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group cohomology, nonabelian group cohomology, Lie group cohomology
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cobordism theory, Introduction
quaternionic projective space$\,\mathbb{H}P^1$
Cohomotopy cohomology theory $\pi^\bullet$ is the (non-abelian) generalized cohomology theory whose cocycle spaces are spaces of maps into an n-sphere, hence whose cohomology classes are homotopy classes of maps into an n-sphere:
So, dually to how homotopy groups
are groups of homotopy classes of maps out of spheres, Cohomotopy sets are sets of homotopy classes of maps into spheres, whence the dual name.
If instead one considers only the stable aspect of Cohomotopy sets, by mapping into the stabilization of the spheres, hence into (some suspension of) the sphere spectrum, then one speaks of stable Cohomotopy, written
In other words, the generalized (Eilenberg-Steenrod) cohomology theory which is represented by the sphere spectrum is stable Cohomotopy.
(terminology)
Therefore “Cohomotopy theory” is really shorthand for “Cohomotopy cohomology theory” and as such is dual to homotopy homology theory, which in the stable case is known as stable homotopy homology theory.
In particular, cohomotopy theory is a concrete particular and not dual to the abstract general of homotopy theory; and is hence also not on par with the abstract general of cohomology theory. Rather, Cohomotopy theory is one instance of a cohomology theory, and as such is a sibling of ordinary cohomology theory (HR-theory)), K-theory, etc.
To emphasize this, one might, in the stable case, say $\mathbb{S}$-theory instead of “stable Cohomotopy theory”; where $\mathbb{S}$ denotes the sphere spectrum. In the unstable case there is no widely adopted notation, but one might consider saying “$\mathbf{\pi}$-theory” (with $\pi$ the established symbol for (co)homotopy groups) or $S$-theory (with “$S$” for n-spheres $S^n$) for unstable Cohomotopy theory.
In any case, to highlight that Cohomotopy theory is a concrete particular and not an abstract general, it makes good sense to capitalize the term and speak of Cohomotopy cohomology theory or just Cohomotopy theory, for short.
The following table indicates the pattern:
As for any generalized cohomology theory there are immediate variants to plain Cohomotopy theory, as shown in the following table:
flavours of Cohomotopy cohomology theory | cohomology (full or rational) | equivariant cohomology (full or rational) |
---|---|---|
non-abelian cohomology | Cohomotopy (full or rational) | equivariant Cohomotopy |
twisted cohomology (full or rational) | twisted Cohomotopy | twisted equivariant Cohomotopy |
stable cohomology (full or rational) | stable Cohomotopy | equivariant stable Cohomotopy |
In some sense Cohomotopy is the most fundamental of all generalized cohomology theories (“The Music of the Spheres”).
Concretely, stable Cohomotopy cohomology theory is the initial object among multiplicative cohomology theories, in that the sphere spectrum is the initial object in (E-infinity) ring spectra. This means that for any other multiplicative cohomology theory $E$ there is an essentially unique multiplicative natural transformation
from Cohomotopy cohomology groups to $E$-cohomology groups – the Boardman homomorphism.
Specifically for $E = K \mathbb{F}$ the algebraic K-theory of a field $\mathbb{F}$ (such as a prime field $\mathbb{F}_p$) there is such a comparison morphism; and another way how stable Cohomotopy is the most fundamental of all K-theories is that it is equivalently the algebraic K-theory over the “absolute base”, namely over the “field with one element” $\mathbb{F}_1$ (see there for more):
For example, for $E =$ KU and in the case of $G$-equivariant cohomology theory (equivariant Cohomotopy theory and equivariant K-theory) the Boardman homomorphism (1) gives the comparison map
from the Burnside ring to the representation ring of the finite group $G$, by forming permutation representations; where we may think of the Burnside ring as being the representation ring over the “field with one element” (see e.g. Chu-Lorscheid-Santhanam 10), as indicated above.
