Homology theory represented by Madsen-Tillmann spectra

Question

The generalized homology theory of the Thom spectrum $MO=\varinjlim\Sigma^nMTO_n$ is bordism theory:\begin{equation*}\pi_k(MO\wedge X)=\Omega^O_k(X)\end{equation*}These groups form the ring of (unoriented) $X$-cobordism classes of (unoriented) manifolds.

But what information do the Madsen-Tillmann spectra $MTO_n$ contain? Does the homotopy ring\begin{equation*}\pi_k(MTO_n\wedge X)=\text{ ?}\end{equation*}yield any useful classification of manifolds?

Answer 1

This is an exercise in understanding the Pontrjagin--Thom correspondence. The group $\pi_k(MTO(n) \wedge X_+)$ is in bijection with tuples of

1. a $(n+k)$-manifold $M$,
2. an $n$-dimensional vector bundle $V \to M$,
3. a stable isomorphism $\varphi: V \oplus \epsilon^k \oplus \epsilon^\ell \cong TM \oplus \epsilon^\ell$ for $\ell \gg 0$, and
4. a continuous map $f : M \to X$.

This data is taken up to cobordism in the obvious way. Note that the spectrum $MTO(n)$ is not connective, which corresponds to the fact that the above makes sense for negative $k$.

On the other hand, for a $d$-dimensional manifold $X$ the cohomology theory $[X,MTO(n)]$ is represented by tuples of

1. a $(d+n)$-dimensional manifold $E$ with a proper map $\pi : E \to X$,
2. an $n$-dimensional vector bundle $V \to E$,
3. a stable isomorphism $\varphi : TE \oplus \epsilon^\ell \cong V \oplus \pi^*(TX) \oplus \epsilon^\ell$,

again taken up to cobordism in the obvious way.

The point that user43326 is referring to is that if $\pi : E^{d+n} \to X^d$ is a smooth fibre bundle with compact $n$-dimensional fibres, then we may define $V := T_\pi E = \mathrm{Ker}(D\pi : TE \to TX)$ to be the vertical tangent bundle and choose a splitting of the short exact sequence $$0 \to T_\pi E \to TE \to \pi^*(TX) \to 0$$ of vector bundles on $E$. This gives an isomorphism $\varphi : TE \cong V \oplus \pi^*(TX)$, and so the tuple $(\pi: E \to X, V, \varphi)$ represents a class in $[X, MTO(n)]$. (However, despite what user43326 said, it is not true that all classes in this cohomology theory arise in this way.)

The reason that $$\mathrm{hocolim}_{n \to \infty} \Sigma^n MTO(n) \simeq MO$$ can be easily seen from the first description above. Concretely, a class in $\pi_k$ of the homotopy colimit is represented by a tuple

1. a $k$-manifold $M$,
2. an $n$-dimensional vector bundle $V \to M$ for some $n \gg0$,
3. a stable isomorphism $\varphi : V \oplus \epsilon^\ell \cong TM \oplus \epsilon^{n-k} \oplus \epsilon^\ell$, for $\ell \gg0$,

taken up to cobordism. By taking $n$ large enough, and destabilising the isomorphism, we get $\varphi: V \cong TM \oplus \epsilon^{n-k}$, and so the last two pieces of data cancel out: we are left with just $k$-manifolds up to cobordism.