Section 10.1: Thom spaces and the collapse construction | Stable homotopy theory and higher algebra

Let us make Pontryagin’s construction from the introduction precise, so as to see what has to be modified in the absence of a framing. Suppose a closed $n$-manifold $M$ is embedded in $\R ^{n+k}$ and that its normal bundle has been trivialized, so that $M$ admits a tubular neighborhood identified with $M \times \R ^k$. Collapsing everything outside this tubular neighborhood gives a pointed map \[ S^{n+k} \longrightarrow (M \times \R ^k)^+ \cong M_+ \wedge S^k \longrightarrow S^k, \] and hence, after passing to stabilizations, an element of the stable stem $\pi _n(\S ) = \pi _0\Hom _{\Sp }(\S [n],\S )$. Without a trivialization there is no map to $S^k$: the collapse map remembers the normal directions of $M$, and these vary from point to point. Thom’s remedy is to replace the target $S^k$ by a pointed space built out of the normal bundle itself, obtained by compactifying each fiber separately and then gluing all of the resulting points at infinity into a single basepoint.

Construction 10.1.1 (Fiberwise one-point compactification). Let $p\colon E \to X$ be a real vector bundle of rank $n$. We define its fiberwise one-point compactification $S^E$ as the topological space obtained from $E$ by forming the one-point compactification fiberwise, replacing $\R ^n$ with $S^n = \R ^n \cup \{\infty \}$ in each fiber. More precisely, given a local trivialization $\{\phi _i\colon U_i \times \R ^n \to p^{-1}(U_i)\}_{i \in I}$ of $E$, we define $S^E$ by gluing together the spaces $U_i \times S^n$ along the transition functions: for $x \in U_i \cap U_j$, the map $\phi _j^{-1} \circ \phi _i\colon \{x\} \times \R ^n \to \{x\} \times \R ^n$ extends uniquely to a homeomorphism $\{x\} \times S^n \to \{x\} \times S^n$ fixing the point at infinity. One can show $S^E$ is independent of the choice of local trivializations of $E \to X$.

The space $S^E$ comes equipped with a map $S^E \to X$. The ‘points at infinity’ assemble into a continuous section $s_{\infty }\colon X \to S^E$.

Definition 10.1.2 (Thom space). Let $p\colon E \to X$ be a real vector bundle. We define the Thom space $\Th (E)$ as the quotient of $S^E$ by the points at infinity: \[ \Th (E) := S^E/s_\infty (X). \]

Remark 10.1.3. Given a metric on $E$, one obtains a homeomorphic description of $\Th (E)$ as the quotient $D(E)/S(E)$, where $D(E)$ and $S(E)$ denote the unit disk and sphere bundles with respect to the metric. The homeomorphism is given by sending a vector $v$ to $v/(1+\|v\|)$ in each fiber. This is often taken as the definition, but has the disadvantage of requiring an auxiliary choice of metric.

Remark 10.1.4. If $X$ is compact Hausdorff, the inclusion of $E$ induces a homeomorphism $E^+\cong \Th (E)$. Indeed, a neighborhood of the collapsed section in $\Th (E)$ has compact complement because $S^E$ is compact Hausdorff, so the quotient topology at the basepoint is the one-point compactification topology.

The key observation is that the Thom space of a trivialized bundle is an honest $n$-fold suspension:

Example 10.1.5 (Suspension as Thom space). Let $E = X \times \R ^n$ be the trivial rank $n$ vector bundle over $X$. Then $\Th (E) \cong X_+ \wedge S^n \cong \Sigma ^n(X_+)$.

For a general rank $n$ bundle $E$, the Thom space $\Th (E)$ differs from $\Sigma ^n(X_+)$ only in the way the fiber spheres are glued to one another. We will therefore think of $\Th (E)$ as a twisted suspension of $X_+$, twisted by $E$.

Example 10.1.6. Let $E \to X$ and $E' \to X'$ be vector bundles. Then their product $E \times E' \to X \times X'$ is again a vector bundle (called their external direct sum) and there is an isomorphism of pointed spaces \[ \Th (E \times E') \cong \Th (E) \wedge \Th (E'). \] To see this, equip $E \times E'$ with the metric $\|(v,w)\| = \max (\|v\|,\|w\|)$. Then the disk bundle $D(E \times E')$ is homeomorphic to the product $D(E) \times D(E')$, and under this homeomorphism the sphere bundle $S(E \times E')$ corresponds to the subspace \[ D(E)\times S(E') \cup S(E) \times D(E') \] of the product. Passing to quotient spaces then proves the claim.

Construction 10.1.7 (Functoriality of Thom spaces). Consider a fiberwise injective morphism of rank $n$ vector bundles, i.e. a commutative diagram

Commutative diagram generated from the LaTeX source

in which $\widetilde {f}$ induces injective $\R $-linear maps $E_x \hookrightarrow E'_{f(x)}$ on fibers. Then the map $\widetilde {f}$ induces a map $S^{E} \to S^{E'}$ on fiberwise one-point compactifications, which preserves the sections at infinity and hence induces a continuous map \[ \Th (\widetilde {f}) \colon \Th (E) \to \Th (E'). \]

We can now carry out the collapse construction in the generality we will need. We isolate it as a separate construction, since it is used again in the proof of Atiyah duality in Section 11.3.

