homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
Where homotopy groups are groups of homotopy classes of maps out spheres, $\pi_n(X)\coloneqq [S^n \to X]$, cohomotopy groups are groups of homotopy classes into spheres, $\pi^n(X) \coloneqq [X \to S^n]$.
If instead one considers mapping into the stabilization of the spheres, hence into (some suspension of) the sphere spectrum, then one speaks of stable cohomotopy. In other words, the generalized (Eilenberg-Steenrod) cohomology theory which is represented by the sphere spectrum is stable cohomotopy.
In this vein, regarding terminology: the concept of cohomology (as discussed there) in the very general sense of non-abelian cohomology, is about homotopy classes of maps into any object $A$ (in some (∞,1)-topos). In this way, general non-abelian cohomology is sort of dual to homotopy, and hence might generally be called co-homotopy. This is the statement of Eckmann-Hilton duality. The duality between homotopy (groups) and co-homotopy proper may then be thought of as being the special case of this where $A$ is taken to be a sphere.
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).
Let $X$ be a 4-manifold which is connected and oriented.
The Pontryagin-Thom construction 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 is $X$ 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)))
Wikipedia, Cohomotopy group
Edwin Spanier, Borsuk’s Cohomotopy Groups, Annals of Mathematics Second Series, Vol. 50, No. 1 (Jan., 1949), pp. 203-245 (jstor)
Laurence Taylor, The principal fibration sequence and the second cohomotopy set, Proceedings of the Freedman Fest, 235251, Geom. Topol. Monogr., 18, Coventry, 2012 (arXiv:0910.1781)
Robion Kirby, Paul Melvin, Peter Teichner, Cohomotopy sets of 4-manifolds, GTM 18 (2012) 161-190 (arXiv:1203.1608)
Last revised on March 16, 2018 at 09:40:02. See the history of this page for a list of all contributions to it.