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Suppose it is given an orientation preserving homotopy equivalence $h:N→M$ between closed oriented connected manifolds. Let $X,Y\subset M$ be diffeomorphic submanifolds, and assume $h$ to be transverse to both $X$ and $Y$. Define $A:=h^{−1}(X)$ and $B:=h^{−1}(Y)$. I would like to know if

sign($A$)=sign($B$) ?

To avoid triviality, assume dim($A$)=dim($B$) to be a multiple of 4. Is there a way to show that (maybe) $A$ and $B$ are oriented cobordant? Any example/counterexample can be useful.

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    $\begingroup$ What do you mean exactly by $h$ being transverse to $X$ and $Y$? $\endgroup$
    – Yonatan Harpaz
    Oct 8, 2015 at 20:25
  • $\begingroup$ I am assuming $h$ to be smooth, like everything else here. Therefore it makes sense to speak of transversality ($dh_a(T_aN)+T_{x}X=T_{x}M$ for every $a\in h^{-1}(x)$). This is the condition which ensures $A$ and $B$ to be closed smooth manifolds. $\endgroup$
    – EdoardoFossati
    Oct 9, 2015 at 8:46

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This turned out to be false. By Example 3.1 in the paper of James Davis "Manifold aspects of the Novikov conjecture", there is a homotopy equivalence $h:S(E')\to S^4\times S^4$ such that $\sigma(h^{-1}(pt\times S^4))=16$, but $\sigma(h^{-1}(S^4\times pt))=\sigma(S^4)=0$ since $h$ preserves the fibers.

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  • $\begingroup$ More simply, take any example of a homotopy equivalence $h:N\to M$ such that some sphere $A\subset M$ has $\sigma(h^{-1}(A))\neq 0$, and choose $B$ to be a sphere in $M$ of the same dimension as $A$ that bounds a disk in a coordinate chart of $M$. $\endgroup$
    – Tom Goodwillie
    Nov 13, 2015 at 17:53
  • $\begingroup$ It sounds correct, clearly. But I wasn't able to find such a concrete example, since the map $h$ should not have the homotopy type of a homeomorpshim and hence it is not immediate (at least for me) to deal with these manifolds. $\endgroup$
    – EdoardoFossati
    Nov 13, 2015 at 18:01

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