3
$\begingroup$

Consider a locally trivial fibration $f: E \to B$ with fiber $F = \mathbb{C}^n$. The Leray spectral sequence with compact supports is

$$ E_2: H^p_c(B, \underline{H^q_c(F)}) \implies H^{p+q}_c(E). $$

Since the cohomology of the fiber is nonzero only at degree $q = 2n$, the spectral sequence degenerates, and we get an isomorphism $H^\bullet_c(B) \cong H^{\bullet+2n}_c(E)$. Is it possible to describe the isomorphism geometrically? I am looking for an explanation like "the dual map in Borel-Moore homology is the one induced by $f$, sending each Borel-Moore cycle on $E$ to its projection on $B$." Is somethink like this true? Where can I read some basic literature about this?

Improve this question
$\endgroup$
2
  • $\begingroup$ I have a feeling that the compact support cohomology of the fiber is related to the ordinary cohomology of the Thom space, so that your map becomes some sort of Thom isomorphism. $\endgroup$
    – user43326
    Commented Oct 7, 2022 at 18:59
  • $\begingroup$ @user43326 Does the Thom space/isomorphism still work if my fibration is not a vector bundle? The fiber is isomorphic to $\mathbb{C}^n$ but the vector structure is not respected. $\endgroup$
    – Eduardo de Lorenzo
    Commented Oct 10, 2022 at 16:57

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .