Blakers-Massey theorem

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###Context###
#### Homotopy theory
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[[!include homotopy - contents]]
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#### $(\infty,1)$-Topos Theory
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[[!include (infinity,1)-topos - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

The _Blakers-Massey theorem_ in the [[homotopy theory]] of [[pointed topological spaces]] is concerned with algebraically describing the first obstruction to [[excision]] for relative [[homotopy groups]]. There is also a weaker version just describing vanishing conditions which should be called the _Blakers-Massey connectivity theorem_.

## Statement

### Traditional

This [[obstruction]] is measured by triad homotopy groups $\pi_m(X;A,B)$ for a pointed space $X$ with two subspaces $A,B$ each containing the base point. Here the group structure is defined for $m \geq 3$ and is abelian for $m \geq 4$. There is an exact sequence

$$\cdots \to \pi_{n+1}(X;A,B) \to \pi_n (A, A \cap B) \to^{\epsilon} \pi_n( X,B) \to \pi_n(X;A,B) \to \cdots  $$
where $\epsilon$ is the excision map. The main result of Blakers and Massey is as follows:

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

Suppose the triad **X** $ =(X;A,B)$ is such that: (i) the interiors of $A,B$ cover $X$; (ii) that $A,B$ and $C=A \cap B$ are connected; (iii) that $C$ is simply connected; (iv) and that $(A,C)$ is $(m-1)$-connected and $(B,C)$ is $(n-1)$-connected, $m,n \geq 3$. Then **X**$=(X;A,B)$ is $(m+n-2)$-connected and if $C$ is simply connected then the morphism given by the generalised Whitehead product
$$ \pi_{m}(A,C) \otimes \pi_{n}(B,C) \to \pi_{m+n-1}(X;A,B)$$
_is an isomorphism_.

=--

([Blakers-Massey 51](#BlakersMassey51), [tomDieck 08, theorem 6.4.1](#tomDieck08)).

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

A more intrinsic statement in the language of [[homotopy theory]] of the connectivity part of the theorem is that for $f_1$ and $f_2$ two [[maps]] out of the same [[domain]] which are $n_1$-[[n-connected morphism|connective]] and $n_2$-connective, respectively, then the canonical map from that domain into the [[homotopy pullback]] of their [[homotopy pushout]]

$$
  \array{
    X &\stackrel{f_1}{\longrightarrow}& Y_1
    \\
    \downarrow^{\mathrlap{f_2}} & \searrow
    \\
    Y_2 && Y_2 \underset{Y_2 \underset{X}{\coprod} Y_1}{\times} Y_1
  }
$$

is $(n_1 + n_2 - 1)$-[[n-connected morphism|connective]].

=--

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

For the special case that $Y_1 \simeq Y_2 \simeq \ast$ are point contractible, the Blakers-Massey theorem reduces to the [[Freudenthal suspension theorem]].

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

Since the tensor product is zero if one of its factors is zero, this result also gives criteria for the excision morphism $\epsilon$ to be an isomorphism in a certain range of dimensions. For this reason the excision consequences of that sequence are  also  called the _excision theorem of Blakers and Massey_ and have been given quite separate proofs for example in ([Hatcher](#Hatcher)), and in ([tom Dieck](#tomDieck08)). The  first non zero triad homotopy group is also called the _critical group_. Note that in _algebraic topology_ one wants _algebraic_ results, not just connectivity results.

=--

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

A natural question is what happens if the conditions  that $m,n \geq 3$ and $C$ simply connected are weakened.   For example in the case $m=n=2$  we have the additional structure that the morphisms $\pi_2(A,C) \to \pi_1(C), \pi_2(B,C) \to \pi_1(C)$ are crossed modules, and so the required relative homotopy groups are in general nonabelian. If $m \geq 3 ,n \geq 3$ then $\pi_m(A,C), \pi_n(B,C)$ are still $\pi_1(C)$-modules.

The extension to the non simply connected case was given by Brown and Loday; one simply replaces the usual tensor product by the nonabelian tensor product of groups which act on each other and on themselves by conjugation. This result is a special case of a Seifert-van Kampen Theorem for $n$-cubes of spaces. Notice that the assumption (i) of the theorem is reminiscent of such a type of theorem. The useful fact is that one gets such a theorem for a certain kind of _structured space_ which allows for the development of algebraic structures which have structures in a range of dimensions.

Thus one of the intuitions is that the Blakers-Massey Theorem, and hence also the FST,  is of the Seifert-van Kampen type, since we are assuming that $X$ is the union of the interiors of $A,B$.

=--

### In higher topos theory

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

The Blakers-Massey connectivity theorem in the form of remark \ref{InTermsOfPushouts} holds in every [[(∞,1)-topos]] of [[(∞,1)-sheaves]].

=--

This is shown in ([Rezk 10, prop. 8.16](#Rezk10)) with reference to [[(∞,1)-sites]].
An intrinsic proof in [[homotopy type theory]] is announced in ([HoTTBook, theorem 8.10.2](#HoTTBook), [Lumsdaine-Finster-Licata 13](#LumsdaineFinsterLicata13)). The fully formal computer-checked version of this proof appears as HoTT-[[Agda]] code in ([Favonia 14](#Favonia)).

This translates to an [[internal language]] proof of Blakers-Massey valid in all [[(∞,1)-toposes]] (including [[elementary (∞,1)-toposes]]). Unwinding of the fully formal HoTT proof to ordinary mathematical language is, for the special case of the [[Freudenthal suspension theorem]], in ([Rezk 14](#Rezk14)).

### Higher cubical BM-theorems and analytic $\infty$-functors
 {#HigherCubical}

There are higher analogs of the BM-theorem with (pushout) squares replaced by higher dimensional cubes. The higher BM-theorem ([Goodwillie 91](#Goodwillie91)) says equivalently that the identity [[(∞,1)-functor]] on [[∞Grpd]] is a 1-[[analytic (∞,1)-functor]]. See ([Munson-Volic 15, section 6](#MunsonVolic15)).

## References

### Classical

The original connectivity  statement:

* {#BlakersMassey51} [[Albert Blakers]], [[William  Massey]], _The homotopy groups of a triad I_ , Annals of Mathematics 53: 161–204, (1951) ([jstor:1969346](https://www.jstor.org/stable/1969346))

The algebraic statement and proof:

* [[Albert Blakers]], [[William Massey]], _The homotopy groups of a triad. {III}_,  Ann. of Math. (2), 58: (1953) 409--417 ([jstor:1969744](https://www.jstor.org/stable/1969744))

Textbook accounts:

* {#tDKP70} [[Tammo tom Dieck]], [[Klaus Heiner Kamps]], [[Dieter Puppe]], Section 15 of: *Homotopietheorie*, Lecture Notes in Mathematics **157** Springer 1970 ([doi:10.1007/BFb0059721](https://link.springer.com/book/10.1007/BFb0059721))

* {#Kochmann96} [[Stanley Kochmann]], theorem 3.2.4 of _[[Bordism, Stable Homotopy and Adams Spectral Sequences]]_, AMS 1996

* {#Hatcher} [[Alan Hatcher]], theorem 4.23 _[Algebraic Topology](http://www.math.cornell.edu/~hatcher/AT/ATpage.html)_

* {#tomDieck08} [[Tammo tom Dieck]],  theorem 6.4.1 _Algebraic Topology_, EMS Textbooks in Mathematics, (2008) ([pdf](http://www.maths.ed.ac.uk/~aar/papers/diecktop.pdf))

The Blakers-Massey's Connectivity Theorem was shown to be a consequence of Farjoun's "cellular inequalities"

* [[Emmanuel Dror Farjoun]], _Cellular spaces, null spaces and homotopy localization_m No. 1621-1622. Springer, 1996]

is Theorem 1.B of

* [[Wojciech Chachólski]], _A generalization of the triad theorem of Blakers-Massey_ Topology 36.6 (1997): 1381-1400

This would constitute a purely [[homotopy theory|homotopy-theoretic]] proof.

The generalisation of the algebraic statement is Theorem 4.3 in:

* [[Ronnie Brown|R. Brown]] and [[Jean-Louis Loday]], Homotopical excision, and Hurewicz theorems, for $n$-cubes of spaces, _Proc. London Math. Soc._
(3) 54 (1987) 176-192.  [pdf](https://groupoids.org.uk/pdffiles/VKTEVANS2.pdf)

which relies essentially on the paper

* [[Ronnie Brown| R. Brown]] and J.-L. Loday, Van Kampen theorems for diagrams of
spaces,  _Topology_ 26 (1987) 311-334,

for the van Kampen Theorem and for the nonabelian tensor product of groups. Here is a link to a bibliography of 170 items on the [nonabelian tensor product](http://groupoids.org.uk/nonabtens.html ).

Further applications are explained in

[[Ronnie Brown|R. Brown]],  Triadic Van Kampen theorems and Hurewicz theorems, _
Algebraic  Topology, Proc. Int. Conf. March 1988_, Edited
M.Mahowald and S.Priddy,  Cont. Math. 96 (1989) 39-57.  [pdf](http://groupoids.org.uk//pdffiles/VKTEVANS2.pdf)

The following paper  applies the methods of the above two Brown-Loday papers to the well known problem of $n$-ad connectivity and to determination of the critical group, see Theorem 3.8 of:

* Ellis, G.J. and Steiner, R. Higher-dimensional crossed modules and the homotopy groups  of $(n+1)$-ads. _J. Pure Appl. Algebra_ 46 (1987) 117--136.

The methods work because of their equivalence between cat$^n$-groups and crossed $n$-cubes of groups. This can be explained by saying that we need two kinds of algebraic categories for calculations with $(n+1)$-types: _broad_ categories for conjecturing and proving theorems, and _narrow_ algebraic categories for calculations and relations with classical ideas. In this case the broad category is that of cat$^n$-groups, and the narrow category is that of crossed $n$-cubes of groups, which are related geometrically to the homotopy groups of $r$-ads and to generalised Whitehead products. The tricky equivalence between the two kinds of categories is one of the engines behind the results, since it enables the use of whichever category is most convenient at any given time. Note also these two categories model weak, pointed, homotopy $(n+1)$-types, as shown by Loday in his paper

* [[Jean-Louis Loday]],  Spaces with finitely many non-trivial homotopy groups, _J. Pure Appl. Algebra_ 24 (1982)  179-202.

Further background to these ideas is in

* [[Ronnie Brown| R. Brown]] "A philosophy of modelling and computing homotopy types" Presentation to CT2015, Aveiro, Portugal, June 14-19. [pdf](https://groupoids.org.uk//pdffiles/aveiro-beamer-handout.pdf) and in "Modelling and computing homotopy types: I" Indag.Math.  29  (2018) 459-482 [pdf](https://www.sciencedirect.com/science/article/pii/S0019357717300460)

Discussion of Blakers-Massey connectivity for [[ring spectra]]/[[E-∞ rings]] and other [[algebras over operads]] is in

* Michael Ching, [[John Harper]], _Higher homotopy excision and Blakers-Massey theorems for structured ring spectra_ ([arXiv:1402.4775](http://arxiv.org/abs/1402.4775))

The higher cubical version of Blakers-Massey connectivity is due to

* {#Goodwillie91} [[Tom Goodwillie]], _Calculus II: Analytic functors_, K-Theory  01/1991; 5(4):295-332. DOI: 10.1007/BF00535644

a textbook account is in

* {#MunsonVolic15} [[Brian Munson]], [[Ismar Volic]], _Cubical homotopy theory_, Cambridge University Press, 2015 [pdf](http://palmer.wellesley.edu/~ivolic/pdf/Papers/CubicalHomotopyTheory.pdf)

### In $\infty$-topos theory and homotopy type theory
 {#ReferencesInHoTT}

A proof of Blakers-Massey connectivity in general [[∞-stack]] [[(∞,1)-toposes]] is in prop. 8.16 of

* {#Rezk10} [[Charles Rezk]], _Toposes and homotopy toposes_ (2010) ([pdf](http://www.math.uiuc.edu/~rezk/homotopy-topos-sketch.pdf))

A general version of the connectivity theorem in [[homotopy type theory]]
(and thus in [[(infinity,1)-topos theory]]) was found by

* {#LumsdaineFinsterLicata13} [[Peter LeFanu Lumsdaine]], [[Eric Finster]], [[Dan Licata]] (to appear)

A fully computer-checked version of this proof in HoTT-[[Agda]] was produced in

* {#Favonia} [[Favonia]], _[BlakersMassey.agda](https://github.com/HoTT/HoTT-Agda/blob/1.0/Homotopy/BlakersMassey.agda)_

the statement appeared also as

* {#HoTTBook} [[Univalent Foundations Project]], theorem 8.10.2 of _[[Homotopy Type Theory -- Univalent Foundations of Mathematics]]_

and an announcement was given in

* {#Lumsdaine13} [[Peter LeFanu Lumsdaine]], _The Blakers-Massey theorem in homotopy type theory_ talk at [Conference on Type Theory, Homotopy Theory and Univalent Foundations](http://www.crm.cat/en/Activities/Pages/ActivityFoldersAndPages/Curs%202013-2014/CHomotopy/chomotopy.aspx) (2013) ([talk abstracts pdf](http://www.crm.cat/en/Activities/Documents/AbstractsTypeTheory.pdf)).

