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Let $(E^\bullet,d_E)$ and $(F^\bullet,d_F)$ be two complexes of real vector bundles on a topological manifold $X$, and let $f^\bullet\colon E^\bullet\to F^\bullet$ be a morphism of complexes, i.e. a ...

(I have asked this question at math stackexchange, it was upvoted but got no answers; maybe you can help.) It is well-known that $\beta_1(R(f))\le\beta_1(X)$, where $\beta_1$ is the first Betti ...