I'm going through Čech cohomology at a gentle pace and as you know we need a colimit to obtain the sought-after sheaf cohomology. In practice, for calculating actual cohomology groups, I would invoke Leray's theorem and choose an acyclic covering, so that there is no need for a colimit, and that's quite clear so far. On sunny days I also have a version of Mayer-Vietoris that makes computations tractable.

My issue is that I assumed quite naively that contractible spaces (say, open disks) have trivial cohomology (say, for the constant presheaf). And I get the expected results so it's not obviously wrong. But how can I prove this?

I'm aware that Čech cohomology is isomorphic to other cohomology theories for "nice enough" space, and that would be a way (albeit an impractical one) to get a proof. But surely there is a simple and elegant argument "internal" to Čech cohomology?

  • $\begingroup$ Čech cohomology computes what you want if you can find an open cover such that intersections of open sets there are contractible. If your space is contractible, you have a very natural choice of such an open cover! $\endgroup$ – Pedro Tamaroff Feb 27 '19 at 11:14
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    $\begingroup$ Isn't Čech cohomology homotopy invariant? Just like every cohomology theory? So it boils down to calculating Čech cohomology of a point. Which is easy to do straight from the definition. $\endgroup$ – freakish Feb 27 '19 at 11:26
  • $\begingroup$ @freakish Perhaps you should give an official answer to clear the question from the "unanswered" queue. $\endgroup$ – Paul Frost Feb 27 '19 at 16:23

Čech cohomology functor is homotopy invariant. Meaning homotopic maps are mapped to equal homomorphisms. This implies that a homotopy equivalence $f:X\to Y$ induces an isomorphism

$$f^k: H^k(Y,\mathscr{F}) \to H^k(X, f^{-1}\mathscr{F})$$

for any $k$ and any "suitable" (i.e. locally constant) sheaf $\mathscr{F}$. Here $f^{-1}\mathscr{F}$ means the inverse image of a presheaf which in case of a constant map (i.e. the contractible case) is a constant sheaf associated with $\mathscr{F}(Y)$. For more details (including proofs) see this paper.

So to calculate Čech cohomology of a contractible space all you have to do is to calculate Čech cohomology of a point. Which can be done straight from the definition. That's because every presheaf is trivial (meaning uniquely determined by a single abelian group) and there is only one open cover containing only one open set. It requires some effort but all calculations are very doable and I encourage you to try it.


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