A writeup finally appeared as:

* {#FFLL16} [[Kuen-Bang Hou]] ([[Favonia]]), [[Eric Finster]], [[Dan Licata]], [[Peter LeFanu Lumsdaine]], *A mechanization of the Blakers-Massey connectivity theorem in Homotopy Type Theory*, LICS '16 (2016) 565–574  [[arXiv.1605.03227](https://arxiv.org/abs/1605.03227), [doi:10.1145/2933575.2934545](https://doi.org/10.1145/2933575.2934545)]

Another unwinding to ordinary mathematical language of the above [code](#Favonia) was meanwhile given in

* {#Rezk14} [[Charles Rezk]], _Proof of the Blakers-Massey theorem_, 2014 [pdf](http://www.math.uiuc.edu/~rezk/freudenthal-and-blakers-massey.pdf).

prompted by online discussion at

* {#Schreiber14} [[Urs Schreiber]], _Explaining the point of HoTT on FOM_, Google+ thread 2014-10-22 ([archived version](https://github.com/DavidMichaelRoberts/Sandbox/blob/master/Schreiber_Gplus_post.md))

  (scroll down a fair bit through the list of replies there to see the exchange between [[Charles Rezk]] and [[Favonia]]).

Further developments along these lines are in

* {#AnelBiedermanFinsterJoyal17a} [[Mathieu Anel]], [[Georg Biedermann]], [[Eric Finster]], [[André Joyal]], _A Generalized Blakers-Massey Theorem_, Journal of Topology **13** 4 (2020) 1521-1553 $[$[arXiv:1703.09050](https://arxiv.org/abs/1703.09050), [doi:10.1112/topo.12163](https://doi.org/10.1112/topo.12163)$]$

  > (generalizing to the case where the [[(n-connected, n-truncated) factorization system]] may be replaced by more general [[modal homotopy type theory|modalities]])

* {#AnelBiedermanFinsterJoyal17b} [[Mathieu Anel]], [[Georg Biedermann]], [[Eric Finster]], [[André Joyal]], _Goodwillie's Calculus of Functors and Higher Topos Theory_ ([arXiv:1703.09632](https://arxiv.org/abs/1703.09632))

### In shape theory

* Šime Ungar, $n$-Connectedness of inverse systems and applications to shape theory, Glasnik Matematički 13 (1978), 371-396 [pdf](http://www.irb.hr/korisnici/zskoda/ungarConnwoabsh.pdf)

> Let (X, A, x) be an n-connected inverse system of CW-pairs such that the restriction (A, x) is m-connected. We prove that there exists an isomorphic inverse system (Y, B, y) having n-connected terms such that the terms of the restriction (B, y) are m-connected. This result is then applied in proving analogues of Hurewicz and Blakers-Massey theorems for homotopy pro-groups and shape groups.


BLG model

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### Context
#### String theory
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[[!include string theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The BLG model is a 3-dimensional [[SCFT]] involving a [[Chern-Simons theory]] coupled to matter. It is argued to be the [[worldvolume]] theory of 2 coincident [[black brane|black]] [[M2-branes]] with 16 manifest [[supersymmetries]]. The generalization to an arbitrary number of M2-branes is supposed to be given by the [[ABJM model]].

[[!include 7d spherical space forms -- table]]

## Properties

### The Lagrangian

(...)

### Boundary conditions

Discussion of [[boundary conditions]] of the BLG model, leading to [[brane intersection]] with [[M-wave]], [[M5-brane]] and [[MO9-brane]] is in ([Chu-Smith 09](#ChuSmith09), [BPST 09](#BPST09)).

### The "$3$-algebra" structure
 {#3AlgebraStructure}

The BLG model Lagrangian involves a trilinear operation on the scalar fields $\phi \in V$

$$
 [-,-,-] : V^{\otimes 3} \to V
  \,,
$$

the _[[M2-brane 3-algebra]]_.

Moreover, the [[supersymmetry]] of the Lagrangian hinges on the fact that this map satisfies a condition that has some similarity to a [[Jacobi identity]] for the binary operation on a [[Lie algebra]].

Therefore, superficially, it looks like this might be the trinary bracket on an _[[L-∞ algebra]]_ structure on the space $V$.

On the one hand, indeed, by the discussion at _[[supergravity C-field]]_ , the M2-brane is charged under a [[circle n-bundle with connection|circle 3-bundle with connection]] whose [[higher gauge theory]] is controlled by [[Lie n-algebra|Lie 3-algebras]] in direct analogy to how the [[higher gauge theory]] of the [[string]] is controled by [[gerbes]]/[[principal 2-bundles]] and their [[Lie 2-algebra]]s and that of charged [[particles]] by ordinary [[Lie algebras]].

Motivated by attempts in [Basu-Harvey 04](#BasuHarvey04) to generalize [[Nahm's equations]] for [[fuzzy funnels]] of [[D2-D4-brane intersections]] to [[M2-M5-brane intersections]], [Bagger-Lambert 06](#BaggerLambert06) introduced the algebraic structure that came to be known the _[[M2-brane 3-algebra]]_, further highlighted by [Gustavsson 07](#Gustavsson07).
This terminology was picked up by many authors In the process, it transmuted sometimes to "[[3-Lie algebra]]" and sometimes even to "Lie 3-algebra".

Beware that the Bagger-Lambert "[[M2-brane 3-algebra]]" is _not- a _[[Lie n-algebra|Lie 3-algebra]]_ in the established sense of an [[L-∞ algebra]] structure on a [[graded vector space]] $V$ concentrated in the lowest three degrees.

The reason is that for the notion of an [[L-∞ algebra]] (as discussed there) it is crucial that $V$ is a $\mathbb{N}$-graded (or $\mathbb{Z}$-graded) vector space and that the $n$-ary brackets respect the degree in a certain way.  But in the [BaggerLambert](#BaggerLamber)-proposal, $V$ is all concentrated in a single degree (is regarded as ungraded). One immediately finds that in this case the $L_\infty$-respect of $[-,-,-]$ for the grading would imply that $V$ is taken to be in degree $1/2$. Since this is not in $\mathbb{N}$, it does not yield an $L_\infty$-algebra.

Notice that the $\mathbb{N}$-grading (or $\mathbb{Z}$-grading) of $L_\infty$-algebras is crucial for the [[homotopy theory|homotopy theoretic]] interpretation of [[L-∞ algebra]]s as higher Lie algebras. None of the good theory of $L_\infty$-algebras survives when this grading is dropped. This grading has its origin in the [[Dold-Kan correspondence]], which establishes integral graded homological structures as models for structures in [[higher category theory]]. Notably, a higher Lie algebra is supposed to have a [[Lie integration]] to a [[smooth ∞-groupoid|smooth $n$-groupoid]]. Under this process, the elements in degree $k$ of the higher Lie algebra become tangents to the space of [[k-morphisms]] of this smooth $n$-groupoid. Clearly, here only integer $k$ do make sense.

On the other hand, it is of course possible to consider the structure on "$L_\infty$-algebras without grading", even if these will not have a good theory. This notion has once been introduced by Filippov (Sib. Math. Zh. No 6 126–140 (195)) under the term _[[n-Lie algebra]]_ .

Beware, therefore, that the innocent-looking difference between the terms

* [[Lie n-algebra]]

* [[n-Lie algebra]]

corresponds, unfortunately, to a major difference in the behaviour of the concepts behind these terms.

In conclusion, it is clear that 2-brane physics is governed by Lie 3-algebraic structures, but it is not yet clear how the trinary operation highlighted by [BaggerLambert](#BaggerLambert) would be an example.

In view of this it might be noteworthy that the equivalent reformulation and generalization of the BLG model by the [[ABJM model]] does not involve any "3-algebras" at all. In fact at least most of the "3-algebras" appearing in the membrane literature may be understood as being data of a plain [[Lie algebra]] with an invariant product and a representation ([MFMR 08](#MFMR)). These authors summarize the state of affiars on p. 3 as

> All this prompts one to question whether the 3-algebras appearing in the constructions [1–3,10,11] play a fundamental role in M-theory or, at least insofar as the effective field theory is concerned, are largely superfluous. The equivalence of [10] and [6] and the abundance of new theories (dual to known M-theory backgrounds) which seem not to involve a 3-algebra might suggest the latter. Nonetheless, given our lack of understanding of how to incorporate in Lie-algebraic terms the expected properties of M-theoretic degrees of freedom, like the entropy scaling laws for M2- and M5-brane condensates, it may be useful to understand the precise relation between the 3-algebras appearing in the recent literature on superconformal Chern–Simons theory and Lie algebras.

The article ([MFMR 08](#MFMR)) provides this relation and under this relation the "3-algebraic" BLG model has then been understood as a special case of the ordinary Lie algebraic [[ABJM theory]]. A review is in ([Bagger-Lambert 12](#BaggerLambert12)).

It has also been suggested that "3-algebras" are to be interpreted in **[[n-plectic geometry|2-plectic structure]]** ([Saemann-Szabo](#SaemannSzabo)).

## Related concepts

* [[M2-brane 3-algebra]]

* [[ABJM theory]]

* [[membrane matrix model]]

[[!include superconformal symmetry -- table]]

## References

### Precursors

* [[John Schwarz]], _Superconformal Chern-Simons Theories_ ([arXiv:arXiv:hep-th/0411077](https://arxiv.org/abs/hep-th/0411077)

The lift of [[Dp-D(p+2)-brane bound states]] in [[string theory]] to [[M2-M5-brane bound states]]/[[E-strings]] in  [[M-theory]], under [[duality between M-theory and type IIA string theory]]+[[T-duality]], via generalization of [[Nahm's equation]] (this eventually motivated the [[BLG-model]]/[[ABJM model]]):

* {#BasuHarvey04} [[Anirban Basu]], [[Jeffrey Harvey]], _The M2-M5 Brane System and a Generalized Nahm's Equation_, Nucl.Phys. B713 (2005) 136-150 ([arXiv:hep-th/0412310](https://arxiv.org/abs/hep-th/0412310))

* {#BaggerLambertMukhiPapageorgakis13} [[Jonathan Bagger]], [[Neil Lambert]], [[Sunil Mukhi]], [[Constantinos Papageorgakis]], Section 2.2.1 of _Multiple Membranes in M-theory_, Physics Reports, Volume 527, Issue 1, 1 June 2013, Pages 1-100 ([arXiv:1203.3546](http://arxiv.org/abs/1203.3546), [doi:10.1016/j.physrep.2013.01.006](https://doi.org/10.1016/j.physrep.2013.01.006))

### General

The original articles are

* {#BaggerLambert06} [[Jonathan Bagger]], [[Neil Lambert]], _Modeling Multiple M2's_, Phys. Rev. D75, 045020 (2007). ([hep-th/0611108](http://arxiv.org/abs/hep-th/0611108)).

* {#BaggerLambert07} [[Jonathan Bagger]], [[Neil Lambert]], _Gauge Symmetry and Supersymmetry of Multiple M2-Branes_, Phys. Rev. D77, 065008 (2008). ([arXiv:0711.0955](http://arXiv.org/abs/0711.0955)).

and with special emphasis on the "[[M2-brane 3-algebra]]"-structure also:

* {#Gustavsson07} [[Andreas Gustavsson]], _Algebraic structures on parallel M2-branes_ Nucl. Phys. B811, 66-76 (2009) ([arXiv:0709.1260](https://arxiv.org/abs/0709.1260))
* [[Andreas Gustavsson]], _One-loop corrections to Bagger-Lambert theory_, Nuclear Physics B 807, n. 1-2 (2009) 315--333 [doi](https://doi.org/10.1016/j.nuclphysb.2008.09.003)

The interpretation in terms of branes at a $\mathbb{Z}/2$-[[ADE-singularities]] with [[discrete torsion]] in the [[supergravity C-field]] is due to

* [[Neil Lambert]], [[David Tong]], _Membranes on an Orbifold_, Phys.Rev.Lett.101:041602, 2008 ([arXiv:0804.1114](https://arxiv.org/abs/0804.1114))

* [[Jacques Distler]], [[Sunil Mukhi]], [[Constantinos Papageorgakis]], [[Mark Van Raamsdonk]], _M2-branes on M-folds_, JHEP 0805:038,2008 ([arXiv:0804.1256](https://arxiv.org/abs/0804.1256))

Review:

* {#BaggerLambert12} [[Jonathan Bagger]], [[Neil Lambert]], [[Sunil Mukhi]], [[Constantinos Papageorgakis]], _Multiple Membranes in M-theory_, Physics Reports, Volume 527, Issue 1, 1 June 2013, Pages 1-100 ([arXiv:1203.3546](http://arxiv.org/abs/1203.3546), [doi:10.1016/j.physrep.2013.01.006](https://doi.org/10.1016/j.physrep.2013.01.006))

* {#Lambert12} [[Neil Lambert]], _M-Theory and Maximally Supersymmetric Gauge Theories_, Annual Review of Nuclear and Particle Science, Vol. 62:285-313 ([arXiv:1203.4244](https://arxiv.org/abs/1203.4244), [doi:10.1146/annurev-nucl-102010-130248](https://doi.org/10.1146/annurev-nucl-102010-130248))

* [[Neil Lambert]], _Lessons from M2's and Hopes for M5's_, _Proceedings of the [LMS-EPSRC Durham Symposium](http://www.maths.dur.ac.uk/lms/):_ _[[Higher Structures in M-Theory 2018]]_ Fortschritte der Physik, 2019 ([arXiv:1903.02825](https://arxiv.org/abs/1903.02825), [slides pdf](http://www.maths.dur.ac.uk/lms/109/talks/1877lambert.pdf), [video recording](http://www.maths.dur.ac.uk/lms/109/movies/1877lamb.mp4))

Discussion of [[boundary conditions]] leading to [[brane intersection]] with [[M-wave]], [[M5-brane]] and [[MO9-brane]] is in

* {#ChuSmith09} [[Chong-Sun Chu]], Douglas J. Smith, _Multiple Self-Dual Strings on M5-Branes_, JHEP 1001:001, 2010 ([arXiv:0909.2333](https://arxiv.org/abs/0909.2333))