Construction 10.1.8 (Pontryagin–Thom collapse map). Let $i\colon N \hookrightarrow N'$ be a smooth embedding of smooth manifolds, with $N$ compact, and let $\nu (i)$ denote its normal bundle. Choose a tubular neighborhood, given by an open neighborhood $U \subseteq N'$ of $N$ and a diffeomorphism from a neighborhood of the zero section in $\nu (i)$ onto $U$. Collapsing the complement of a smaller tubular neighborhood to the basepoint gives a pointed map \[ \PT (i)\colon N'_+ \longrightarrow \Th (\nu (i)), \] called the Pontryagin–Thom collapse map. Its homotopy class is independent of the auxiliary choices. If $N'$ is equipped with a basepoint away from $N$, we may choose the tubular neighborhood of $N$ to avoid this basepoint, giving a pointed map \[ \PT (i)\colon N' \longrightarrow \Th (\nu (i)). \]

$The Pontryagin–Thom collapse map of an embedding $N \hookrightarrow N'$, which crushes the complement of a tubular neighborhood of $N$ to the basepoint. In the case drawn, $N$ is a circle in a surface with trivial normal bundle. Cutting the band $U$ along $N$ exhibits $\Th (\nu (i))$ as two cones on $N$ glued along $N$, that is, as a sphere with equator $N$ whose two cone points have both become the basepoint $\ast $.$

Figure 10.1: The Pontryagin–Thom collapse map of an embedding $N \hookrightarrow N'$, which crushes the complement of a tubular neighborhood of $N$ to the basepoint. In the case drawn, $N$ is a circle in a surface with trivial normal bundle. Cutting the band $U$ along $N$ exhibits $\Th (\nu (i))$ as two cones on $N$ glued along $N$, that is, as a sphere with equator $N$ whose two cone points have both become the basepoint $\ast $.

For a framed embedding, the normal bundle is trivialized and $\Th (\nu (i)) \cong \Sigma ^k(N_+)$, so that we recover Pontryagin’s construction. In general the target depends on $\nu (i)$. The next step is to make it independent of the manifold by passing to the universal bundle.

Example 10.1.9. Let us consider the Grassmannian $\Gr _n(\R ^\infty )$ of $n$-dimensional subspaces of $\R ^\infty $. By the real analogue of the Grassmannian classification theorem recorded in Remark 9.2.22, this space is a model for the classifying space $BO(n)$ and classifies rank $n$ real vector bundles over paracompact Hausdorff spaces. It is a cell complex by its Schubert cell decomposition. The universal bundle $\gamma _n$ over $\Gr _n(\R ^\infty )$ has as total space \[E_n = \{(V,v) \in \Gr _n(\R ^\infty ) \times \R ^\infty \mid v \in V\}\] with the projection map sending $(V,v)$ to $V$. The fiber over a point $V \in \Gr _n(\R ^\infty )$ is precisely the $n$-dimensional vector space $V$ itself. The Thom space $\Th (\gamma _n)$ is the space of pairs $(V,v)$ where $v \in V$ has length $\leq 1$, modulo those pairs where $v$ has length exactly $1$.

The Thom spaces $\Th (\gamma _n)$ are particularly important: every rank $n$ vector bundle over a paracompact Hausdorff space is pulled back from $\gamma _n$ by Remark 9.2.22, so its Thom space admits a map to $\Th (\gamma _n)$ via Construction 10.1.7. Composing this with the collapse map from Construction 10.1.8 thus produces maps from manifolds into these Thom spaces. They are moreover compatible as $n$ varies, in the following sense.

Construction 10.1.10 (The prespectrum $\MO ^{\pre }$). Consider the canonical inclusion \[ \Gr _n(\R ^{\infty }) \hookrightarrow \Gr _{n+1}(\R \times \R ^{\infty }) \cong \Gr _{n+1}(\R ^{\infty }), \quad V \mapsto \R \oplus V. \] The pullback of $\gamma _{n+1}$ along this inclusion is the product bundle $\gamma _n \times \ul {\R }$, and in particular we obtain a pointed continuous map \[ \Sigma (\Th (\gamma _n)) \cong \Th (\gamma _n \times \ul {\R }) \to \Th (\gamma _{n+1}). \] By adjunction, this defines a map $\sigma _n\colon \Th (\gamma _n) \to \Omega (\Th (\gamma _{n+1}))$, turning the sequence of pointed animae $(\Pi _{\infty }\Th (\gamma _n))$ into a prespectrum, which we will denote by $\MO ^{\pre }$.

Applying spectrification (Proposition 4.3.29) to $\MO ^{\pre }$ produces a spectrum, and this is the classical construction of $\MO $ going back to Thom (1954). Instead of taking this as our definition, the next section gives a construction of the Thom spectrum which requires no auxiliary choices and applies to virtual bundles; we shall identify the two in Theorem 10.2.23.

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