So far, this applies to stable Cohomotopy theory, which historically has received almost all the attention. But, while stabilization makes the immensely rich nature of homotopy theory a tad more tractable, it is only an approximation (just the first Goodwillie derivative!) of full unstable/non-abelian cohomology. Hence the one concept more fundamental than stable Cohomotopy theory is actual Cohomotopy theory.
For example, the classification of Yang-Mills instantons on $\mathbb{R}^4$ is typically regarded in the non-abelian cohomology theory represented by the classifying space $B SU(N)$ of the special unitary group (for $N \geq 2$, starting with SU(2))
But since the one-point compactification of 4d Euclidean space is the 4-sphere $\big( \mathbb{R}^4\big)^{cpt} \simeq S^4$, this classification factors through one in unstable Cohomotopy theory, via the “unstable Boardman homomorphism” $S^4 \longrightarrow B SU(N)$ representing the generator of the 4th homotopy group of $B SU(N)$ (see there)
(see SS 19, p. 9-10) This is the tip of an iceberg. Which needs to be discussed elsewhere.
Let $n \in \mathbb{N}$ be a natural number and $X \in Mfd$ be a connected orientable closed manifold of dimension $n$. Then the $n$th cohomotopy classes $\left[X \overset{c}{\to} S^n\right] \in \pi^n(X)$ of $X$ are in bijection to the degree $deg(c) \in \mathbb{Z}$ of the representing functions, hence the canonical function
from $n$th cohomotopy to $n$th integral cohomology is a bijection.
(e.g. Kosinski 93, IX (5.8))
relation to the Freudenthal suspension theorem (Spanier 49, section 9)
For $X$ a compact smooth manifold, there is a smooth function $X \to S^n$ representing every cohomotopy class (with respect to the standard smooth structure on the sphere manifold).
For $X$ a closed smooth manifold of dimension $D$, the assignment of Cohomotopy charge (the Pontryagin-Thom construction, in this unstable and framed version due to Pontrjagin 55, see review in Kosinski 93, IX.5, Milnor 97) identifies the set
of cobordism classes of closed and normally framed submanifolds $\Sigma \overset{\iota}{\hookrightarrow} X$ of dimension $d$ inside $X$ with the cohomotopy $\pi^{D-d}(X)$ of $X$ in degree $D- d$
(e.g. Kosinski 93, IX Theorem (5.5))
In particular, by this bijection the canonical group structure on cobordism groups in sufficiently high codimension (essentially given by disjoint union of submanifolds) this way induces a group structure on the cohomotopy sets in sufficiently high degree.
$\,$
graphics grabbed form Sati-Schreiber 19
Here the normal framing of the submanifolds plays the role of the charge in Cohomotopy which they carry:
$\,$
graphics grabbed form Sati-Schreiber 19
For example:
$\,$
graphics grabbed form Sati-Schreiber 19
This construction generalizes to equivariant cohomotopy, see there.
With the equivariant Hopf degree theorem the above example has the following $\mathbb{Z}_2$-equivariant version (see there):
$\,$
graphics grabbed form Sati-Schreiber 19
Further by the equivariant Hopf degree theorem (see there), this example generalizes to equivariant cohomotopy of toroidal orientifolds:
$\,$
graphics grabbed form Sati-Schreiber 19
Under the Pontryagin-Thom isomorphism (above) the product in Cohomotopy, i.e. the composition operation
corresponds to forming product manifolds of submanifolds, in that (see also Kosinski 93, Section IX 6.1):
This is exhibited by the following pasting diagram of pullbacks/fiber products (repeatedly using the pasting law):
Here the vertical inclusions are the defining ones of the submanifolds or of their normal bundles, identified with some choice of tubular neighbourhoods.
The Cohomotopy charge map is the function that assigns to a configuration of points their total charge as measured in Cohomotopy-cohomology theory.
This is alternatively known as the “electric field map” (Salvatore 01 following Segal 73, Section 1, see also Knudsen 18, p. 49) or the “scanning map” (Kallel 98).
