Bordism and the Pontryagin–Thom theorem

Definition 10.3.1 (Normal \(B\)-structure). Let \(M\) be a compact smooth \(n\)-manifold. Its stable normal bundle \(\ul {\R ^n}-T_M\) is a virtual vector bundle of rank \(0\), with classifying map \(\nu _M\colon M\to \bB O\). A normal \(B\)-structure on \(M\) is a lift of this map along \(\phi \):

Concretely, this consists of a map \(\sigma \colon M \to B\) together with a homotopy \(\phi \circ \sigma \simeq \nu _M\). If \(M\) has boundary, then the stable normal bundle of \(\partial M\) is identified with the restriction of the stable normal bundle of \(M\) using the outward normal direction. Consequently, a normal \(B\)-structure on \(M\) restricts to one on \(\partial M\). A diffeomorphism of manifolds with normal \(B\)-structures is understood to be equipped with an identification in the anima of such lifts.

Example 10.3.2. For \(B = \bB O\) with \(\phi \) the identity, a normal \(B\)-structure is no data at all. For \(B = \bB SO\) it is an orientation of the stable normal bundle, and for \(B = \pt \) it is a trivialization of it, that is, a stable framing of \(M\).

Definition 10.3.3 (\(B\)-bordism). Let \(X\) be a topological space and let \(n \geq 0\). A \(B\)-bordism cycle of degree \(n\) over \(X\) is a triple \((M,f,\sigma )\) consisting of a closed smooth \(n\)-manifold \(M\), a continuous map \(f\colon M \to X\), and a normal \(B\)-structure \(\sigma \) on \(M\). Two cycles \((M_0,f_0,\sigma _0)\) and \((M_1,f_1,\sigma _1)\) are bordant if there exist compact smooth \((n+1)\)-manifolds \(W_0\) and \(W_1\) with normal \(B\)-structures, together with maps \(F_i\colon W_i \to X\), and a diffeomorphism \[ M_0 \amalg \partial W_0 \; \cong \; M_1 \amalg \partial W_1 \] of manifolds with normal \(B\)-structures over \(X\).

We write \(\Omega _n^B(X)\) for the set of bordism classes. Disjoint union defines an addition, with the empty manifold as zero. For \(n<0\), we set \(\Omega _n^B(X)=0\).

Remark 10.3.4. We used the formulation of Switzer (1975), Chapter 12. Note it is an equivalence relation: reflexivity takes \(W_0 = W_1 = \varnothing \), symmetry is built into the statement, and transitivity follows by taking disjoint unions. It is also visibly additive, so \(\Omega _n^B(X)\) is a commutative monoid in which every class of the form \([\partial W]\) vanishes.

Inverses are obtained via the cylinder construction. Pulling back \(\sigma \) along the projection equips \(M \times [0,1]\) with a normal \(B\)-structure whose restriction to one end is \(\sigma \); write \(\sigma '\) for the structure induced on the second end. Then \[ (M,\sigma ) \amalg (M,\sigma ') \cong \partial (M \times [0,1]) \] as manifolds with normal \(B\)-structures, and hence \[ [(M,f,\sigma )] + [(M,f,\sigma ')] \; = \; 0 \] in \(\Omega _n^B(X)\). Thus \(\Omega _n^B(X)\) is an abelian group. For \(B = \bB O\) there is no additional structure, so \(\sigma ' = \sigma \) and every class is its own inverse; for \(B = \bB SO\) the structure \(\sigma '\) is the reversed orientation, and the groups are not \(2\)-torsion in general.

Construction 10.3.5 (Pontryagin–Thom transformation). Let \((M,f,\sigma )\) be a \(B\)-bordism cycle of degree \(n\) over \(X\). The closed smooth manifold \(M\) has the homotopy type of a finite cell complex by [Milnor (1963), Theorem 3.5 and Corollary 6.7], so the Thom-spectrum constructions above apply to its vector bundles. Choose a smooth embedding \(i\colon M \hookrightarrow \R ^{n+k}\) for some \(k\), compactify the ambient space to \(S^{n+k}\) with basepoint at infinity, and let \(\nu \) be the rank \(k\) normal bundle. Applying \(\Sigma ^{\infty }\) to the pointed collapse map of Construction 10.1.8 and shifting gives a map of spectra \[ \S [n] \longrightarrow \th (\nu )[-k]. \] By Lemma 10.2.21, the target is the Thom spectrum of the stable normal bundle. The map \((\sigma ,f)\colon M \to B\times X\) therefore induces a map \[ \th (\nu )[-k] \longrightarrow MB \otimes \S [X]. \] The composite determines an element of \(\pi _n(MB \otimes \S [X])\). The standard arguments proving independence of the embedding and bordism invariance are included in the Pontryagin–Thom theorem quoted below. They show that the construction defines a natural homomorphism \[ \PT \colon \Omega _n^B(X) \longrightarrow \pi _n(MB\otimes \S [X]). \]

Theorem 10.3.6 (Pontryagin–Thom theorem). For every cell complex \(X\) and every \(n \in \Z \), the Pontryagin–Thom transformation is a natural isomorphism \[ \Omega _n^B(X) \xrightarrow {\cong } \pi _n(MB\otimes \S [X]). \] In particular, the groups \(\Omega _n^B(X)\) are the absolute groups of the generalized homology theory represented by \(MB\).

Remark 10.3.7. The spectrum \(MB\) also represents a cohomology theory \(MB^*(X)\), traditionally called cobordism. We do not develop a geometric cocycle model for it; see Atiyah (1961) for the relation with bordism.

Remark 10.3.8 (Coefficient rings). The Pontryagin–Thom theorem turns computations of bordism groups into computations of homotopy groups of spectra. For \(\MO \) and \(\MU \), Construction 10.2.25 constructs the multiplication on the Thom spectrum from Whitney sum. Under the Pontryagin–Thom isomorphism, this multiplication corresponds to cartesian product of bordism classes. We will not prove this geometric compatibility here.

Thom computed the unoriented coefficient ring as \[ \pi _*(\MO ) \cong \F _2[a_i \mid i \geq 1,\ i \neq 2^j-1], \qquad \abs {a_i}=i; \] see Thom (1954). The complex bordism spectrum has coefficient ring \[ \pi _*(\MU ) \cong \Z [x_1,x_2,\ldots ], \qquad \abs {x_i}=2i, \] a theorem of Milnor and Novikov; see Ravenel (1986). By Theorem 10.3.6 this is the ring of complex bordism classes. The spectrum \(\MU \) is central to chromatic homotopy theory, where its additional structure and the algebra of its cooperations organize stable homotopy according to height. At the other extreme, take \(B = \pt \) with the constant map to \(\bB O\), so that \(MB = \S \). By Example 10.3.2 the cycles are then stably framed manifolds, and the Pontryagin–Thom theorem becomes \[ \Omega _n^{\mathrm {fr}}:=\Omega _n^{\pt }(\pt ) \cong \pi _n(\S ), \] recovering Pontryagin’s construction from the beginning of Section 10.1. In degree one, a circle with its nonbounding stable framing represents the nonzero class. Under the displayed isomorphism, this class corresponds to the stable Hopf map \(\eta \in \pi _1(\S )\) of Example 4.4.26.