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 smooth manifold of dimension $n \in \mathbb{N}$ and let $k \leq n$. Then the Pontryagin-Thom construction induces a bijection
from the cohomotopy sets of $X$ to the cobordism group of $(n-k)$-dimensional submanifolds with normal framing up to normally framed cobordism.
In particular, the natural group structure on cobordism group (essentially given by disjoint union of submanifolds) this way induces a group structure on the cohomotopy sets.
This is made explicit for instance in Kosinski 93, chapter IX.
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)))
Original articles include
The relation between cohomotopy classes of manifolds to the cobordism group is discussed for instance in
Further discussion includes
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)
See also
Wikipedia, Cohomotopy group
Last revised on May 18, 2018 at 06:32:37. See the history of this page for a list of all contributions to it.