For $D \in \mathbb{N}$ the Cohomotopy charge map is the continuous function
from the configuration space of points in the Euclidean space $\mathbb{R}^D$ to the $D$-Cohomotopy cocycle space vanishing at infinity on the Euclidean space(which is equivalently the space of pointed maps from the one-point compactification $S^D \simeq \big( \mathbb{R}^D \big)$ to itself, and hence equivalently the $D$-fold iterated based loop space of the D-sphere), which sends a configuration of points in $\mathbb{R}^D$, each regarded as carrying unit charge to their total charge as measured in Cohomotopy-cohomology theory (Segal 73, Section 3).
This has evident generalizations to other manifolds than just Euclidean spaces, to spaces of labeleed configurations and to equivariant Cohomotopy. The following graphics illustrates the Cohomotopy charge map on G-space tori for $G = \mathbb{Z}_2$ with values in $\mathbb{Z}_2$-equivariant Cohomotopy:
graphics grabbed from SS 19
In some situations the Cohomotopy charge map is a weak homotopy equivalence and hence exhibits, for all purposes of homotopy theory, the Cohomotopy cocycle space of Cohomotopy charges as an equivalent reflection of the configuration space of points.
(group completion on configuration space of points is iterated based loop space)
from the full unordered and unlabeled configuration space (here) of Euclidean space $\mathbb{R}^D$ to the $D$-fold iterated based loop space of the D-sphere, exhibits the group completion (here) of the configuration space monoid
(Cohomotopy charge map is weak homotopy equivalence on sphere-labeled configuration space of points)
For $D, k \in \mathbb{N}$ with $k \geq 1$, the Cohomotopy charge map (2)
is a weak homotopy equivalence from the configuration space (here) of unordered points with labels in $S^k$ and vanishing at the base point of the label space to the $D$-fold iterated loop space of the D+k-sphere.
The May-Segal theorem generalizes from Euclidean space to closed smooth manifolds if at the same time one passes from plain Cohomotopy to twisted Cohomotopy, twisted, via the J-homomorphism, by the tangent bundle:
Let
$X^n$ be a smooth closed manifold of dimension $n$;
$1 \leq k \in \mathbb{N}$ a positive natural number.
Then the Cohomotopy charge map constitutes a weak homotopy equivalence
between
the J-twisted (n+k)-Cohomotopy space of $X^n$, hence the space of sections of the $(n + k)$-spherical fibration over $X$ which is associated via the tangent bundle by the O(n)-action on $S^{n+k} = S(\mathbb{R}^{n} \times \mathbb{R}^{k+1})$
the configuration space of points on $X^n$ with labels in $S^k$.
(Bödigheimer 87, Prop. 2, following McDuff 75)
In the special case that the closed manifold $X^n$ in Prop. is parallelizable, hence that its tangent bundle is trivializable, the statement of Prop. reduces to this:
Let
$X^n$ be a parallelizable closed manifold of dimension $n$;
$1 \leq k \in \mathbb{N}$ a positive natural number.
Then the Cohomotopy charge map constitutes a weak homotopy equivalence
between
$(n+k)$-Cohomotopy space of $X^n$, hence the space of maps from $X$ to the (n+k)-sphere
the configuration space of points on $X^n$ with labels in $S^k$.
The assignment of scattering amplitudes of monopoles in SU(2)-Yang-Mills theory is a diffeomorphism
identifying the moduli space of monopoles of number $k$ with the space of complex-rational functions form the Riemann sphere to itself, of degree $k$ (hence the cocycle space of complex-rational 2-Cohomotopy).
(Atiyah-Hitchin 88, Theorem 2.10)
$\,$
This is a non-abelian analog of the Dirac charge quantization of the electromagnetic field, with ordinary cohomology replaced by Cohomotopy cohomology theory.
Cohomotopy sets of projective spaces are computed in West 71
Let $X$ be a 4-manifold which is connected and oriented.
