In the previous post, we were trying to show that any homology class of a space $X$ in dimension at most six can be represented by a smooth oriented manifold mapping to $X$. This statement is a geometric one, but it can be proved via homotopy-theoretic means. In the previous post, we interpreted it in terms of homotopy theory, and we showed that

$\displaystyle MSO_*(X)_{(p)} \rightarrow H_*(X; \mathbb{Z}_{(p)})$

was a surjection in degrees ${\leq 6}$ (actually, in degrees ${\leq 7}$) for ${p}$ either ${2}$ or an odd prime ${p>3}$. In this post, we will handle the case ${p = 3}$. Namely, we will produce an approximation to ${MSO}$ in the first few homotopy groups (essentially, we’ll work out the first couple of pieces of a Postnikov decomposition). This will give a criterion for when a homology class in low degrees is in the image of ${MSO_*}$, and we’ll see that it is always satisfied in degrees ${\leq 6}$. This will complete the proof of:

Theorem 1 For any space ${X}$, the map ${MSO_i(X) \rightarrow H_i(X; \mathbb{Z})}$ is surjective for ${ i \leq 6}$: that is, any homology class of dimension ${\leq 6}$ is representable by a smooth manifold.

In the case of an odd prime ${>3}$, we used ${H \mathbb{Z}_{(p)} \oplus H \mathbb{Z}_{(p)}[4]}$ as a 7-approximation to ${MSO_{(p)}}$. This is not going to work at ${3}$, because the cohomologies are off. Namely, the cohomology of ${MSO}$ at ${3}$ has two generators in degrees ${\leq 8}$ (namely, the Thom class ${t}$ and ${p_1 t}$ for ${p_1}$ the first Pontryagin class). However, ${H \mathbb{Z}_{(3)} \oplus H \mathbb{Z}_{(3)}[4] }$ has four generators in mod ${3}$ cohomology in these dimensions: ${\iota_0, \iota_4, \mathcal{P}^1 \iota_0, \beta \mathcal{P}^1 \iota_0}$ for ${\iota_0, \iota_4}$ the tautological classes. So the Postnikov decomposition is going to look somewhat different.

Most of this material described in the past few posts comes from a variety of sources: Thom’s original paper (Quelques propriétés globales), Rudyak’s On Thom Spectra, Orientability, and Cobordism, and Stong’s Notes on Cobordism Theory.

A classical problem in topology was whether, on a (suitably nice) topological space ${X}$, every homology class can be represented by a manifold. In other words, given a homology class ${x \in H_n(X; \mathbb{Z})}$, is there an ${n}$-dimensional oriented manifold ${M}$ together with a continuous map ${f: M \rightarrow X}$ such that

$\displaystyle f_*([M]) = x,$

for ${[M] \in H_n(M; \mathbb{Z})}$ the fundamental class?

The question can be rephrased in more modern language as follows. There is a spectrum ${MSO}$, which yields a homology theory (“oriented bordism”) ${MSO_*}$ on topological spaces. There is a morphism of spectra ${MSO \rightarrow H \mathbb{Z}}$ corresponding to the Thom class in ${MSO}$, which means that for every topological space ${X}$, there is a map

$\displaystyle MSO_*(X) \rightarrow H_*(X; \mathbb{Z}).$

Since ${MSO_*(X)}$ can be described (via the Thom-Pontryagin construction) as cobordism classes of manifolds equipped with a map to ${X}$, we find that that the representability of a homology class ${x \in H_n(X; \mathbb{Z})}$ is equivalent to its being in the image of ${MSO_n(X) \rightarrow H_n(X; \mathbb{Z})}$.

The case where ${\mathbb{Z}}$ is replaced by ${\mathbb{Z}/2}$ is now straightforward: we have an analogous map of spectra

$\displaystyle MO \rightarrow H \mathbb{Z}/2$

which corresponds on homology theories to the map ${MO_*(X) \rightarrow H_*(X; \mathbb{Z}/2)}$ sending a manifold ${M \rightarrow X}$ to the image of the ${\mathbb{Z}/2}$-fundamental class in homology. Here we have:

Theorem 1 (Thom) The map ${MO_*(X) \rightarrow H_*(X; \mathbb{Z}/2)}$ is a surjection for any space ${X}$: that is, any mod 2 homology class is representable by a manifold.

This now follows because ${MO}$ itself splits as wedge of copies of ${H \mathbb{Z}/2}$, so the Thom class ${MO \rightarrow H \mathbb{Z}/2}$ actually turns out to have a section in the homotopy category of spectra. It follows that ${MO \wedge X \rightarrow H \mathbb{Z}/2 \wedge X}$ has a section for any space ${X}$, so taking homotopy groups proves the claim.

The analog for realizing ${\mathbb{Z}}$-homology classes is false: that is, the map ${MSO_*(X) \rightarrow H_*(X; \mathbb{Z})}$ is generally not surjective for a space ${X}$. Nonetheless, using the tools we have so far, we will be able to prove:

Theorem 2 Given a space ${X}$ and a homology class ${x \in H_n(X; \mathbb{Z})}$ for ${n \leq 6}$, ${x}$ is representable by an oriented manifold. In general, any homology class has an odd multiple which is representable by a manifold. (more…)