* {#BPST09} [[David Berman]], Malcolm J. Perry, [[Ergin Sezgin]], Daniel C. Thompson, _Boundary Conditions for Interacting Membranes_, JHEP 1004:025, 2010 ([arXiv:0912.3504](https://arxiv.org/abs/0912.3504))

Discussion in [[Horava-Witten theory]] reducing M2-branes to [[heterotic strings]] is in

* {#Lambert15} [[Neil Lambert]], _Heterotic M2-branes_, Physics Letters B
Volume 749, 7 October 2015, Pages 363-367 ([arXiv:1507.07931](http://arxiv.org/abs/1507.07931))

The interpretation of at least most of the "[[M2-brane 3-algebra]]" appearing in the membrane literature in terms of plain [[metric Lie algebras]] is due to

* {#MFMR} [[Paul de Medeiros]], [[José Figueroa-O'Farrill]], [[Elena Méndez-Escobar]], [[Patricia Ritter]], _On the Lie-algebraic origin of metric 3-algebras_, Commun. Math. Phys. 290: 871-902, 2009 ([arXiv:0809.1086](http://arxiv.org/abs/0809.1086))

See also

* [[José Figueroa-O'Farrill]], _Triple systems and Lie superalgebras_ in _M2-branes, ADE and Lie superalgebras_, talk at IPMU 2009 ([pdf](http://www.maths.ed.ac.uk/~jmf/CV/Seminars/Hongo.pdf), [[FigueroaOFarrillM2Branes02.pdf:file]])

The suggestion that BGL "3-algebras" are to be interpreted in [[2-plectic geometry]] appears in

* {#SaemannSzabo} [[Christian Saemann]], [[Richard Szabo]], _Quantization of 2-Plectic Manifolds_ ([arXiv:1106.1890](http://arxiv.org/abs/1106.1890))

[[!redirects BLG-model]]


blob homology

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## Idea

The _blob complex_ may be thought of as a generalization of the [[Hochschild complex]] to [[higher category theory|higher categories]] and higher dimensional [[manifold]]s.

One thinks of the Hochschild complex as associated to a 1-[[category]] and a 1-[[manifold]] (the circle).  It's a fairly small complex, analogous to cellular homology.  The blob complex for the same input data (1-category and circle) yields a [[quasi-isomorphism|quasi-isomorphic]] but much much larger [[chain complex]], analogous to [[singular homology]].

Its advantage over the Hochschild complex is that it is "local".  In higher dimensions this locality means that it is easy to (well-) define the blob complex of an [[n-category]] + n-manifold without choosing any sort of decomposition of the n-manifold.

## Definition

There are two definitions.  The second is more general, and is homotopy equivalent to the first when both are available.  See references below for more details.

### First definition

* Start with a linear [[n-category]] $C$ with [[strong duality]] -- a [[blob n-category]] -- (e.g. [[pivotal 2-category]]) and an $n$-[[manifold]] $M$.

* Define $B_0(M)$ to be finite linear combinations of
  "$C$-[[string diagram]]s" drawn on $M$.

* Define $B_1(M)$ to be finite linear combinations of triples $(B, u, r)$, where $B \subset M$ is a ball, $r$ is a [[string diagram]] on $M\setminus B$, and $u$ is a linear combination of  string diagrams on which evaluates to zero.

* Define $B_2(M)$ to be ...

### Second definition

...

## Examples

...

## Related concepts

* [[blob n-category]]

Blob homology has some similarities with

* [[factorization algebra]]

* [[local net]]s in [[AQFT]].

It should be closely related to

* [[topological chiral homology]]/[[factorization homology]]

## References

A preprint is here:

* [[Scott Morrison]], [[Kevin Walker]], _The blob complex_, ([arxiv/1009.5025](http://arxiv.org/abs/1009.5025)) ([website](http://tqft.net/blobs))
{#MorrisonWalker}

Notes from talks can be found [here](http://canyon23.net/math/talks) and [here](http://tqft.net/UCR-blobs1).

Some notes appeared in the context of the

* [[Oberwolfach Workshop, June 2009 -- Strings, Fields, Topology]]

See the

* Oberwolfach report No 28, 2009, [pdf](http://www.mfo.de/document/0924/OWR_2009_28.pdf)

[[!redirects blob homology]]
[[!redirects Blob homology]]
[[!redirects blob complex]]
[[!redirects blob complexes]]
[[!redirects blob chain complex]]
[[!redirects blob chain complexes]]


blob n-category

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## Idea

The notion of _blob $n$-category_ captures the notion of an [[(infinity,n)-category with duals|n-category with all duals]]. It is formulated in the style of [[hyperstructure]]: without any distinction between source and targets.

The definition is well-adapted to describing the [[(∞,n)-category of cobordisms]] in the spirit of [[blob homology]].

## Definition

Let $n \in \mathbb{N}$ be a [[natural number]].

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

A **blob $n$-graph** $C$ is given by

* for every $k \in \mathbb{N}$ a [[functor]]

  $$
    C_k : core(Ball_k) \to Set
  $$

  from the [[groupoid]] of [[topological space]]s [[homeomorphism|homeomorphic]] to a $k$-[[ball]] and [[homeomorphism]]s between them to [[Set]].

=--

We think of $C(B^k)$ as the set of [[k-morphism]]s in the $n$-graph $C$. This means that the [[geometric shape for higher structures]] used here is the [[globe]]. Therefore the term _blob_ .

We define now a notion of composition on $k$-cells of a blob $n$-graph by induction over $k$. Given a blob $n$-graph with composition for $k$-cells, it can be extended from balls to arbitrary manifolds by the definition [extension to general shapes](#ExtensionToGeneralShapes) below.

+-- {: .num_defn}
###### Definition (roughly)

Say that a blob $n$-graph is a **blob $n$-graph with composition for 0-cells**.

Assume we have a blob $n$-graph $C$ with composition for $(k-1)$-cells for $k \geq 1$. Then **composition of $k$-cells** on $C$ is a choice of the following structure

* a [[natural transformation]] -- **boundary restriction**
  (source/target)

  $$
    \partial : C_k(X) \to \underset{\to}{C}_{k-1}(\partial X)
    \,,
  $$

  where on the right we have the extension to $(k-1)$ spheres of $C_{k-1}$
  described [below](#ExtensionToGeneralShapes);

* for all balls $B = B_1 \cup_{B_1 \cap B_2} B_2$ and $E := \partial (B_1 \cap B_2)$ a natural transformation -- **composition**

  $$
    \circ : C(B_1) \times_{C(B_1 \cap B_2)} C(B_2) \to C(B)
  $$

  satisfying some compatibility conditions

* for all balls $X$, $D$ a natural map -- **identity** --

  $$
    C(X) \to C(X \times D)
  $$

  satisfying some compatibility conditions.

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+-- {: .num_defn #ExtensionToGeneralShapes}
###### Definition (roughly)
**(extension to general shapes)**

For $C$ a blob $n$-graph with composition for $(k-1)$-cells and $X$ any $(k-1)$-[[dimension]]al [[manifold]] with $k \lt n$, define $\underset{\to}{C}_{k-1}(X)$ to be  the [[colimit]]

$$
  \underset{\to}{C}_{k-1}(X) := {\lim_{\to}}_{({\coprod_i U_i \to X})}
   \left( fiber\;product\;of\;C_{k-1}(U_i)s\;over\;joint\;boundary\;labels \right)
$$

over the category of _permissible decompositions_ (...) of $X$, where the composition operation in $C$ is used to label refinements of permissible decompositions.

=--

This is ([MorrisonWalker, def. 6.3.2](#MorrisonWalker)).

## Examples

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###### Definition

For $X$ a [[topological space]], its **fundamental blob $n$-category** $\Pi_{\leq n}(X)$ is the blob $n$-category which sends a $k$-ball for $k \lt n$ to the set of [[continuous map]]s of the ball into $X$, and an $n$-ball to the set of [[homotopy]]-classes of such maps, relative boundary.

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This is ([MorrisonWalker, example 6.2.1](#MorrisonWalker))).

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###### Definition

For $n \in \mathbb{N}$ the **blob $n$-category of $n$-dimensional [[cobordism]]s** $Bord_n$ is the blob $n$-category that sends a $k$-ball $B$ for $k \lt n$ to the set of $k$-dimensional submanifolds $W \hookrightarrow B \times \mathbb{R}^\infty$ such that the projection $W \to B$ is transverse to $\partial B$. An $n$-ball is sent to homeomorphism classes rel boundary of such submanifolds.

=--

This is ([MorrisonWalker, example 6.2.6](#MorrisonWalker))).

## Related concepts

* [[blob homology]]

* [[(∞,n)-category with adjoints]], [[(∞,n)-category with duals]]

## References

Section 6 of

* [[Scott Morrison]], [[Kevin Walker]], _The blob complex_, ([arxiv/1009.5025](http://arxiv.org/abs/1009.5025)) ([website](http://tqft.net/blobs))
{#MorrisonWalker}


Bloch group

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## Idea

The _Bloch group_ $\mathcal{B}(\mathbb{C})$ is the [[quotient]] of the degree-3 [[group homology]] $H_3^{grp}(\flat PSL(2,\mathbb{C}), \mathbb{Z})$ of the [[complex numbers|complex]] [[projective linear group]] $PSL(2,\mathbb{C})$ (with [[discrete topological space|discrete topology]]) by its [[torsion subgroup]], which is $\mathbb{Q}/\mathbb{Z}$.

Accordingly $H_3^{grp}(\flat PSL(2,\mathbb{C}), \mathbb{Z})$ itself is also called the _extended Bloch group_ ([Neumann 04](#Neumann04)).

## Properties

### Relation to hyperbolic 3-manifolds
 {#RelationToHyperbolic3Manifolds}

The [[fundamental class]] of a [[hyperbolic 3-manifold]] $X$ canonically gives an element in the Bloch group: let $X = \mathbb{H}^3/\Gamma$ for $\Gamma\hookrightarrow PSL(2,\mathbb{C})$, then the [[homotopy type]] of $X$ is that of $B\Gamma$ and hence the fundamental class maps forward under

$$
  H_3(X,\mathbb{Z})
  \simeq
  H_3(B \Gamma, \mathbb{Z})
  \simeq
  H_3^{grp}(\Gamma, \mathbb{Z})
  \longrightarrow
  H_3^{grp}(\flat PSL(2,\mathbb{C}), \mathbb{Z})
  \longrightarrow
  H_3^{grp}(\flat PSL(2,\mathbb{C}), \mathbb{Z})/(\mathbb{Q}/\mathbb{Z})
  \,.
$$

(e.g. [Neumann 11, section 2.2](#Neumann11))

### Cheeger-Simons class and complex volume
 {#RelationToCheegerSimonsClass}

There is a group [[homomorphism]]

$$
  \hat c
  \colon
  H_3(\flat PSL(2,\mathbb{C}), \mathbb{Z})
  \longrightarrow
  \mathbb{C}/\pi^2 \mathbb{Z}
$$

such that when applied to a [[fundamental class]] $[X]$ of a
[[hyperbolic 3-manifold]] according to the [above](#RelationToHyperbolic3Manifolds), then it yields its [[complex volume]]

$$
  \hat c([X]) = cs(X) + i vol(X)
  \,.
$$

This is called the _[[Cheeger-Simons class]]_.

(e.g. [Neumann 11, section 2.3](#Neumann11))

### Relation to algebraic K-theory
 {#RelationToAlgebraicKTheory}

Up to [[torsion subgroups]], the extended Bloch group of a is isomorphic to the third [[algebraic K-theory]] group $K_3^{ind}(\mathbb{C})$ ([Suslin90](#Suslin90), [Zickert 09](#Zickert09)).

### Relation to the Borel regulator
 {#RelationToTheBorelRegulator}

For $k$ an [[algebraic number field]] and $\{\sigma_i \colon k \hookrightarrow \mathbb{C}\}$ its complex embeddings, write

$$
  vol_i \coloneqq vol \circ (\sigma_i)_\ast \colon
  H_3(PSL(2,k),\mathbb{Z})
  \to
  \mathbb{R}
$$

for the induced volume measures, using the [[Cheeger-Simons class]] from [above](#RelationToCheegerSimonsClass). The [[direct product]] of these volumes is called the _[[Borel regulator]]_

$$
  Borel \coloneqq (vol_1, \cdots, vol_{r_2})
  \colon
  H_3(PSL(2,\mathbb{C}), \mathbb{Z})
  \longrightarrow
  \mathbb{R}^{r_2}
  \,.
$$

(e.g. [Zickert 09, (1.2)](#Zickert09))

Now the point is that restricted to the Bloch group proper the Borel regulator is [[injective]].

(e.g. [Neumann 11, theorem 2.4](#Neumann11))

## Related concepts

* [[hyperbolic manifold]]

* [[Cheeger-Simons class]], [[complex volume]]

* [[Borel regulator]]

## References

* {#Neumann04} [[Walter Neumann]], _Extended Bloch group and the Cheeger-Chern-Simons class_, Geom. Topol. 8 (2004) 413-474 ([arXiv:math/0307092](http://arxiv.org/abs/math/0307092))

The relation to [[algebraic K-theory]] and the [[Borel regulator]] is due to

* Chih-Han Sah, _Homology of classical Lie groups made discrete. III._, J. Pure Appl. Algebra, 56(3):269–312, 1989.

* {#Suslin90} [[Andrei Suslin]]. _$K_3$ of a field, and the Bloch group_. Trudy Mat. Inst. Steklov., 183:180–199, 229, 1990. Translated in Proc. Steklov Inst. Math. 1991, no. 4, 217–239, Galois theory, rings, algebraic groups and their applications (Russian).

