Contents

# Contents

## Idea

a manifold of dimension 4.

## Properties

#### Cohomotopy

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

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

where $\pi^\bullet$ 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):

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

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)))

## 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.