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Sorry if this question is to naive or badly phrased. I am curious about the following problem, given a manifold $M$, how "close" can we find a smooth or PL manifold, $N$, with a map $f:N\to M$. The manifolds in this case would preferably be closed compact and of the same dimension. Close in this case is given by $(f)_q=\sum_i Coker(f|_{H_i(M)})q^{i}$, where $q$ is arbitrary but fixed.

In the case of $M$ admitting such a structure it is zero, but I'm unable to establish more substantive results. If the manifold is of dimension five for example, I thinking passing to a covering space will allow us to kill the fundamental group, and thus the degree 4 cohomology, which allows us to kill the Kirby-Siberman class. So there is abound of $dim(H^4(M))q$. I'm also interested in the case of $f:M\to N$ with the cokernal replaced with the kernel, but it seems less amenable.

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  • $\begingroup$ I'm having a hard time understanding the definition of $(f)_q$: is $q$ a real number? Are you summing ranks/ cardinalities of kernels? $\endgroup$
    – Marco Golla
    Commented Jul 25, 2015 at 9:18
  • $\begingroup$ $q$ should be an arbitrary positive real number, assume to be large. I used bad notation, I meant the dimension the homology in $\mathbb{Z}/(2)$-coefficients. $\endgroup$
    – Pax
    Commented Jul 25, 2015 at 9:24
  • $\begingroup$ About cokernel versus kernel, note that if the manifolds are $n$-dimensional and $f$ induces an isomorphism in $H_n$ then it induces a surjection in $H_j$ for all $j$. $\endgroup$
    – Tom Goodwillie
    Commented Jul 25, 2015 at 15:02

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