nLab
4-manifold

Contents

Contents

Idea

a manifold of dimension 4.

Examples

Properties

Cohomotopy

Let XX be a 4-manifold which is connected and oriented.

The Pontryagin-Thom construction as above gives for nn \in \mathbb{Z} the commuting diagram of sets

π n(X) 𝔽 4n(X) h n h 4n H n(X,) H 4n(X,), \array{ \pi^n(X) &\overset{\simeq}{\longrightarrow}& \mathbb{F}_{4-n}(X) \\ {}^{ \mathllap{h^n} } \downarrow && \downarrow^{ h_{4-n} } \\ H^n(X,\mathbb{Z}) &\underset{\simeq}{\longrightarrow}& H_{4-n}(X,\mathbb{Z}) \,, }

where π \pi^\bullet denotes cohomotopy sets, H H^\bullet denotes ordinary cohomology, H H_\bullet denotes ordinary homology and 𝔽 \mathbb{F}_\bullet is normally framed cobordism classes of normally framed submanifolds. Finally h nh^n is the operation of pullback of the generating integral cohomology class on S nS^n (by the nature of Eilenberg-MacLane spaces):

h n(α):XαS ngeneratorB n. h^n(\alpha) \;\colon\; X \overset{\alpha}{\longrightarrow} S^n \overset{generator}{\longrightarrow} B^n \mathbb{Z} \,.

Now

  • h 0h^0, h 1h^1, h 4h^4 are isomorphisms

  • h 3h^3 is an isomorphism if XX is “odd” in that it contains at least one closed oriented surface of odd self-intersection, otherwise h 3h^3 becomes an isomorphism on a /2\mathbb{Z}/2-quotient group of π 3(X)\pi^3(X) (which is a group via the group-structure of the 3-sphere (SU(2)))

(Kirby-Melvin-Teichner 12)

References

All PL 4-manifolds are simple branched covers of the 4-sphere:

On cohomotopy of 4-manifolds:

Last revised on March 19, 2019 at 02:03:49. See the history of this page for a list of all contributions to it.