* {#Zickert09} [[Christian Zickert]], _The extended Bloch group and algebraic K-theory_ ([arXiv:0910.4005](http://arxiv.org/abs/0910.4005))

Review is in

* {#Neumann11} [[Walter Neumann]], section 2.4 _Realizing arithmetic invariants of hyperbolic 3-manifolds_, Contemporary Math 541 (Amer. Math. Soc. 2011), 233--246 ([arXiv:1108.0062](http://arxiv.org/abs/1108.0062))

[[!redirects Bloch groups]]


Bloch region

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## Idea

In [[quantum physics]] and specifically in [[quantum computation]]/[[quantum information]], _Bloch region_ is the name of the [[convex space]] of [[density matrices]] on a given [[Hilbert space]].

Equivalently this is the space of [[mixed states|mixed]] [[quantum states]] for a [[quantum mechanical system]] with the lines in given [[Hilbert space]] as its [[pure states]].

The [[superoperators]] that preserve Bloch regions (also for products, "complete positivity") are called _[[quantum operations]]_.

## Properties

### Relation to the simplex

For $(n+1) \in \mathbb{N}$ a natural number, the Bloch region in the $(n+1)$-dimensional [[Hilbert space]] $\mathbb{C}^{n+1}$ is a [[quantum physics]]-analog of the space of [[probability measures]] on the [[finite set]] of $(n+1)$ elements. This in turn is equivalently the standard $n$-[[simplex]] $\Delta^n$.

## References

A quick introduction is in

* [[Greg Kuperberg]], section 1.4 of _A concise introduction to quantum probability, quantum mechanics, and quantum computation_, 2005 ([pdf](http://www.math.ucdavis.edu/~greg/intro-2005.pdf))
 {#Kuperberg05}

[[!redirects Bloch regions]]


Bloch's theorem

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## Idea

(...)

## Related concepts

* [[solid state physics]]

* [[electron band theory]]

## References

### For free electrons

The original article:

* [[Felix Bloch]], *Über die Quantenmechanik der Elektronen in Kristallgittern*, Zeitschrift für Physik **52** 7-8 (1929) 555-600 $[$[doi:10.1007/BF01339455](https://www.scopus.com/record/display.uri?eid=2-s2.0-34250940025&origin=inward&featureToggles=FEATURE_NEW_DOC_DETAILS_EXPORT:1)$]$

Textbook account:

* [[Michael Reed]] and [[Barry Simon]], Sec. XIII.16 *Schrödinger operators with periodic potentials*, of:  *[[Methods of Modern Mathematical Physics]] -- IV: Analysis of Operators*, Academic Press (1978)
([ISBN:9780080570457](https://www.elsevier.com/books/iv-analysis-of-operators/reed/978-0-08-057045-7))

Textbook account in [[semiconductor]]-theory:

* [[Sheng San Li]], Section 4.3 of: *Semiconductor Physical Electronics*, Springer (2006)   $[$[doi:10.1007/0-387-37766-2_4](https://doi.org/10.1007/0-387-37766-2_4)$]$

In the context of [[topological phases of matter]]:

* [[Jérôme Cayssol]], [[Jean-Noël Fuchs]], Section IV.B of: *Topological and geometrical aspects of band theory*, J. Phys. Mater. **4** (2021) 034007 ([arXiv:2012.11941](https://arxiv.org/abs/2012.11941), [doi:10.1088/2515-7639/abf0b5](https://doi.org/10.1088/2515-7639/abf0b5))

See also:

* Wikipedia, *[Bloch's theorem](https://en.wikipedia.org/wiki/Bloch%27s_theorem)*

Variant for [[hyperbolic spaces]]:

* Joseph Maciejko, Steven Rayan, *Hyperbolic band theory*, Science Advances **7** 36 (2021) [[doi:10.1126/sciadv.abe9170](https://doi.org/10.1126/sciadv.abe9170)]

* Adil Attar, Igor Boettcher, *Selberg trace formula in hyperbolic band theory*, Phys. Rev. E **106** 034114 (2022) [[arXiv:2201.06587](https://arxiv.org/abs/2201.06587), [doi:10.1103/PhysRevE.106.034114](https://doi.org/10.1103/PhysRevE.106.034114)]

### For interacting electrons

* Yuejin Guo, Jean-Marc Langlois and William A. Goddard , *Electronic Structure and Valence-Bond Band Structure of Cuprate Superconducting Materials*, New Series, **239** 4842 (1988) 896-899 $[$[jstor:1700316](https://www.jstor.org/stable/1700316)$]$

* Jingsan Hu, Jianfei Gu, Weiyi Zhang, *Bloch’s band structures of a pair of interacting electrons in simple one- and two-dimensional lattices*, Physics Letters A **414** (2021) 127634 $[$[doi:10.1016/j.physleta.2021.127634](https://doi.org/10.1016/j.physleta.2021.127634)$]$

[[!redirects Bloch theorem]]

[[!redirects Bloch theory]]

[[!redirects Bloch-Floquet theory]]

[[!redirects Bloch state]]
[[!redirects Bloch states]]

[[!redirects Bloch wave]]
[[!redirects Bloch waves]]


block design

Given positive integers $v,\beta,r,k,\lambda$, a __block 2-design__ of type $(v,\beta,r,k,\lambda)$ is a combinatorial structure consisting of

* a finite set $X$ whose elements are declared _points_
* set $B$ of $k$-element subsets of $X$, called _blocks_

such that

* (i) for each $x\in X$ the number of $v\in B$ such that $x\in b$ is independent of $x$ and equals $r$
* (ii) for each pair of distinct elements $x,y\in X$, the number of $b\in B$ such that $\{x,y\}\subset b$ is independent of $(x,y)$ and equals $\lambda$

__Block $s$-designs__ of type $(v,\beta,r,k,\lambda)$ for $s\gt 2$ have (ii) changed to

(ii') for each subset $\{x_1,\ldots,x_s\}\subset X$ of cardinality precisely $s$, the number of $b\in B$ such that $\{x_1,\ldots,x_s\}\subset b$ is independent of the choice of such a subset and equals $\lambda$.

Examples: [[Steiner system]]s, finite projective planes.

Applications: cryptography, codes, finite geometries, finite group theory.

* Wikipedia [block design](http://en.wikipedia.org/wiki/Block_design)


blockchain

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\section{Idea}
 {#Idea}

In [[computer science]], the notion of _blockchains_ --- originating in concepts of cryptocurrency, with [Wei Dai (1996)](#WeiDai96) (*[B-money](https://en.bitcoin.it/wiki/B-money)*) and [Nakamoto (2008)](#NakamotoBitcoinAPeerToPeerElectronicCashSystem) (*[bitcoin](https://en.wikipedia.org/wiki/Bitcoin)*) --- refers to [[linear order|linearly ordered]] log files shared in a [[undirected graph|network]] with multiple nodes and without any central authority, which can be added to but are extremely difficult to otherwise modify --- as long as a majority of nodes act non-maliciously (see [below](#FaultTolerance)).

Blockchain can be considered as certain data structure -- chain of blocks -- which are cryptographically "sealed" by hashes which are for each block added to the next block and which is incremented by one of many parties called nodes
which follow a commonly accepted algorithm and each contains a copy of the blockchain (full node) or its essential part. Who adds a block, or whose added block is respected by other nodes is a part of the algorithm (blockchain protocol) and is desirable to be byzantine fault tolerant. The securest algorithm in nontrusted environment (public blockchain) is the proof of work algorithm. Regarding that blockchain is a tool to establish noncentralized record verification it is common to use blockchains to carry records of assets, including cryptocurrencies and other token, off chain assets, credential records, verification proofs and so on. These data and transaction steps belong and are authorized by accounts. Modern blockchain protocols
assume also that code handling assets in deterministic manner may be part of the account, when the nodes add new blocks they should also execute the code of the contract, within a blockchain virtual machine; such accounts are called [[smart contract]]s and realize in a specific way the idea of smart contracts outlined in theoretical computer science before the blockchain era.

The phrase _distributed ledger_ is sometimes viewed as a synonym. Technically, distributed ledger is the same idea realized in a more general setup where the record may be organized in non-linear structure, for example when the blocks form a DAG (directed acyclic graph) rather than a single chain, with appropriate protocol.

\section{Proof of work}

The key idea is that of _proof of work_. Think of a blockchain simply as a [[list]] $[ B_{1}, B_{2}, \ldots ]$ of 'blocks'. A _block_ $B$ is an _unencrypted block_ of content, which could be anything, together with a _cryptographic hash_ of it ('signature') with certain properties.

The key points are the following.

1. The cryptographic hash with the required properties is hard to create, where 'hard' means requiring significant computational resources and taking time (e.g. something like [[prime]] factorisation). Use of these computational resources to create the hash is referred to as 'proof of work'.

1. The hash of $B_{i+1}$ depends on the hash of $B_{i}$.

1. The hash of a block depends exactly on its contents: any change in contents will lead to a significantly different hash.

1. It is easy (requires little resources and time) to check that a particular hash has the required properties.

Suppose that we have $n$ nodes in a network. Each node accepts as the current blockchain the longest valid blockchain which it knows of. If a node $X$ adds a block $B$ to a blockchain $L = [ B_1, \ldots, B_m ]$, i.e. succeeds in a creating a hash of some unencrypted block of content, it sends $L' = [B_1, \ldots, B_m, B]$ out to all nodes listening to/connected to it. If the longest blockchain $L''$ which a node $Y$ knows of has length $m$, then upon receiving $L'$ it drops any attempts it is currently making to add a block to $L''$, checks the validity of $L'$, and if it is valid accepts it as the current blockchain.

In such a system, it is extremely difficult for any node to tamper with the  blockchain. This is because if it wishes to change the content of a particular block, then because of 2. and 3. it must not only re-create the hash of the tampered block, but also the hash of all blocks after it, as well as the hash of a new block. If any honest node creates the hash of just one new block in the meantime and sends it out to the network, the tampering attempt now has still another block to hash. Because of the computational resources required to create even one valid hash, it is thus highly unlikely that tampering will succeed, and thus a tamperer may soon use up great amounts of resources for no gain. If successful adding of blocks is incentivised appropriately, it is thus more sensible to act honestly.

There are various further details, but this is the rough idea.

\section{Fault tolerance}
  {#FaultTolerance}

The intended functioning of blockchain networks relies on most but not necessarily all actors acting reliably/non-maliciously towards the protocol, a situation known as *Byzantine fault tolerance* (BFT), see eg. [Wang et al (2022)](#WangEtAl22)

\section{Related concepts}

* [[distributed computing]]

* [[smart contract]]

* [[Ethereum]]

* [[computer science]]

* [[high performance distributed ledger]]

* [[Telegram Open Network]]

* [[virtual machine]]

## References

### General

The idea originates with:

* {#WeiDai96} Wei Dai *bmoney* (1996) [[web](http://www.weidai.com/bmoney.txt)]

* {#NakamotoBitcoinAPeerToPeerElectronicCashSystem} Satoshi Nakamoto, _Bitcoin: A peer-to-peer electronic cash system_, 2008 [pdf](https://bitcoin.org/bitcoin.pdf)

A foundational/mathematical introductory book from the point of view of (building) a distributed computing system is

* Roger Wattenhofer, _The science of the blockchain_, Inverted Forest Publishing (2016)

A comprehensive guide for architects of blockchain applications with an attempt at certain level of abstraction of central concepts is

* Xiwei Xu, Ingo Weber, Mark Staples, _Architecture for blockchain applications_, Springer 2019

The following systematic book written by a several authors offers pretty rigorous handbook on blockchain security issues

* Sachin S. Shetty, Charles A. Kamhoua, Laurent L. Njilla (eds.) _Blockchain for distributed systems security_, 324 pp. IEEE Press, SMTE Books 2019

See also:

* {#WangEtAl22} Wang et al., *BFT in Blockchains: From Protocols to Use Cases*, ACM Computing Surveys **54** 10s (2022) 209 1–37 [[pdf](https://bchainzhang.github.io/files/bft-csur22.pdf), [doi:10.1145/3503042](https://doi.org/10.1145/3503042)]

* [[Melanie Swan]], [[Renato P dos Santos]], [[Frank Witte]], Between Science and Economics, Volume 2: *Quantum Computing Physics, Blockchains, and Deep Learning Smart Networks*, World Scientific 2020 ([doi:10.1142/q0243](https://doi.org/10.1142/q0243))

Some key papers in [[zoranskoda:distributed consensus]] include

* M. J. Fischer, N. Lynch, M. S. Paterson, _Impossibility of distributed con-
sensus with one faulty process_, JACM 32:2, 374-382 (1985)
* M. Castro, B. Liskov, et al., _Practical byzantine fault tolerance_, Proc. 3rd
Symposium on Operating Systems Design and Implementation (1999) 173–186.
* L. Lamport, R. Shostak, M. Pease, _The byzantine generals problem_, ACM
Transactions on Programming Languages and Systems, 4/3 (1982), p. 382–401.

Other references on distributed ledgers

* [[Vitalik Buterin]], _Ethereum: A next-generation smart contract and decentralized application platform_, [pdf](https://github.com/ethereum/wiki/wiki/White-Paper.pdf)
* Gavin Wood, _Ethereum: a secure decentralised generalised transaction
ledger_, Ethereum, Tech. Rep. 2017
* Gavin Wood, _PolkaDot: vision for a heterogeneous multi-chain framework_ [pdf](https://github.com/w3f/polkadot-white-paper/raw/master/PolkaDotPaper.pdf)
* Joseph Poon, Vitalik Buterin, _Plasma: scalable autonomous smart contracts_,
[pdf](https://www.plasma.io/plasma.pdf)
* Joseph Poon, Tadge Dryja, _Lightning network_, [pdf](https://lightning.network/lightning-network-paper.pdf) Mar 2015.

