I'm reading the books by Bott & Tu and Milnor & Stasheef simultaneously. The following is my doubt:

The Thom isomorphism in Bott & Tu is obtained as $H_{cv}^{*+n}(E)\rightarrow H^*(M)$, where $\pi\colon E\to M$ is $n$ plane bundle over the manifold of dimension $m$ manifold $M$ and the isomorphism is given by the integration along fiber map $\pi_*$. By the projection formula, the Thom isomorphism $\mathscr{T}$, inverse to $\pi_*$, is then explicitly given by $\omega\mapsto \pi^*(\omega)\wedge\Phi$, where $\Phi=\mathscr{T}(1)$ is the Thom class of $E$.

In Milnor & Stasheef, the isomorphism is $\mathscr{T}\colon H^*(M)\to H^{*+n}(E,E_0)$, where $E_0$ is the complement of zero section and the map factors through $H^*(E)$ i.e. $H^*(M)\xrightarrow{\pi^*} H^*(E)\xrightarrow{\smile\text{fundamental class}} H^{*+n}(E,E_0)$.

I know that $H_{c}^{*+n}(E)=\varinjlim H_{c}^{*+n}(E, E-K)$ over the directed set of compact subsets of $E$. My question is, how can I relate $H^{*+n}(E,E_0)$ and $H_{cv}^{*+n}(E)$.

Any hint would be helpful. Thanks.


1 Answer 1


Compactly supported homology can be seen to be equal to the reduced homology of the one point compactification, essentially by definition of the topology on the one point compactification.

A similar argument shows that vertical compactly supported homology is the reduced homology of the space obtained by compactifying the fibers and then identifying each of the points at infinity to a single point.

Excision then relates this to the relative homology $H^* (E, E_0)$. $H^* (E, E_0) \cong H^*(D(E),S(E)) \cong \bar{H}^*(D(E)/S(V))$. Here $D(E),S(E)$ are the disk and sphere bundles. This quotient can be seen to be homeomorphic to the space obtained by compactifying the fibers and then identifying the points at infinity. This space is called the Thom space of E. This then yields the isomorphism between vertical compactly supported homology and the relative homology of the total space and the total space with deleted zero section

  • $\begingroup$ On the off chance you see this, would you be willing to expand your first paragraph? It's not clear to me (but I'm not much good at topology!) $\endgroup$ Mar 27, 2021 at 5:58
  • $\begingroup$ @AlekosRobotis First off, let me mention everything I wrote should be in cohomology not homology. Now, suppose that our space $X$ behaves nice enough so that there is a well behaved contractible neighborhood of the point at infinity in the compactification (maybe this is always true). Excision type results will imply that the inclusion of cochains that are zero valued in some neighborhood of infinity into cochains that are zero valued at infinity is an isomorphism on cohomology. But such a complex is isomorphic to the complex of compactly supported cochains, just by forgetting $\infty$. $\endgroup$ Mar 27, 2021 at 6:30

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