Let's suppose that

$f:X\rightarrow X$ is a continuous map such that

- $H_{\ast}(f): H_{\ast}(X)\rightarrow H_{\ast}(X)$ is a homology isomorphism (with integral coefficients)
- $X$ is a finite connected CW-complex.
- $\pi_{1}(f): \pi_{1}(X)\rightarrow \pi_{1}(X)$ is an isomorphism of fundamental groups.
- $\pi_{1}(X)$ is a finitely presented group.
- $\pi_{n}(f)=0$ for $n>1$.
- the homotopy colimit $$hocolim(X\rightarrow_{f} X\rightarrow_{f} X\dots)$$ is homotopy equivalent to a finite CW-complex.

Does it imply that $f$ has to be a weak homotopy equivalence ?

My guess is that the answer should be **no** but I don't have a counterexample.