The Pontryagin-Thom construction as above gives for $n \in \mathbb{Z}$ the commuting diagram of sets
where $\pi^\bullet$ denotes cohomotopy sets, $H^\bullet$ denotes ordinary cohomology, $H_\bullet$ denotes ordinary homology and $\mathbb{F}_\bullet$ is normally framed cobordism classes of normally framed submanifolds. Finally $h^n$ is the operation of pullback of the generating integral cohomology class on $S^n$ (by the nature of Eilenberg-MacLane spaces):
Now
$h^0$, $h^1$, $h^4$ are isomorphisms
$h^3$ is an isomorphism if $X$ is “odd” in that it contains at least one closed oriented surface of odd self-intersection, otherwise $h^3$ becomes an isomorphism on a $\mathbb{Z}/2$-quotient group of $\pi^3(X)$ (which is a group via the group-structure of the 3-sphere (SU(2)))
flavours of Cohomotopy cohomology theory | cohomology (full or rational) | equivariant cohomology (full or rational) |
---|---|---|
non-abelian cohomology | Cohomotopy (full or rational) | equivariant Cohomotopy |
twisted cohomology (full or rational) | twisted Cohomotopy | twisted equivariant Cohomotopy |
stable cohomology (full or rational) | stable Cohomotopy | equivariant stable Cohomotopy |
(equivariant) cohomology | representing spectrum | equivariant cohomology of the point $\ast$ | cohomology of classifying space $B G$ |
---|---|---|---|
(equivariant) ordinary cohomology | HZ | Borel equivariance $H^\bullet_G(\ast) \simeq H^\bullet(B G, \mathbb{Z})$ | |
(equivariant) complex K-theory | KU | representation ring $KU_G(\ast) \simeq R_{\mathbb{C}}(G)$ | Atiyah-Segal completion theorem $R(G) \simeq KU_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {KU_G(\ast)} \simeq KU(B G)$ |
(equivariant) complex cobordism cohomology | MU | $MU_G(\ast)$ | completion theorem for complex cobordism cohomology $MU_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {MU_G(\ast)} \simeq MU(B G)$ |
(equivariant) algebraic K-theory | $K \mathbb{F}_p$ | representation ring $(K \mathbb{F}_p)_G(\ast) \simeq R_p(G)$ | Rector completion theorem $R_{\mathbb{F}_p}(G) \simeq K (\mathbb{F}_p)_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {(K \mathbb{F}_p)_G(\ast)} \!\! \overset{\text{<a href="https://ncatlab.org/nlab/show/Rector+completion+theorem">Rector 73</a>}}{\simeq} \!\!\!\!\!\! K \mathbb{F}_p(B G)$ |
(equivariant) stable cohomotopy | $K \mathbb{F}_1 \overset{\text{<a href="stable cohomotopy#StableCohomotopyIsAlgebraicKTheoryOverFieldWithOneElement">Segal 74</a>}}{\simeq}$ S | Burnside ring $\mathbb{S}_G(\ast) \simeq A(G)$ | Segal-Carlsson completion theorem $A(G) \overset{\text{<a href="https://ncatlab.org/nlab/show/Burnside+ring+is+equivariant+stable+cohomotopy+of+the+point">Segal 71</a>}}{\simeq} \mathbb{S}_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {\mathbb{S}_G(\ast)} \!\! \overset{\text{<a href="https://ncatlab.org/nlab/show/Segal-Carlsson+completion+theorem">Carlsson 84</a>}}{\simeq} \!\!\!\!\!\! \mathbb{S}(B G)$ |
Original articles:
K. Borsuk, Sur les groupes des classes de transformations continues, CR Acad. Sci. Paris 202.1400-1403 (1936): 2 (doi:10.24033/asens.603)
Edwin Spanier, Borsuk’s Cohomotopy Groups, Annals of Mathematics Second Series, Vol. 50, No. 1 (Jan., 1949), pp. 203-245 (jstor:1969362)
