Do exact homology sequences induce exact cohomology sequences? For example, If $E\rightarrow S^n$ is a fiber bundle over an $n$-dimensional sphere with fiber $F$. Then the sequence of homology groups $$H_m(E)\rightarrow H_{m-n}(F)\rightarrow H_{m-1}(F)\rightarrow H_{m-1}(E)\rightarrow H_{m-n-1}(F)$$ is exact. Does $$H_{dR}^m(E)\rightarrow H_{dR}^{m-n}(F)\rightarrow H_{dR}^{m-1}(F)\rightarrow H_{dR}^{m-1}(E)\rightarrow H_{dR}^{m-n-1}(F)$$ is exact?
2026-03-30 01:32:37.1774834357
Exact homology sequences induce exact cohomology sequences
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