Thom spectra and bordism theory

In this chapter, we study a family of (co)homology theories called (co)bordism theory. Their representing spectra are known as the Thom spectra.

The origins behind these theories lie in Thom’s paper [Thom (1954)]. Thom introduced the relation of cobordism on the set of closed smooth \(n\)-manifolds, with \(M\) and \(N\) being identified if their disjoint union \(M \sqcup N\) is the boundary of some compact \((n+1)\)-manifold \(W\). He completely determined unoriented bordism and made decisive progress on two problems posed by Steenrod: which homology classes of a space are represented by maps from closed manifolds, and which closed smooth manifolds are boundaries.

Given a space \(X\), we may equip the manifolds with maps to \(X\), producing the unoriented bordism groups \(\Omega _n^O(X)\). These groups assemble into a homology theory, a viewpoint developed systematically by Atiyah (1961). The spectrum \(\MO \) representing this theory can be constructed using the notion of the Thom space of a vector bundle. Thom invented this notion in order to generalize the result of Pontryagin (1959) that the groups of framed manifolds up to cobordism are isomorphic to the stable stems \(\pi _*(\S )\). Thom realized that one could dispense with the framings at the cost of replacing the spheres in Pontryagin’s computation by the Thom spaces of certain vector bundles. Forming the Thom spaces of the universal rank \(k\) vector bundles for all \(k\) produces the Thom spectrum \(\MO \).

While Thom’s constructions are geometric in nature, the homotopical perspective on Thom spectra was given its modern \(\infty \)-categorical form by Ando et al. (2014): they may be thought of as twisted suspension spectra. Given a rank \(n\) vector bundle \(E \to X\), forming fiberwise one-point compactifications determines an \(X\)-indexed family of pointed spheres \(S^{E_x}\), and (the underlying anima of) its Thom space is the colimit of this family. When \(E\) is a trivial bundle, this family is the constant functor \(X \to \An _*\) with constant value \(S^n\), and this colimit is the \(n\)-th suspension \(\Sigma ^n(X_+) = X_+ \wedge S^n\). There is an analogous perspective on Thom spectra as colimits of functors \(X \to \Sp \).

The modern perspective also gives a uniform notion of orientation for vector bundles \(E \to X\) with respect to a commutative ring spectrum \(R\): one considers the associated spherical fibration \(\Pi _{\infty }(X) \to \Sp , x \mapsto \Sigma ^{\infty }S^{E_x}\), and asks for a trivialization after tensoring with \(R\). Passing to homotopy groups, this recovers the classical Thom isomorphism.

We start the chapter by recalling the classical Thom spaces in Section 10.1. The modern perspective on Thom spaces/spectra is the topic of Section 10.2. In Section 10.3 we introduce the geometric bordism groups and record the Pontryagin–Thom theorem that shows they form the homology theory represented by the Thom spectrum. Finally, Section 10.4 treats orientations and the Thom isomorphism, following Ando et al. (2014). The technical proof that one-point compactification is coherently symmetric monoidal is postponed to Section 10.5, which may be skipped on a first reading.

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