Franklin P. Peterson, Some Results on Cohomotopy Groups, American Journal of Mathematics Vol. 78, No. 2 (Apr., 1956), pp. 243-258 (jstor:2372514)
Jan Jaworowski, Generalized cohomotopy groups as limit groups, Fundamenta Mathematicae 50 (1962), 393-402 (doi:10.4064/fm-50-4-333-340, pdf)
See also
Wikipedia, Cohomotopy group
Further discussion of Cohomotopy:
The unstable and framed version of the Pontrjagin-Thom construction, namely the original Pontryagin construction, identifying cobordism classes of normally framed submanifolds with their Cohomotopy charge in unstable Cohomotopy sets, is due to:
Lev Pontrjagin, Classification of continuous maps of a complex into a sphere, Dokl. Akad. Nauk SSSR 19 (1938), 361-363
Lev Pontrjagin, Smooth manifolds and their applications in Homotopy theory, Trudy Mat. Inst. im Steklov, No 45, Izdat. Akad. Nauk. USSR, Moscow, 1955 (AMS Translation Series 2, Vol. 11, 1959) (doi:10.1142/9789812772107_0001, pdf)
Textbook accounts:
Antoni Kosinski, chapter IX of: Differential manifolds, Academic Press 1993 (pdf, ISBN:978-0-12-421850-5)
John Milnor, Chapter 7 of: Topology from the differential viewpoint, Princeton University Press, 1997. (ISBN:9780691048338, pdf)
Also:
Discussion of the early history:
Lecture notes:
The stable and structured version of Pontryagin’s construction – the Pontryagin-Thom collapse – relating abstract (instead of embedded) cobordism classes to homotopy classes of maps to Thom spectra:
Textbook accounts:
Robert Stong, Notes on Cobordism theory, 1968 (toc pdf, publisher page)
Stanley Kochman, section 1.5 of: Bordism, Stable Homotopy and Adams Spectral Sequences, AMS 1996
Yuli Rudyak, On Thom spectra, Orientability and Cobordism, Springer 1998 (pdf)
Lecture notes:
John Francis, Topology of manifolds course notes (2010) (web), Lecture 3: Thom’s theorem (pdf), Lecture 4 Transversality (notes by I. Bobkova) (pdf)
Cary Malkiewich, Section 3 of: Unoriented cobordism and $M O$, 2011 (pdf)
Tom Weston, Part I of An introduction to cobordism theory (pdf)
See also:
Cohomotopy sets of projective spaces:
Cohomotopy sets of 4-manifolds:
Nobuo Shimada, Homotopy classification of mappings of a 4-dimensional complex into a 2-dimensional sphere, Nagoya Math. J., Volume 5 (1953), 127-144 (euclid:nmj/1118799399)
Daniel Freed, Karen Uhlenbeck, Appendix B of: Instantons and Four-Manifolds, Mathematical Sciences Research Institute Publications, Springer 1991 (doi:10.1007/978-1-4613-9703-8)
Robion Kirby, Paul Melvin, Peter Teichner, Cohomotopy sets of 4-manifolds, Geometry & Topology Monographs 18 (2012) 161–190 (arXiv:1203.1608, doi:10.2140/gtm.2012.18.161)
Cohomotopy sets of Thom spaces:
In relation to quaternionic line bundles over 5-manifolds:
Cohomotopy sets of spin-manifolds in co-degree 1:
Discussion of Cohomotopy cocycle spaces (i.e. spaces of maps into an n-sphere):
Vagn Lundsgaard Hansen, The homotopy problem for the components in the space of maps on the $n$-sphere, Quart. J. Math. Oxford Ser. (3) 25 (1974), 313-321 (DOI:10.1093/qmath/25.1.313)
Vagn Lundsgaard Hansen, On Spaces of Maps of $n$-Manifolds Into the $n$-Sphere, Transactions of the American Mathematical Society
Vol. 265, No. 1 (May, 1981), pp. 273-281 (jstor:1998494)
