bordism of Manifolds with Oriented Boundaries_, Proceedings of the American Mathematical Society Vol. 47, No. 1 (Jan., 1975), pp. 208-214 ([doi:10.2307/2040234](https://doi.org/10.2307/2040234))

Analogous discussion for [[MOFr]] is in

* {#Stong68} [[Robert Stong]], p. 102 of: _Notes on Cobordism theory_, Princeton University Press, 1968 ([toc pdf](http://pi.math.virginia.edu/StongConf/Stongbookcontents.pdf), [ISBN:9780691649016](http://press.princeton.edu/titles/6465.html), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/stongcob.pdf))

See also

* {#Smith71} [[Larry Smith]], _On characteristic numbers of almost complex manifolds with framed boundaries_, Topology Volume 10, Issue 3, August 1971, Pages 237-256 (doi:10.1016/0040-9383(71)90008-5)

Generalization to [[manifolds with corners]] and relation to the [[f-invariant]]:

* Gerd Laures, _On cobordism of manifolds with corners_, Trans. Amer. Math. Soc. 352 (2000) ([doi:10.1090/S0002-9947-00-02676-3](https://doi.org/10.1090/S0002-9947-00-02676-3))

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[[!redirects (U,fr)-manifolds]]

[[!redirects (U,fr)-bordism]]
[[!redirects (U,fr)-bordisms]]
[[!redirects (U,fr)-bordism ring]]
[[!redirects (U,fr)-bordism rings]]

[[!redirects (U,fr)-cobordism]]
[[!redirects (U,fr)-cobordisms]]
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[[!redirects (U,fr)-cobordism rings]]


Muhammad Ilyas

* [spire page](https://inspirehep.net/authors/2065897)

## Selected writings

On [[topological phases of matter]] and [[topological quantum computation]]:

* [[Muhammad Ilyas]], *Quantum Field Theories, Topological Materials, and Topological Quantum Computing* [[arXiv:2208.09707](https://arxiv.org/abs/2208.09707)]

category: people


Muhammad Shafique

* [institute page](https://nyuad.nyu.edu/en/academics/divisions/engineering/faculty/muhammad-shafique.html)

* [GoogleScholar page](https://scholar.google.com/citations?user=xSR276UAAAAJ&hl=en)

## Selected writings

On [[quantum computation|quantum]] [[quantum machine learning|machine learning]]:

* Kamila Zaman, [[Alberto Marchisio]], Muhammad Abdullah Hanif, [[Muhammad Shafique]], *A Survey on Quantum Machine Learning: Current Trends, Challenges, Opportunities, and the Road Ahead* [[arXiv:2310.10315](https://arxiv.org/abs/2310.10315)]

category: people


Mukund Rangamani

* [webpage](https://mukund.physics.ucdavis.edu/)

## Selected writings

Introducing the [[fluid/gravity correspondence]]:

* [[Sayantani Bhattacharyya]], [[Veronika Hubeny]], [[Shiraz Minwalla]], [[Mukund Rangamani]], _Nonlinear Fluid Dynamics from Gravity_, JHEP 0802:045, 2008 ([arXiv:0712.2456](https://arxiv.org/abs/0712.2456))

More on the [[fluid/gravity correspondence]]:

* [[Veronika Hubeny]], [[Shiraz Minwalla]], [[Mukund Rangamani]], _The fluid/gravity correspondence_ ([arXiv:1107.5780](https://arxiv.org/abs/1107.5780))

* [[Mukund Rangamani]], _Gravity & Hydrodynamics: Lectures on the fluid-gravity correspondence_, Class. Quant. Grav. 26 : 224003, 2009 ([arXiv:0905.4352](https://arxiv.org/abs/0905.4352))

Textbook on [[holographic entanglement entropy]]:

* [[Mukund Rangamani]], [[Tadashi Takayanagi]], *Holographic Entanglement Entropy*, Lecture Notes in Physics **931**, Springer (2017) [[doi:10.1007/978-3-319-52573-0](https://doi.org/10.1007/978-3-319-52573-0), [arXiv:1609.01287](https://arxiv.org/abs/1609.01287)]

category: people