* [[Vitalik Buterin]], _Chain interoperability_, R3 reports, Sep 2016 [pdf](https://www.r3.com/wp-content/uploads/2018/04/Chain_Interoperability_R3.pdf)

* Tier Nolan, _Re: Alt chains and atomic transfers_, 2013, bitcointalk discussion [post](https://bitcointalk.org/index.php?topic=193281.msg2224949#msg2224949)
* [[Nikolai Durov]], _Telegram Open Network_, 132 pp. Dec 2017 [pdf](https://toncoin.tech/TON-official-whitepaper.pdf), mirror [pdf](https://www.kriptovaluta.hr/wp-content/uploads/2018/03/TON-Technology.pdf); _Telegram Open Network Blockchain_, Sep 2018, 121 pp. [pdf](https://www.docdroid.net/qY4sQEv/telegram-open-network-blockchain-september-5-2018.pdf); _Telegram Open Network Virtual Machine_, Sep. 2018, 148 pp. [pdf](https://www.docdroid.net/R3vEKBY/telegram-open-network-virtual-machine-september-5-2018.pdf); _Fift: a brief introduction_ (about a new programming language on TON), May 23, 2019, 87 pp. pdf is at channel [t.me/Tgram/170](https://t.me/Tgram/170)
* L. Baird, _The Swirlds hashgraph consensus algorithm: fair, fast, byzantine fault tolerance_, [pdf](https://www.swirlds.com/downloads/SWIRLDS-TR-2016-01.pdf) (May 2016); _Hashgraph consensus: detailed examples_ [pdf](https://www.swirlds.com/downloads/SWIRLDS-TR-2016-02.pdf); _Overview of Swirlds Hashgraph: An Introduction To The Hashgraph (SDK Available Now)_ [online](https://steemit.com/technology/@swirlds/overview-of-swirlds-hashgraph-an-introduction-to-the-hashgraph-sdk-available-now); [Hedera](https://www.hedera.com/whitepaper) Hashgraph whitepaper [pdf](https://www.hedera.com/hh-whitepaper-v1.5-190219.pdf); intro to whitepaper [web](https://www.hedera.com/whitepaper)
* Aggelos Kiayias, Andrew Miller, Dionysis Zindros, _Non-interactive proofs of proof-of-work_, IACR preprint 2017 [pdf](https://eprint.iacr.org/2017/963.pdf) (see also [here](https://ambcrypto.com/pewdiepie-forays-into-cryptospace-by-partnering-with-dlive-a-blockchain-based-live-streaming-platform) and Cardano, IOHK)
* Mike Hearn, _Corda -- a distributed ledger_ (technical whitepaper, 2016) [pdf](https://docs.corda.net/_static/corda-technical-whitepaper.pdf); Brown et al. _Corda - an introduction_, whitepaper 2016, [pdf](https://docs.corda.net/_static/corda-introductory-whitepaper.pdf); Brown, _Corda - an introduction_ platform whitepaper 2018 [pdf](https://www.corda.net/content/corda-platform-whitepaper.pdf)
* Ittay Eyal, Emin Gün Sirer, _Majority is not enough: Bitcoin mining is vulnerable_, in: Financial Cryptography and Data Security 436–454, Springer 2014 [pdf](http://fc14.ifca.ai/papers/fc14_submission_82.pdf)

For an extended list of references see [[zoranskoda:blockchain]] (at zoranskoda).

### Formal verification
 {#References}

On [[formal methods]] for [[software verification|verification]] of blockchain consensus algorithms (with [[proof assistants]] such as [[Coq]] or [[Agda]]):

* [Hedera](https://hedera.com) blog: *[Formal Methods: The Importance of Being Fault Tolerant in a World with Bad Actors](https://medium.com/hedera/formal-methods-the-importance-of-being-abft-in-a-world-with-bad-actors-7308a4997fdd)* (2018)

* *[1st Workshop on Formal Methods for Blockchains](https://sites.google.com/view/fmbc/)* (October 2019)

* Musab A. Alturki, Jing Chen, Victor Luchangco, Brandon Moore, Karl Palmskog, Lucas Peña, Grigore Roşu, *Towards a Verified Model of the Algorand Consensus Protocol in Coq*, Formal Methods. FM 2019 International Workshops. Lecture Notes in Computer Science **12232** (2019) 362-367 [[arXiv:1907.05523](https://arxiv.org/abs/1907.05523), [doi:10.1007/978-3-030-54994-7_27](https://doi.org/10.1007/978-3-030-54994-7_27)]

* [Quantstamp](https://quantstamp.com/) blog: *[Formally Verifying Hedera Hashgraph's Stablecoin Framework](https://quantstamp.com/blog/quantstamp-stablecoin-case-study-formally-verifying-hedera-hashgraphs-stablecoin-framework)* (2020)

* Karl Crary, *Verifying the Hashgraph Consensus Algorithm* [[arXiv:2102.01167](https://arxiv.org/abs/2102.01167), [pdf](https://www.cs.cmu.edu/~crary/papers/2021/hashgraph.pdf)]

* Harold Carr, Christopher Jenkins, Mark Moir, Victor Cacciari Miraldo, Lisandra Silva, *Towards Formal Verification of HotStuff-based Byzantine Fault Tolerant Consensus in Agda: Extended Version*, in: *NASA Formal Methods: 14th International Symposium, NFM 2022* Proceedings  (2022) 616–635 [[doi:10.1007/978-3-031-06773-0_33](https://doi.org/10.1007/978-3-031-06773-0_33), [arXiv:2203.14711](https://arxiv.org/abs/2203.14711)]

* Sudhani Verma; Divakar Yadav; Girish Chandra, *Introduction of Formal Methods in Blockchain Consensus Mechanism and Its Associated Protocols*, IEEE Access **10** (2022) [[doi:10.1109/ACCESS.2022.3184799](https://doi.org/10.1109/ACCESS.2022.3184799), [pdf](https://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=9801830)]

* [Quantstamp](https://quantstamp.com/) blog: *[Applying lightweight formal methods and SAT solvers to build better blockchain applications](https://quantstamp.com/blog/towards-satisfactory-web3-software-engineering)* (July 2023)

category: computer science, applications

[[!redirects blockchains]]

[[!redirects proof of work]]


blueprint

* [[Oliver Lorscheid]], _The geometry of blueprints. Part I: Algebraic background and scheme theory_, [arxiv/1103.1745](https://arxiv.org/abs/1103.1745); _The geometry of blueprints. Part II: Tits-Weyl models of algebraic groups_, [arxiv/1201.1324](https://arxiv.org/abs/1201.1324); _Blueprints - towards absolute arithmetic_, [arxiv/1204.3129](https://arxiv.org/abs/1204.3129);
_A blueprinted view on $\mathbb{F}_1$-geometry_, [arxiv/1301.0083](https://arxiv.org/abs/1301.0083)
* Javier López Peña, Oliver Lorscheid, _Projective geometry for blueprints_, [arxiv/1203.1665](http://front.math.ucdavis.edu/1203.1665)
* Claudio Bartocci, Andrea Gentili, Jean-Jacques Szczeciniarz, _Some remarks on blueprints and $\mathbb{F}_1$-schemes_, ([arXiv:1905.01183](https://arxiv.org/abs/1905.01183))

Spectra of blueprints glue to __blue schemes__ similarly like spectra of rings glue to [[algebraic scheme]]s. Blue schemes interpolate between the usual schemes and schemes over $\mathbf{F}_1$, i.e. [[absolute geometry]].

* some speculations on application of Reineke's theorem on quiver Grassmanians for the world of blue schemes and $\mathbf{F}_1$-schemes: [Lieven's blog](http://www.neverendingbooks.org/quiver-grassmannians-and-mathbbf_1-geometry)

category: algebraic geometry
[[!redirects blue scheme]]


BMN matrix model

+-- {: .rightHandSide}
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###Context###
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _[[BMN matrix model]]_ ([BMN 02](#BerensteinMaldacenaNastase02), [DSJVR 02](#DSJVR02)) is a [[deformation]] of the [[BFSS matrix model]] by [[mass]]-terms and by a [[Chern-Simons theory|Chern-Simons]]-term. It arises as the [[light-cone gauge quantization]] of the [[M2-brane]] not on [[Minkowski spacetime]] as for the [[BFSS matrix model]], but on its [[deformation]] to a [[pp-wave spacetime]]; the [[mass]]-terms in the model are proportional to the [[B-field]] [[flux density]] in the pp-wave [[string theory vacuum|vacuum]].

## Properties

### Ground states: $D0\perp D2$ and $M2/M5$ bound states
 {#GroundStates}

While the [[BFSS matrix model]] notoriously suffers from issues with the definition of its ground state (see [there](BFSS+matrix+model#ProblemsGroundState)) this problem is fixed in the BMN matrix model by the mass term:

The [[supersymmetry|supersymmetric]] [[classical field theory|classical]] solutions are configurations of  [[fuzzy 2-spheres]], corresponding to [[M2-brane]]-[[giant gravitons]] ([BMN 02 (5.4)](#BerensteinMaldacenaNastase02),
[DSJVR 02, Section 4.2](#DSJVR02)), which under the [[duality between M-theory and type IIA string theory]] become [[D0-D2 brane bound states]] ([Lin04](#Lin04)).

In fact, [[duality in string theory|dually]] these gound states encode also transversal [[M5-branes]] and [[M2-M5 brane bound states]], see [below](#M2M5BraneBoundStatesInTheBMNMatrixModel).

### Relation to black M2/M5-brane backgrounds

The [[pp-wave spacetime]]-background for the BMN matrix model arises as the [[Penrose limit]] ([[Inönü-Wigner contraction]]) of the [[anti de Sitter spacetime|AdS4]][[product manifold|x]][[7-sphere]] [[near horizon geometry]] of both the [[black brane|black]] [[M2-brane]] spacetime $AdS_4 \times S^7$, as well as the [[anti de Sitter spacetime|AdS7]]$\times$[[4-sphere]] [[near horizon geometry]] of [[black brane|black]] [[M5-branes]] ([BMN 02, Section 2](#BerensteinMaldacenaNastase02)).

\linebreak

[[!include chord diagrams as multi-trace observables in BMN matrix model -- example]]

\linebreak

[[!include M2-M5 brane bound states in the BMN matrix model -- subsection]]

\linebreak

## Related concepts

[[!include brane matrix models -- content]]

## References

### General

The original articles:

* {#BerensteinMaldacenaNastase02} [[David Berenstein]], [[Juan Maldacena]], [[Horatiu Nastase]], Sections 5 and Appendix B of: _Strings in flat space and pp waves from $\mathcal{N} = 4$ Super Yang Mills_, JHEP **0204** (2002) 013 [[arXiv:hep-th/0202021](https://arxiv.org/abs/hep-th/0202021), [doi:10.1088/1126-6708/2002/04/013](https://doi.org/10.1088/1126-6708/2002/04/013)]

  > (which is otherwise about [[single trace operators]] in the [[AdS/CFT correspondence]]).

* {#DSJVR02} [[Keshav Dasgupta]], [[Mohammad Sheikh-Jabbari]], [[Mark Van Raamsdonk]], _Matrix Perturbation Theory For M-theory On a PP-Wave_, JHEP 0205:056, 2002 ([arXiv:hep-th/0205185](https://arxiv.org/abs/hep-th/0205185))

* [[Nakwoo Kim]], [[Jan Plefka]], _On the Spectrum of PP-Wave Matrix Theory_, Nucl. Phys. B643:31-48, 2002 ([arXiv:hep-th/0207034](https://arxiv.org/abs/hep-th/0207034))

* [[Keshav Dasgupta]], [[Mohammad Sheikh-Jabbari]], [[Mark Van Raamsdonk]], _Protected Multiplets of M-Theory on a Plane Wave_, JHEP 0209:021, 2002 ([arXiv:hep-th/0207050](https://arxiv.org/abs/hep-th/0207050))

* [[Nakwoo Kim]], [[Jeong-Hyuck Park]], _Superalgebra for M-theory on a pp-wave_, Phys.Rev. D66 (2002) 106007 ([arXiv:hep-th/0207061](https://arxiv.org/abs/hep-th/0207061))

Review:

* {#Ydri18} [[Badis Ydri]], Section 3.10 in: _Review of M(atrix)-Theory, Type IIB Matrix Model and Matrix String Theory_ ([arXiv:1708.00734](https://arxiv.org/abs/1708.00734)), published as: _Matrix Models of String Theory_, IOP 2018 ([ISBN:978-0-7503-1726-9](https://iopscience.iop.org/book/978-0-7503-1726-9))

In relation to the [[Myers effect]] and [[D0-D2 brane bound states]]:

* {#Lin04} [[Hai Lin]], _The Supergravity Dual of the BMN Matrix Model_, JHEP 0412:001, 2004 ([arXiv:hep-th/0407250](https://arxiv.org/abs/hep-th/0407250))

Discussion of the [[large N limit]]:

* Corneliu Sochichiu, _Continuum limit(s) of BMN matrix model: Where is the (nonabelian) gauge group?_, Phys.Lett. B574 (2003) 105-110 ([arXiv:hep-th/0206239](https://arxiv.org/abs/hep-th/0206239))

Computer simulation ([[lattice gauge theory]]):

* Hrant Gharibyan, [[Masanori Hanada]], Masazumi Honda, Junyu Liu, _Toward simulating Superstring/M-theory on a quantum computer_ ([arXiv:2011.06573](https://arxiv.org/abs/2011.06573))

and comparison in the [[large N limit]] with computations in [[supergravity]]:

* S. Pateloudis et al., *Precision test of gauge/gravity duality in $D0$-brane matrix model at low temperature*,  J. High Energ. Phys. **2023** 71 (2023)  [[arXiv:2210.04881](https://arxiv.org/abs/2210.04881), doi:10.1007/JHEP03(2023)071]

See also

* [[Jens Hoppe]]. Ki-Myeong Lee, _New BPS Configurations of BMN Matrix Theory_, JHEP 0806:041, 2008 ([arXiv:0712.3616](https://arxiv.org/abs/0712.3616))

* Masanori Hanada, Lorenzo Mannelli, Yoshinori Matsuo, _Four-dimensional $\mathcal{N}=1$ super Yang-Mills from matrix model_, Phys. Rev. D 80, 125001 ([arXiv:0905.2995](https://arxiv.org/abs/0905.2995))

* [[Yolanda Lozano]], [[Carlos Nunez]], Salomon Zacarias, _BMN Vacua, Superstars and Non-Abelian T-duality_, JHEP 09 (2017) 008 ([arXiv:1703.00417](https://arxiv.org/abs/1703.00417))

* Yuhma Asano, Samuel Kováčic, Denjoe O'Connor, _The Confining Transition in the Bosonic BMN Matrix Model_ ([arXiv:2001.03749](https://arxiv.org/abs/2001.03749))

* Minos Axenides, Emmanuel Floratos, Dimitrios Katsinis, Georgios Linardopoulos, *Cascade of instabilities in the classical limit of the BMN matrix model* ([arXiv:2109.01088](https://arxiv.org/abs/2109.01088))

* Shota Komatsu, Adrien Martina, João Penedones, Noé Suchel, Antoine Vuignier, Xiang Zhao, *Gravity from quantum mechanics of finite matrices* [[arXiv:2401.16471](https://arxiv.org/abs/2401.16471)]

### Thermodynamics

Discussion of [[thermodynamics]], [[phase transitions]], [[confinement]] and [[thermal field theory]] of the BMN matrix model:

* Hyeonjoon Shin, Kentaroh Yoshida, _Thermodynamic Behavior of Fuzzy Membranes in PP-Wave Matrix Model_, Phys.Lett. B627 (2005) 188-196 ([arXiv:hep-th/0507029](https://arxiv.org/abs/hep-th/0507029))

* Curtis T. Asplund, [[David Berenstein]], Diego Trancanelli, _Evidence for fast thermalization in the plane-wave matrix model_, 	Phys. Rev. Lett. 107, 171602 (2011) ([arXiv:1104.5469](https://arxiv.org/abs/1104.5469))

* Lucas Brady, Vatche Sahakian, _Scrambling with Matrix Black Holes_, Phys Rev D 88, 046003 (2013) ([arXiv:1306.5200](https://arxiv.org/abs/1306.5200))

* Miguel S. Costa, Lauren Greenspan, Joao Penedones, [[Jorge Santos]], _Thermodynamics of the BMN matrix model at strong coupling_, JHEP 03 (2015) 069 ([arXiv:1411.5541](https://arxiv.org/abs/1411.5541))

* Samuel Pramodh, Vatche Sahakian
_From Black Hole to Qubits: Evidence of Fast Scrambling in BMN theory_, JHEP 7, July 2015, 67 ([arXiv:1412.2396](https://arxiv.org/abs/1412.2396))

* Yuhma Asano, Veselin G. Filev, Samuel Kováčik, Denjoe O'Connor, _The non-perturbative phase diagram of the BMN matrix model_, JHEP 1807 (2018) 152 ([arXiv:1805.05314](https://arxiv.org/abs/1805.05314))

* Yuhma Asano, Samuel Kováčic, Denjoe O'Connor, _The Confining Transition in the Bosonic BMN Matrix Model_ ([arXiv:2001.03749](https://arxiv.org/abs/2001.03749))

Discussion of [[chaos]] in the [[M2-brane]] physics described by the BMN matrix model:

* Minos Axenides, Emmanuel Floratos, Georgios Linardopoulos, _M2-brane Dynamics in the Classical Limit of the BMN Matrix Model_, Phys. Lett. B773 (2017) 265 ([arXiv:1707.02878](https://arxiv.org/abs/1707.02878))

See also:

Discussion of [[entanglement]] and [[entropy]]:

* Cameron Gray, Vatche Sahakian, William Warfield, _Emergent geometry through quantum entanglement in Matrix theories_ ([arXiv:2103.06941](https://arxiv.org/abs/2103.06941))

### KK-compactification from $D=4$ $\mathcal{N}=4$ SYM

On the [[BMN matrix model]] as a [[KK-compactification]] of [[D=4 N=4 super Yang-Mills theory]] on the [[3-sphere]]:

* [[Nakwoo Kim]], [[Thomas Klose]], [[Jan Plefka]], _Plane-wave Matrix Theory from N=4 Super Yang-Mills on $\mathbb{R} \times S^3$_, Nucl. Phys. B671:359-382, 2003 ([arXiv:hep-th/0306054](https://arxiv.org/abs/hep-th/0306054))

Using the [[KK-compactification]] of [[D=4 N=4 super Yang-Mills theory]] to the [[BMN matrix model]] for [[lattice gauge theory]]-computations in [[D=4 N=4 SYM]] and for numerical checks of the [[AdS-CFT correspondence]]:

* Masazumi Honda, Goro Ishiki, Sang-Woo Kim, Jun Nishimura, Asato Tsuchiya, _Direct test of the AdS/CFT correspondence by Monte Carlo studies of N=4 super Yang-Mills theory_, JHEP 1311 (2013) 200 ([arXiv:1308.3525](https://arxiv.org/abs/1308.3525))

[[!include M5-branes in the BMN matrix model -- references]]

[[!include pp-waves as Penrose limits of AdS spacetimes -- references]]

[[!redirects BMN matrix models]]


Bn-geometry

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
#### Differential cohomology
+--{: .hide}
[[!include differential cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In the context of [[T-duality]] and in particular of [[differential T-duality]] one considers (as discussed in detail there) [[fiber products]] of two [[torus]]-[[fiber bundles]] together with a [[circle 2-bundle]] on this, ([[circle n-bundle with connection|with connection]]).

In some disguise, this has been called _$B_n$-geometry_ ([Baraglia](#Baraglia)). The T-duality interpretation is made explicit in [Bouwknegt](Bouwknegt)

Here "$B_n$" refers to the [[special orthogonal group]] of the form $SO(n+1,n)$, which appears as the structure group of a [[generalized tangent bundle]] tensored with a [[line bundle]] (the [[Poincare line bundle]] of the T-duality correspondence).

## Properties

### Interpretation in higher differential geometry

We give the interpretation of $B_n$-geometry in [[smooth infinity-groupoid|higher differential geometry]].

For $c_{conn},c'_{conn} \colon X \to \mathbf{B}U(1)$ modulating two circle principal bundles with conection, a [[differential T-duality]] structure is a choice of trivialization of their [[cup product]] class. From this we get the [[pasting diagram]] of [[homotopy pullbacks]] of smooth $\infty$-stacks

$$
  \array{
    P \times_X \hat P&\stackrel{\tau}{\to}& \mathbf{B}^2 U(1) &\to& *
    \\
    \downarrow && \downarrow && \downarrow
    \\
    X
     &\to&
    fib(\cup)
     &\to&
    (\mathbf{B}U(1)) \times (\mathbf{B}U(1))
    \\
    &\searrow & \downarrow && \downarrow^{\mathrlap{\cup}}
    \\
    && * &\to& \mathbf{B}^3 U(1)
  }
  \,.
$$

Here $\tau$ is the morphism that modulates the circle 2-bundle on the fiber product of the two circle bundles.

(...)

## Related concepts

* [[differential T-duality]]

* [[generalized complex geometry]]

* [[exceptional generalized geometry]]

## References

The term $B_n$-geometry was introduced in

* {#Baraglia} [[David Baraglia]], _Leibniz algebroids, twistings and exceptional generalized geometry_ ([arXiv:1101.0856](http://arxiv.org/abs/1101.0856))

A review is in

* {#Hitchin} [[Nigel Hitchin]], talks at StringMath2012,  ([pdf](http://www.hcm.uni-bonn.de/fileadmin/stringmath2012/plenary_talks/03-Hitchin.pdf))

The relation to T-duality is made clear around slide 80 of

* {#Bouwknegt} [[Peter Bouwknegt]], _Courant Algebroids and Generalizations of Geometry_, talk at StringMath2011 ([pdf](http://www.math.upenn.edu/StringMath2011/notes/Bouwknegt_StringMath2011_talk.pdf))

A discussion of the [[infinity-Lie theory|higher Lie theoretic]] aspects is in

* [[Ernesto Lupercio]], Camilo Rengifo, [[Bernardo Uribe]], _T-duality and exceptional generalized geometry through symmetries of dg-manifolds_ ([arXiv:1208.6048](http://arxiv.org/abs/1208.6048))

[[!redirects geometry of type Bn]]


BN-pair

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[[!include group theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

A [[pair]] $B,N$ of [[subgroups]] of a [[group]] $G$ is a **BN-pair** if the following holds:

* the [[union]] $B\cup N$ generates $G$,

* the [[intersection]] $B\cap N$ is a [[normal subgroup]] of $N$,

* the [[quotient group]] $W \coloneqq N/(B\cap N)$ is generated by a set $S$ of [[involutions]],

* for each $s\in S,w\in W$ we have $B s B \cdot B w B\subseteq B s w B\cup B w B$

* for $s\in S$, $B s B\cdot B s B \neq B$

Note that the [[set]] $S$ is uniquely determined by these [[axioms]], and that $(W,S)$ is a [[Coxeter group|Coxeter system]].

## References

* Anders Bjorner, Francesco Brenti, _Combinatorics of coxeter groups_, Springer 2005 ([doi:10.1007%2F3-540-27596-7](https://link.springer.com/book/10.1007%2F3-540-27596-7))

[[!redirects BN-pairs]]


Bo Feng

* [webpage](https://person.zju.edu.cn/en/0007089)

## Selected writings

On [[discrete torsion]]:

* [[Bo Feng]], [[Amihay Hanany]], [[Yang-Hui He]], Nikolaos Prezas, _Discrete Torsion, Non-Abelian Orbifolds and the Schur Multiplier_, JHEP 0101:033, 2001 ([arXiv:hep-th/0010023](https://arxiv.org/abs/hep-th/0010023))

On IIA [[NS5-branes]] and their [[T-duality|T-dual]] [[orientifolds]] with [[RR-field tadpole cancellation]]:

* {#FengHeKarchUranga01} [[Bo Feng]], [[Yang-Hui He]], [[Andreas Karch]], [[Angel Uranga]], _Orientifold dual for stuck NS5 branes_, JHEP 0106:065, 2001 ([arXiv:hep-th/0103177](https://arxiv.org/abs/hep-th/0103177))

On [[toric duality]]:

* [[Bo Feng]], [[Amihay Hanany]], [[Yang-Hui He]], _D-brane gauge theories from toric singularities and toric duality, Nucl. Phys. B 595 (2001) 165 ([arXiv:hep-th/0003085](https://arxiv.org/abs/hep-th/0003085))

* [[Bo Feng]], [[Amihay Hanany]], [[Yang-Hui He]], _Phase structure of D-brane gauge theories and toric duality_ , J. High Energy Phys. 08 (2001) 040 ([hep-th/0104259](https://arxiv.org/abs/hep-th/0104259))

* [[Bo Feng]], [[Amihay Hanany]], [[Yang-Hui He]], [[Angel Uranga]], _Toric duality as Seiberg duality and brane diamonds, J. High Energy Phys. 12 (2001) 035 ([hep-th/0109063](https://arxiv.org/abs/hep-th/0109063))

* [[Bo Feng]], S. Franco, [[Amihay Hanany]], [[Yang-Hui He]], _Unhiggsing the del Pezzo_, J. High Energy Phys. 08 (2003) 058 ([hep-th/0209228](https://arxiv.org/abs/hep-th/0209228))

On [[double trace operators]] in [[matrix models]]:

* [[Vijay Balasubramanian]], [[Jan de Boer]], [[Bo Feng]], [[Yang-Hui He]], Min-xin Huang, Vishnu Jejjala, Asad Naqvi, _Multi-Trace Superpotentials vs. Matrix Models_, Commun. Math. Phys. 242:361-392, 2003 ([arXiv:hep-th/0212082](https://arxiv.org/abs/hep-th/0212082))

category: people


Bo Peng

* [institute page](https://www.tcm.phy.cam.ac.uk/profiles/bp432/)

## Selected writings

On [[braid group statistics|anyonic braiding]] of nodal points in the [[Brillouin zone]] of [[semi-metals]] ("braiding in momentum space"):

* [[Bo Peng]], [[Adrien Bouhon]], [[Robert-Jan Slager]], [[Bartomeu Monserrat]], *Multi-gap topology and non-Abelian braiding of phonons from first principles*, Phys. Rev. B **105** (2022) 085115 ([arXiv:2111.05872](https://arxiv.org/abs/2111.05872), [doi;10.1103/PhysRevB.105.085115](https://doi.org/10.1103/PhysRevB.105.085115))

*  *[[Bo Peng]], [[Adrien Bouhon]], [[Bartomeu Monserrat]], [[Robert-Jan Slager]], *Phonons as a platform for non-Abelian braiding and its manifestation in layered silicates*, Nature Communications volume 13, Article number: 423 (2022) ([doi:10.1038/s41467-022-28046-9](https://doi.org/10.1038/s41467-022-28046-9))

category: people


Boardman homomorphism

+-- {: .rightHandSide}
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###Context###
#### Stable Homotopy theory
+--{: .hide}
[[!include stable homotopy theory - contents]]
=--
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Given any [[homotopy commutative ring spectrum]] $(E, \mu, e)$, then the _Boardman homomorphism_ is the [[homomorphism]] from [[stable homotopy groups]] (hence from [[stable homotopy homology theory]]) to $E$-[[generalized homology]] groups that is induced by [[smash product]] with the unit map $e \colon \mathbb{S} \longrightarrow E$ from the [[sphere spectrum]]:

$$
  \pi_\bullet(-)
  \simeq
  \pi_\bullet(\mathbb{S} \wedge (-))
  \longrightarrow
  \pi_\bullet(E \wedge (-))
  =
  E_\bullet(-)
  \,.
$$

For $E = H \mathbb{Z}$ the [[Eilenberg-MacLane spectrum]] for [[ordinary homology]], then this reduces to the [[Hurewicz homomorphism]] $\pi_\bullet(-) \to H_\bullet(-)$.