Victor Vassiliev, Twisted homology of configuration spaces, homology of spaces of equivariant maps, and stable homology of spaces of non-resultant systems of real homogeneous polynomials (arXiv:1809.05632)
Discussion of cocycle spaces for rational Cohomotopy (see also at rational model of mapping spaces):
Discussion of Cohomotopy-sets in computational topology:
Martin Čadek, Marek Krčál, Jiří Matoušek, Francis Sergeraert, Lukáš Vokřínek, Uli Wagner, Computing all maps into a sphere, Journal of the ACM, Volume 61 Issue 3, May 2014 Article No. 1 (arxiv:1105.6257)
Lukáš Vokřínek, Decidability of the extension problem for maps into odd-dimensional spheres, Discrete Comput Geom (2017) 57: 1 (arXiv:1401.3758)
Application of cohomotopy similar to that of persistent homology, but improving on homological well groups:
Peter Franek, Marek Krčál, On Computability and Triviality of Well Groups, Discrete Comput Geom (2016) 56: 126 (arXiv:1501.03641, doi:10.1007/s00454-016-9794-2)
Peter Franek, Marek Krčál, Persistence of Zero Sets, Homology, Homotopy and Applications, Volume 19 (2017) Number 2 (arXiv:1507.04310, doi:10.4310/HHA.2017.v19.n2.a16)
Peter Franek, Marek Krčál, Cohomotopy groups capture robust Properties of Zero Sets via Homotopy Theory, talk at ACAT meeting 2015 (pfd slides)
Peter Franek, Marek Krčál, Hubert Wagner, Solving equations and optimization problems with uncertainty, J Appl. and Comput. Topology (2018) 1: 297 (arxiv:1607.06344, doi:10.1007/s41468-017-0009-6)
Discussion of the stable cohomotopy (framed cobordism cohomology theory) in the equivariant cohomology-version of cohomotopy (equivariant cohomotopy):
Arthur Wasserman, section 3 of Equivariant differential topology, Topology Vol. 8, pp. 127-150, 1969 (pdf)
James Cruickshank, Twisted homotopy theory and the geometric equivariant 1-stem, Topology and its Applications Volume 129, Issue 3, 1 April 2003, Pages 251-271 (arXiv:10.1016/S0166-8641(02)00183-9)
and in the twisted cohomology-version (twisted cohomotopy)
Discussion of M-brane physics in terms of rational equivariant cohomotopy:
and in terms of twisted cohomotopy:
Domenico Fiorenza, Hisham Sati, Urs Schreiber, Section 3 of Twisted Cohomotopy implies M-theory anomaly cancellation (arXiv:1904.10207)
Domenico Fiorenza, Hisham Sati, Urs Schreiber, Twisted Cohomotopy implies level quantization of the full 6d Wess-Zumino-term of the M5-brane (arXiv:1906.07417)
Hisham Sati, Urs Schreiber, Equivariant Cohomotopy implies orientifold tadpole cancellation (arXiv:1909.12277)
Discussion of smooth functions into the 4-sphere in the context of Connes-Lott models in spectral non-commutative geometry:
Ali Chamseddine, Alain Connes, Viatcheslav Mukhanov, Quanta of Geometry: Noncommutative Aspects, Phys. Rev. Lett. 114 (2015) 9, 091302 (arXiv:1409.2471)
Ali Chamseddine, Alain Connes, Viatcheslav Mukhanov, Geometry and the Quantum: Basics, JHEP 12 (2014) 098 (arXiv:1411.0977)
Alain Connes, section 4 of Geometry and the Quantum, in Foundations of Mathematics and Physics One Century After Hilbert, Springer 2018. 159-196 (arXiv:1703.02470, doi:10.1007/978-3-319-64813-2)
Last revised on January 10, 2021 at 02:08:05. See the history of this page for a list of all contributions to it.