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 collection of degreewise morphism of vector bundles $f^n\colon E^n\to F^n$ compatible with the differentials. Then at every point $x$ of $X$ we have a morphism $f^\bullet_x\colon E_x^\bullet\to F_x^\bullet$ of chain complexes of finite dimensional real vector spaces, and we say that $f^\bullet$ is a pointwise homotopy equivalence at $x$ if there exist a morphism of chain complexes of vector spaces $g_x^\bullet\colon F_x^\bullet \to E_x^\bullet$ and morphisms of graded vector spaces $H_x^\bullet\colon E_x^\bullet\to E_x^{\bullet-1}$ and $K_x^\bullet\colon F_x^\bullet\to F_x^{\bullet-1}$ such that $g_x^\bullet\circ f_x^\bullet= Id-[d,H_x^\bullet]$ and $f_x^\bullet\circ g_x^\bullet= Id-[d,K_x^\bullet]$.

Similarly, we say that $f^\bullet$ is a homotopy equivalence on an open subset $U$ of $X$ if in the above one can choose $g,H,K$ to be morphisms of chain complexes of vector bundles and of graded vector bundles over $U$, respectively. In other words, if we restrict our attention to a trivializing open set $U$ for our bundles, then $f^\bullet$ is a homotopy equivalence on $U$ if we can choose the pointwise defined functions $g_x^\bullet,H_x^\bullet,K_x^\bullet$ to vary continuously over $U$. Clearly, if $f^\bullet$ is a homotopy equivalence over $U$, then it is a pointwise quasi-isomorphism at every point $x$ of $U$, but one would expect the converse not to be true. That is, I would expect that the pointwise existence of a solution $g_x^\bullet,H_x^\bullet,K_x^\bullet$ to the homotopy equivalence equations alone cannot imply that one can find also a continuous solution $\tilde{g}_x^\bullet,\tilde{H}_x^\bullet,\tilde{K}_x^\bullet$ (possibly different from the given pointwise one).

However, I'm finding it much harder than expected to produce a counterexample. Namely, the equations $g_x^\bullet,H_x^\bullet,K_x^\bullet$ are just linear equations once $f^\bullet$ and the differentials are given, so that the problem is a particular instance of the problem of determining the existence of a continuous solution for a parameter-dependent linear system $A(x)\cdot v_x=b_x$ (with $A(x)$ and $b_x$ continuously depending on $x$) once one knows that for every $x$ there exist at least one solution. And it is very easy to produce examples of $A(x)$ and $b_x$ where no continuos solution can exist. Yet,I have been so far failing in my attempts of producing an explicit example with no continuous solution of the specific form given by the homotopy equivalence equations.

What has been frustrating my very low-rank few-terms complexes attempts so far has been the existence of partition of units. So I guess that despite what I was expecting, pointwise homotopy equivalence could actually imply local homotopy equivalence in the continuous setting (and then arguably also in the smooth setting but not the real analytic setting).

Apart from my low-rank few-terms I have been searching the literature, but also here I have not been successful so far.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.