Dually, there is the Boardman homomorphism from [[stable cohomotopy]] to [[generalized cohomology]] induced under forming [[mapping spectra]] into the unit map of $E$:

$$
  \pi^\bullet(-)
  \simeq
  \pi_\bullet([(-),\mathbb{S}])
   \longrightarrow
  \pi_\bullet([(-),E])
   =
  E^\bullet(-)
  \,.
$$

Unifying these two cases, there is the bivariant Boardman homomorphism ([Adams 74, p. 58](#Adams74))

$$
  [X, Y]_\bullet
    \;\simeq\;
  [X, Y \wedge \mathbb{S}]_\bullet
    \longrightarrow
  [X,Y \wedge E]_\bullet
  \,.
$$

Since [[generalized homology]]/[[generalized cohomology]] is typically more tractable than [[homotopy groups]]/[[cohomotopy]] (in particular when [[homology spectra split]]), the Boardman homomorphism is often used to partially reduce computations of the latter in terms of computations of the former.

One example is the computation of the homotopy groups of [[MU]] via the [[homology of MU]] ([[Quillen's theorem on MU]]), see [below](#ForComplexOrientedCohomologyTheories).

## Examples

### In ordinary cohomology

Consider the [[unit]] morphism

$$
  \mathbb{S} \longrightarrow H \mathbb{Z}
$$

from the [[sphere spectrum]] to the [[Eilenberg-MacLane spectrum]] of the [[integers]]. For any [[topological space]]/[[spectrum]] postcomposition with this morphism induces [[Boardman homomorphisms]] of [[cohomology groups]] (in fact of [[commutative rings]])

\[
  \label{BoardmandCohomotopyToOrdinaryCohomology}
  b^n
  \;\colon\;
  \pi^n(X)
  \longrightarrow
  H^n(X, \mathbb{Z})
\]

from the [[stable cohomotopy]] of $X$ in degree $n$ to its [[ordinary cohomology]] in degree $n$.

+-- {: .num_prop #BoundsOnCoKernelOfBoardmandFromStableCohomotopyToOrdinaryCohomology}
###### Proposition
**(bounds on ([[cokernel|co-]])[[kernel]] of [[Boardman homomorphism]] from [[stable cohomotopy]] to [[integral cohomology]])**

If $X$ is a [[CW-spectrum]] which

1. is [[n-connected object of an (infinity,1)-topos|(m-1)-connected]]

1. of dimension $d \in \mathbb{N}$

then

1. the [[kernel]] of the [[Boardman homomorphism]] $b^n$ (eq:BoardmandCohomotopyToOrdinaryCohomology) for

   $$
     m \leq n\leq d -1
   $$

   is a $\overline{\rho}_{d-n}$-[[torsion subgroup|torsion group]]:

   $$
     \overline{\rho}_{d-n} ker(b^n) \;\simeq\; 0
   $$

1. the [[cokernel]] of the [[Boardman homomorphism]] $b^n$ (eq:BoardmandCohomotopyToOrdinaryCohomology) for

   $$
     m \leq n  \leq d - 2
   $$

   is a $\overline{\rho}_{d-n-1}$-[[torsion subgroup|torsion group]]:

   \[
     \label{TorsionEstimateCokernel}
     \overline{\rho}_{d-n-1} coker(b^n) \;\simeq\; 0
   \]

where

$$
  \overline{\rho}_{i}
   \;\coloneqq\;
   \left\{
   \array{
     1 &\vert& i\leq 1
     \\
     \underoverset{j = 1}{i}{\prod}
       exp\left(
         \pi_j\left( \mathbb{S}\right)
       \right)
     &\vert&
     \text{otherwise}
   }
   \right.
$$

is the [[multiplication|product]] of the [[exponent of a group|exponents]] of the [[stable homotopy groups of spheres]] in [[positive number|positive]] degree $\leq i$.

=--

([Arlettaz 04, theorem 1.2](#Arlettaz04))

+-- {: .num_example #ExampleForEstimatesOfTorsionOfCokernelOfBeta}
###### Example
**(estimates for torsion of cokernel of Boardman homomorphism)**

Let $X$ be a [[manifold]]

* of [[dimension]] $d = 6$

* [[simply connected topological space|simply connected]] with $\pi_2(X) \neq 0$

then Prop. \ref{BoundsOnCoKernelOfBoardmandFromStableCohomotopyToOrdinaryCohomology} asserts that the [[cokernel]] of the [[Boardman homomorphism]]

$$
  \beta^4
  \;\colon\;
  \mathbb{S}^4(X)
  \longrightarrow
  H^4( X, \mathbb{Z} )
$$

in

* degree $n = 4$

is [[torsion subgroup|2-torsion]]:

$$
  2 coker(\beta^4) \;=\; 0
  \,.
$$

This is because in this case (eq:TorsionEstimateCokernel) gives that the relevant torsion degree is

$$
  \begin{aligned}
    \overline{\rho}_{d-n-1}
    & =
    \overline{\rho}_{1}
    \\
    & = \exp( \pi_1(\mathbb{S}) )
    \\
    & = \exp( \mathbb{Z}/2 )
    \\
    & = 2
  \end{aligned}
  \,.
$$

Similarly, if instead the manifold has dimension $d = 7$ but sticking to degree $n = 4$, then the estimate is that the cokernel is [[torsion subgroup|4-torsion]],

$$
  4 coker(\beta^4) \;=\; 0
  \,.
$$

since then

$$
  \begin{aligned}
    \overline{\rho}_{d-n-1}
    & =
    \overline{\rho}_{2}
    \\
    & = \exp( \pi_1(\mathbb{S}) ) \cdot \exp( \pi_2(\mathbb{S}) )
    \\
    & = \exp( \mathbb{Z}/2 ) \cdot \exp( \mathbb{Z}/2 )
    \\
    & = 2 \cdot 2
    \\
    & = 4
  \end{aligned}
  \,.
$$

Next for $d = 8$ we

$$
  \begin{aligned}
    \overline{\rho}_{d-n-1}
    & =
    \overline{\rho}_{3}
    \\
    & =
      \exp( \pi_1(\mathbb{S}) )
      \cdot \exp( \pi_2(\mathbb{S}) )
      \cdot \exp( \pi_3(\mathbb{S}) )
    \\
    & = \exp( \mathbb{Z}/2 )
      \cdot \exp( \mathbb{Z}/2 )
      \cdot \exp( \mathbb{Z}/{24} )
    \\
    & = 2 \cdot 2 \cdot 6
    \\
    & = 24
  \end{aligned}
  \,.
$$

=--

+-- {: .num_prop #BoardmanIsoOn7SphereMod2I}
###### Proposition
**([[Boardman homomorphism|Boardman]] [[isomorphism]] on [[2-sphere]] mod [[binary icosahedral group]])

Consider the  [[binary icosahedral group]] $2 I$ and its [[action]] on the [[7-sphere]] induced via the identification $S^7 \simeq S(\mathbb{H} \times \mathbb{H})$ from the [[diagonal action|diagonal]] of the canonical action of $2I$ on the [[quaternions]] $\mathbb{H}$ induced via it being a [[finite subgroup of SU(2)]].

On the [[quotient space]] $S^7/2 I$ the [[Boardman homomorphism]] in degree 4 is an [[isomorphism]]

$$
  \mathbb{S}^4\left(
     S^7/2I
  \right)
  \underoverset{\simeq}{\beta}{\longrightarrow}
  H^4\left( S^7/2I , \mathbb{Z} \right)
$$

from [[stable cohomotopy]] in degree 4 to [[integral cohomology]] in degree 4.

=--

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

In terms of the [[Atiyah-Hirzebruch spectral sequence]] for [[stable cohomotopy]] it is sufficient to see that the two differentials

$$
  H^4\left(
    S^7/2I, \pi^0_s = \pi^s_0 =\mathbb{Z}
  \right)
  \overset{d_3}{\longrightarrow}
  H^6\left(
    S^7/2I, \pi^{-1}_s = \pi^s_{1} =\mathbb{Z}/2
  \right)
$$

and

$$
  H^4\left(
    S^7/2I, \pi^0_s = \pi^s_0 =\mathbb{Z}
  \right)
  \overset{d_3}{\longrightarrow}
  H^7\left(
    S^7/2I, \pi^{-2}_s = \pi^s_{2} =\mathbb{Z}/2
  \right)
$$

both vanish (all higher differentials on $H^4(-,\pi^0_s)$ vanish simply for dimensional reasons as $S^7$ is of [[dimension]] 7, while there are no differentials into $H^4(-,\pi^0_s)$ simply because the [[sphere spectrum]] is [[connective spectrum|connective]], so that the [[stable homotopy groups of spheres]] vanish in [[negative number|negative]] degree).

For $d_2$ to vanish, it is sufficient that

$$
  H^6\left(
    S^7/2I, \pi^{-1}_s = \pi^s_{1} =\mathbb{Z}/2
  \right)
  \;\simeq\;
  0
$$

We now first show that this is the case:

First, by the [[Gysin sequence]] for the [[spherical fibration]]

$$
  \array{
    S^7 &\longrightarrow& S^7/SI
    \\
    && \downarrow
    \\
    && B (2 I)
  }
$$

we have

$$
  H^6\left(
    S^7/2I, \, \mathbb{Z}/2
  \right)
  \;\simeq\;
  H^6\left(
    B(2I),\, \mathbb{Z}/2
  \right)
  \,,
$$

where $B (2 I) \simeq \ast \sslash (2I)$ is the [[classifying space]] of $2I$ (see e.g. at [[infinity-action]]).

Moreover, by the [[universal coefficient theorem]] ([this Prop.](universal+coefficient+theorem#OrdinaryStatementInTopology)) we have a [[short exact sequence]]

$$
  0
    \to
  Ext^1(H_{5}\big(B(2I), \mathbb{Z}), \mathbb{Z}/2\big)
    \longrightarrow
  H^6\big(B(2I), \mathbb{Z}/2\big)
    \longrightarrow
  Hom_{Ab}\big( H_6( B(2I), \mathbb{Z}) , \mathbb{Z}/2 \big)
    \to
  0
  \,.
$$

This means that it is sufficient to see that

$$
  H_{5}\big(B(2I), \mathbb{Z}) \simeq 0
  \phantom{AAA}
  H_{6}\big(B(2I), \mathbb{Z}) \simeq 0
$$

But for every [[finite subgroup of SU(2)]] $G_{ADE} \subset SU(2)$ we have (by [this Prop.](https://ncatlab.org/nlab/show/finite+rotation+group#GroupCohomologyOfFiniteSubgroupsOfSU2))

$$
  H_{5}\big(B(2I), \mathbb{Z}) \simeq G^{ab}_{ADE}
  \phantom{AAA}
  H_{6}\big(B(2I), \mathbb{Z}) \simeq 0
$$

where $G^{ab}_{ADE}$ is the [[abelianization]] of $G_{ADE}$. Specifically for $G_{ADE} = 2I$ this does vanish: the [[binary icosahedral group]] is a [[perfect group]] ([this Prop.](icosahedral+group#2IIsPerfect)).

This shows that $d_2$ vanishes on $H^4(-, \pi^0)$.

Now by a standard argument, the AHSS-differentials between ordinary cohomology groups are stable [[cohomology operations]], and thus, if non-trivial, must be the  [[Steenrod operations]] $Sq^n$ (e.g. [here](https://mathgroove.wordpress.com/2016/11/30/filling-the-details-5-two-words-about-the-atiyah-hirzebruch-spectral-sequence/), but let's add a more canonical reference).

This means first of all that if $d_2$ is not trivial then $d_2 = Sq^2$. But since that vanishes on $H^4(-,\pi^0)$ by the above argument, and on $H^7(-,\pi^2)$ for dimension reasons, so that the relevant entries pass as ordinary cohomology groups to the third page of the [[spectral sequence]], it follows similarly that  $d_3 = Sq^3$.

But by the [[Adem relation]] $Sq^3 = Sq^1 \circ Sq^2$, the vanishing of $Sq_2$ on $H^4(-,\pi^0)$ then also implies the vanishing of $d_3$ on this entry.

=--

### In complex oriented cohomology
 {#ForComplexOrientedCohomologyTheories}

Used for [[complex oriented cohomology theories]] and proof of [[Quillen's theorem on MU]] via the [[homology of MU]] (...)

