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Let $f: X\to Y$ be a smooth map between smooth manifolds, both connected. Let $Y=\cup_{i=1}^k Y_i$ be a finite union of disjoint locally closed submanifold $Y_i$ such that $f^{-1}(Y_i)\to Y_i$ is smooth fibre bundle for $i=1,2, \cdots, k$ and the induced map $f_*:H_*(f^{-1}(Y_i);\mathbb{Z})\to H_*(Y_i;\mathbb{Z})$ is an isomorphism for all $i$. Then $f_*:H_*(X;\mathbb{Z})\to H_*(Y;\mathbb{Z})$ is also isomorhphism.

Thanks in advance!!

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    $\begingroup$ can you reduce it to the case that $n=2$? $\endgroup$
    – kiran
    Commented Feb 22, 2022 at 17:34
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    $\begingroup$ Yes proof when $Y=Y_1\cup Y_2$ will be very helpful. Can you give some idea ?@kiran $\endgroup$
    – tota
    Commented Feb 23, 2022 at 5:50
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    $\begingroup$ I am confused but if Y is connected how it can be written as a finite union of disjoint submanifolds? Do you allow manifolds to have boundaries? $\endgroup$
    – user43326
    Commented Feb 23, 2022 at 8:51
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    $\begingroup$ Y is disjoint union of locally closed submanifolds. If you are not comfortable you can take $Y_i$ to be locally closed set only. $\endgroup$
    – tota
    Commented Feb 23, 2022 at 12:45
  • $\begingroup$ @tota Maybe apply a tubular-neighborhood theorem to turn Y_1 and Y_2 into an open cover, then try to apply mayer vietoris? $\endgroup$
    – kiran
    Commented Feb 24, 2022 at 19:28

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