# Bott and Tu, Thom Class pg 64

Prop 6.18,pg 64:The Thom class $$\Phi$$ on a rank $$N$$ oriented vector bundle $$E$$ can be uniquely characterized as the cohomology class in $$H^n_{cv}(E)$$ which restricts to the generator of $$H^n_c(F)$$ on each fiber $$F$$.

The proof wrote that

If $$\Phi'$$ is a coho. class in $$H^n_{cv}(E)$$ that restricts to a generator on each fiber then for $$w \in H^n(M)$$, $$w \wedge \pi_* \Phi' = w.$$

I do not see how this holds from the definition of $$\pi_*$$

Definitions:

From Thom Isomorphism for an oriented bundle $$E$$ over $$M$$, we have isomorphism between de Rham coho. and compact vertical coho. $$H^*(M) \rightarrow H^{*-n}_{cv}(E)$$ We define $$\Phi$$ the Thom class to be the image of $$1$$ in $$H^0(M)$$ under the above Thom isomorphism.

A form $$w \in \Omega^*_{cv}(E)$$ is locally of type (I) or (II), the map \begin{align*} (I) \quad & \pi^* \phi f(x, t_1, \ldots, t_n) dt_{i_1} \ldots d_{t_r} \mapsto 0 , \quad r< n \\ (II) \quad & \pi^* \phi f(x,t_1, \ldots, t_n) dt_1 \ldots dt_n \mapsto \phi \int_{\Bbb R^n} f(x,t_1, \ldots, t_n) dt_1 \ldots dt_n \end{align*} is our $$\pi_* :\Omega^{*}_{cv}(E) \rightarrow \Omega^{*-n}(M)$$. $$E$$ a vector bundle over $$M$$.

• Can you include some more details in your question? I'm not sure which definition for a Thom class you wish to use (as it stands the stated theorem is a perfectly suitable definition). And what is $\Phi'$? And what is $\omega$? – Tyrone Nov 12 '18 at 10:55
• I have added the details, but more can be found in the text. Tell me if things are still unclear. – CL. Nov 12 '18 at 22:52
• I'm not sure I quite agree with the exact wording of the statement in the book, since you could always multiply such a $\Phi'$ by a non-zero real number to get another class which restricts to a generator in each fibre. I suggest working locally and studying the equation $\pi_*(\pi^*\omega\wedge \Phi')$ which appears in the book, using the description of the map $\pi_*$ to see the result. – Tyrone Nov 13 '18 at 14:24