([Adams 74, pages 60-62](#Adams74), [Lurie 10, lecture 7](#Lurie10))

### In topological modular forms

See _[[Boardman homomorphism in tmf]]_

## Related concepts

* [[Hopf degree theorem]]

## References

According to [Hunton 95](#Hunton95), the concept was introduced, in print, in:

* [[John Michael Boardman]], _The eightfold way to BP-operations_, in: _Current trends in algebraic topology_, pp. 187–226, Canadian Mathematical Society Proceedings, 2, Part 1.  Providence 1982 ([ISBN:978-0-8218-6003-8](https://bookstore.ams.org/cmsams-2))

Further discussion:

* {#Adams74} [[John Frank Adams]], part II, section 6 of: _[[Stable homotopy and generalised homology]]_ (1974)

* {#Hunton95} John Hunton, _The Boardman homomorphism_, Contemporary Mathematics 181, 251-251, 1995 ([GoogleBooks](https://books.google.ae/books?hl=en&lr=&id=hcYaCAAAQBAJ&oi=fnd&pg=PA251&dq=John+Hunton,+The+Boardman+homomorphism&ots=_ZI5SEcU28&sig=cK1Xd7AZwZP7WeTiy4dGv5v9BJk&redir_esc=y#v=onepage&q=John%20Hunton%2C%20The%20Boardman%20homomorphism&f=false))

* {#Kochmann96} [[Stanley Kochmann]], section 4.3 of: _[[Bordism, Stable Homotopy and Adams Spectral Sequences]]_, AMS 1996

* {#Arlettaz04} [[Dominique Arlettaz]], _The generalized Boardman homomorphisms_, Central European Journal of Mathematics March 2004, Volume 2, Issue 1, pp 50-56 ([doi:10.2478/BF02475949](https://doi.org/10.2478/BF02475949))

* {#Lurie10} [[Jacob Lurie]], lecture 7 of _[[Chromatic Homotopy Theory]]_, 2010, ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture7.pdf))

* [[Akhil Mathew]], _Torsion exponents in stable homotopy and the Hurewicz homomorphism_, Algebr. Geom. Topol. 16 (2016) 1025-1041 ([arXiv:1501.07561](https://arxiv.org/abs/1501.07561))

* Hadi Zare, _On the image of the unstable Boardman map_ ([arXiv:1806.07079](https://arxiv.org/abs/1806.07079))

On the [[Boardman homomorphism]] (generalized [[Hurewicz homomorphism]]) to [[tmf]]:

* [[Mark Behrens]], [[Mark Mahowald]], J.D. Quigley, _The 2-primary Hurewicz image of $tmf$_ ([arXiv:2011.08956](https://arxiv.org/abs/2011.08956))

[[!redirects Boardman homomorphisms]]


Boardman homomorphism in tmf

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

#Contents#
* table of contents
{:toc}

## Definition

Write $\mathbb{S}$ for the [[sphere spectrum]] and _[[tmf]]_ for the [[connective spectrum]] of [[topological modular forms]].

Since [[tmf]] is an [[E-∞ ring|E-∞]][[ring spectrum]], there is an essentially unique homomorphism of [[E-∞ ring|E-∞]][[ring spectra]]

$$
  \mathbb{S}
  \overset{e_{tmf}}{\longrightarrow}
  tmf
  \,.
$$

Regarded as a morphism of [[generalized homology]]-theories, this is called the [[Hurewicz homomorphism]], or rather the [[Boardman homomorphism]] for $tmf$

## Properties

+-- {: .num_prop}
###### Proposition
**(Boardman homomorphism in $tmf$ is 6-connected)**

The [[Boardman homomorphism in tmf]]

$$
  \mathbb{S}
  \overset{e_{tmf}}{\longrightarrow}
  tmf
$$

induces an [[isomorphism]] on [[stable homotopy groups]]
(hence from the [[stable homotopy groups of spheres]] to the stable homotopy groups of tmf), up to degree 6:

$$
  \pi_{\bullet \leq 6}(\mathbb{S})
  \underoverset{\simeq}{\pi_{\bullet \leq 6}(e_{tmf})}{\longrightarrow}
  \pi_{\bullet\leq 6}(tmf)
  \,.
$$

=--

([Hopkins 02, Prop. 4.6](#Hopkins02), [DFHH 14, Ch. 13](#DFHH14))

## Related concepts

* [[stable cohomotopy]]

## References

* {#Hopkins02} [[Michael Hopkins]], section 4 of _Algebraic topology and modular forms_, Proceedings of the ICM, Beijing 2002, vol. 1, 283--309 ([arXiv:math/0212397](http://arxiv.org/abs/math/0212397))

* {#DFHH14} [[Christopher Douglas]], [[John Francis]], [[André Henriques]], [[Michael Hill]], Chapter 13 of: _Topological Modular Forms_, Mathematical Surveys and Monographs Volume 201, AMS 2014  ([ISBN:978-1-4704-1884-7](https://bookstore.ams.org/surv-201))


Boardman-Vogt resolution

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

#Context#
* table of contents
{:toc}

## Idea

A _Boardman-Vogt resolution_ or _W-construction_ is a particular choice of cofibrant [[resolution]] of [[topological operads]] (or [[simplicial operads]] or similar). It is closely related to the operation of forming the [[dendroidal homotopy coherent nerve]]. Its restriction to [[Top]]-[[enriched categories]] / [[simplicial categories]] is accordingly closely related to the ordinary [[homotopy coherent nerve]].

The construction is a variant of the _[[free operad]]_ construction, where the free composites of operations are further labeled by "distances" such that for vanishing distance the free composite is replaced with the actual composite.

The original Boardman-Vogt resolution over [[Top]] generalizes naturally to a cofibrant [[resolution]] in the [[model structure on operads]] for operads enriched over any suitable [[monoidal model category]] that is equipped with a suitable [[comonoid|comonoidal]] [[interval object]]. In this general form the construction subsumes

* the W-construction on topological operads ([BoardmanVogt](#BoardmanVogt));

* the cobar-bar resolution of chain complex operads ([GezlerJones](#GetzlerJones), [GinzburgKapranov](#GinzburgKapranov));

* the Godement simplicial resolution ([Godement](#Godement))

## Definition  {#Definition}

### Over $Top$

Over [[Top]] the BV-resolution works as follows:

for $P$ a [[topological operad]], its [[free operad]] $F_*(P)$ has as $n$-ary operations all [[trees]] with $n$ inputs, with each vertex of valence $k+1$ labeled by an element in $P(k)$. Composition is given by grafting of trees.

The operad $W(P)$ is obtained from this by in addition

* labeling the inner edges of any tree by [[real number]]s $\ell \in [0,1]$;

* identifying trees one of whose edges has length 0 with the tree with that edge removed and with the correspnding operad-operation labels composed.

* equipping the set of labeled trees with the corresponding quotient of the product topology, to make it into a [[topological space]].

There is an evident operad morphism $F_*(P) \to W(P)$ obtained by regarding each edge of a tree as being of length 1, and there is an evident morphism $W(P) \to P$ obtained by forgetting all trees and sending their operad-operation-labels to their composite.

The composition

$$
  F_*(P) \to W(P) \to P
$$

is the [[unit of an adjunction|counit]] of the [[free functor|free]]/forgetful [[adjunction]] between operads and their underlying [[collection]]s and if $P$ is degreewise sufficiently nice, this factors that counit as a cofibration followed by a weak equivalence and exhibits $W(P)$ as a cofibrant [[resolution]] of $P$.

### General

Write $\mathbb{T}$ for the [[groupoid]] of planar [[trees]] and non-planar [[isomorphism]].

Fix a suitable [[interval object]] $H$, as described at [[model structure on operads]].

For $T$ a [[tree]], write

$$
  H(T) := \bigotimes_{e \in E(T)}H
  \,,
$$

where the tensor product runs over all _internal_ edges of $T$. For $D \subset E(T)$ a subset of internal edges, let

$$
  H_D(T) = \bigotimes_{ E(T)\setminus D} H
  \,.
$$

The acyclic cofibration $0 \to H$ induces an acyclic cofibration

$$
  H_D(T) \hookrightarrow H(T)
$$

and, by the [[pushout-product axiom]], an acyclic cofibration

$$
  H^-(T) := \coprod_{D \neq \emptyset} H_D(T) \hookrightarrow H(T)
  \,.
$$

In a similar fashion, for $P$ an operad, write $P(T)$ for the tensor product of one copy of its objects of $n$-ary operation for each $n$-ary vertex in $T$, and $P^-(T)$ for the coproduct over all such tensor products where at least one, maybe more, _unary_ vertices are omitted. Also the canonical

$$
  P^-(T) \hookrightarrow P(T)
$$

is a cofibration. Consider for each $T$ the [[pushout]]

$$
  \array{
    H^-(T) \otimes P^-(T) &\to& H^-(T) \otimes P(T)
    \\
    \downarrow && \downarrow
    \\
    H(T) \otimes P^-(T) &\to&
    (H \otimes P)^-(T)
  }
  \,.
$$

This induces a univesal morphism

$$
  (H \otimes P)^-(T) \to H(T) \otimes P(T)
$$

and by the [[pushout-product axiom]] in the [[monoidal model category]] $\mathcal{E}$ this, too, is a cofibration.

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

Define $W(H,P)$ by induction. Start with setting

$$
  W_0(H,P) := P
  \,.
$$

Assume that in each induction step we are given morphisms

$$
  (H(S) \otimes P(S)) \otimes_{Aut(S)} I[\Sigma_n] \to W_{k-1}(H,P)(n)
$$

for all trees $S$ with less than $k$ internal edges. Using the  composition operation in the operad $P$ to compose two operation when the edge connecting them carries no $H$-label, we obtain from this a morphism

$$
  \alpha_T^- : (H \otimes P)^-(T) \otimes_{Aut(T)} I[\Sigma_n] \to W_{k-1}(H,P)(n)
$$

Then in the induction step we define for each $k \in \mathbb{N}$ the object $W_n(H,P)$ by the [[pushout]]

$$
  \array{
     \coprod_{[T], T \in \mathbb{T}(n,k)}
      (H \otimes P)^-(T)
      \otimes_{Aut(T)}
       I[\Sigma_n]
     &\stackrel{\coprod \alpha_T^-}{\to}&
     W_{k-1}(H,P)(n)
     \\
     \downarrow && \downarrow
     \\
     \coprod_{[T], T \in \mathbb{T}(n,k)}
      (H \otimes P)(T)
      \otimes_{Aut(T)}
       I[\Sigma_n]
     &\stackrel{\coprod \alpha_T^-}{\to}&
     W_{k}(H,P)(n)
  }
 \,,
$$

where $\mathbb{T}$ is the subcategory of trees with precisely $n$ inputs and $k$ internal edges.

The bottom morphism we feed back into the induction procedure.

This gives a sequence of collections, and the W-resolution is its [[colimit]]

$$
  W(H,P) := {\lim_{\to}}_k W_k(H,P)
  \,.
$$

=--

One shows that this collection naturally carries the structure of an operad, etc. pp.

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

The object $(H \otimes P)(T)$ is to be thought of as the space whose points are tuples consisting of one operation in $P$ per vertex in $T$, of that arity, and of labels in $H$ assigned to the inner edges in $T$.

The object $(H \otimes P)^-(T)$ is a similar space, but where some of the labels on the inner edges are omitted.

The above pushout identifies points that contain lables of inner edges that are 0 with points in one $W$-stratum below where that edge (or rather its label) is simply omitted and the corresponding operations composed.

=--

## Examples

* [[Jim Stasheff]]'s [[A-∞ operad]] is the relative Boardman-Vogt resolution $W([0,1], I_* \to Assoc)$ of [[Assoc]] in [[Top]] where $I_*$ is the operad for [[pointed object]]s ([Berger-Moerdijk](#BergerMoerdijkAlgebras)).

## References

The W-construction on topological operads is in

* {#BoardmanVogt} [[Michael Boardman]] and [[Rainer Vogt]], _Homotopy invariant algebraic structures on topological spaces_ , Lect. Notes Math. 347 (1973).

The cobar-bar resolution of chain complex operads is in

* {#GetzlerJones} [[Ezra Getzler]], J.D.S. Jones, _Operads, homotopy algebra, and iterated integrals for double loop spaces_ , (1995)

* {#GinzburgKapranov} [[Victor Ginzburg]], [[Mikhail Kapranov]], _Koszul duality for operads_ , Duke Math. J. 76 (1994) 203–272.

The Godement simplicial resolution is in

* {#Godement} R. Godement, _Topologie algébrique et théorie des faisceaux_ , no. 13, Publ. Math. Univ. Strasbourg, Hermann, Paris, 1958.

The generalization to [[operad]]s enriched in any monoidal category with a suitable interval object is in

*{#BergerMoerdijkResolution}  [[Clemens Berger]], [[Ieke Moerdijk]], _The Boardman-Vogt resolution of operads in monoidal model categories_ , Topology 45 (2006), 807–849. ([pdf](http://math.unice.fr/~cberger/BV.pdf))

* {#BergerMoerdijkAlgebras} [[Clemens Berger]], [[Ieke Moerdijk]], _Resolution of coloured operads and rectification of homotopy algebras_ ([arXiv:math/0512576](http://arxiv.org/abs/math/0512576))

The Boardman-Vogt-resolution for the operad for [[local nets of observables]] in [[homotopical algebraic quantum field theory]] ([Benini-Schenkel-Woike 17](homotopical+algebraic+quantum+field+theory#BeniniSchenkelWoike17)) is discussed in

* {#Yau18} [[Donald Yau]], _Homotopical Quantum Field Theory_ ([arXiv:1802.08101](https://arxiv.org/abs/1802.08101))

[[!redirects Boardman-Vogt resolutions]]

[[!redirects W-construction]]
[[!redirects W-constructions]]

[[!redirects relative Boardman-Vogt resolution]]

[[!redirects relative Boardman-Vogt resolutions]]