4
$\begingroup$

Given a map $f : S \to M^4$ from a compact closed not necessarily connected oriented surface to a compact oriented 4-manifold, such that $f_*([S])$ is zero in $H_2(M)$, is there a compact oriented 3-manifold $W$ with boundary $S$ and a map $F : W \to M$ that extends $f$?

$\endgroup$

1 Answer 1

7
$\begingroup$

Yes, there is. This follows from the definition of the (singular oriented) bordism group $\Omega_2(M)$ and the fact that the natural map $\Omega_2(M)\to H_2(M;\mathbb{Z})$ is an isomorphism. In fact the conclusion holds for much more general spaces $M$ (such as CW-complexes).

In more detail, $\Omega_2(M)$ denotes equivalence classes of continuous maps $f: S\to M$, where $S$ is a closed oriented surface (not necessarily connected), under the relation of bordism: two such maps $f_1: S_1\to M$ and $f_2: S_2\to M$ are declared bordant if there is a $3$-manifold $W$ with boundary $\partial W = S_1\sqcup S_2$ and a map $F:W\to M$ extending $f_1\sqcup f_2: S_1\sqcup S_2\to M$ (everything up to diffeomorphism). This defines an abelian group with addition given by disjoint union. The zero element is represented by any $f:S\to M$ which bounds a map from a $3$-manifold (as in your question).

The natural map $\Omega_2(M)\to H_2(M;\mathbb{Z})$ sends $[f:S\to M]$ to $f_\ast([S])$. The easiest (but perhaps not the most elementary) way to see that this is an isomorphism is to examine the Atiyah-Hirzebruch spectral sequence for bordism (see Conner and Floyd's "Differentiable Periodic Maps", section 7), armed with the additional data that $\Omega_1 = \Omega_2 = 0$, i.e all oriented $1$ and $2$-manifolds bound.

$\endgroup$
0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.