4
$\begingroup$

For the purpose this question, let us only consider those topological spaces which are paracompact and Hausdorff. For a space $X$, we define the Cech cohomology as $\check{H}(X,\mathbb Z):=\displaystyle\lim_{\to} H^\bullet(\mathcal V,\mathbb Z)$ where $\mathcal V$ runs over all open covers of $X$ (ordered by refinement) and $H^\bullet(\mathcal V,\mathbb Z)$ denotes the simplicial cohomology of the nerve of $\mathcal V$. It is well-known that this coincides with the (derived functor) cohomology $H^\bullet(X,\underline{\mathbb Z}_X)$, where $\underline{\mathbb Z}_X$ is the constant sheaf with stalks $\mathbb Z$ on $X$.

In his book "Lectures on Algebraic Topology", Dold defines Cech cohomology (only for locally compact subsets of Euclidean Neighborhood Retracts) $X$ as follows. First choose an embedding $\iota: X\subset E$ into an ENR $E$, and then define $\check{H}(X,\mathbb Z):=\displaystyle\lim_{\to} H^\bullet(U,\mathbb Z)$ where $U$ ranges over the collection of open neighborhoods of $\iota(X)$ in $E$ (ordered by reverse inclusion), and $H^\bullet(U,\mathbb Z)$ is the singular cohomology of $U$.

How can I see that the two definitions are the same? I am ok with a proof even in the special case when $E$ is a topological manifold, as this is the case I am interested in at the moment. I'm also willing to accept the fact that for locally contractible spaces, the Cech cohomology (first definition) is naturally isomorphic with singular cohomology.

$\endgroup$
  • $\begingroup$ Are you allowed to use the fact that $\varprojlim (U,i) \cong \iota(X)$, where $\varprojlim$ is the usual inverse limit and $i$ is the obvious inclusion map? If so, you can then use the fact that usual Cech cohomology is a continuous contravariant function, so $\check{H}(X) \cong \check{H}(\varprojlim (U,i)) \cong \varinjlim (\check{H}(U),i^*)$ and then you're only left with showing that you can replace the final $\check{H}$ with the usual singular cohomology $H$ using some fact about the neighbourhoods - $\endgroup$ – Dan Rust Sep 7 '17 at 12:02
  • $\begingroup$ something like eventually locally contractible in the limit, so that Cech cohomology coincides with cellular cohomology. $\endgroup$ – Dan Rust Sep 7 '17 at 12:03
  • $\begingroup$ Can you tell me a reference for the fact that Cech cohomology is a continuous contravariant functor? I have heard of this before, but don't know a proof. $\endgroup$ – Mohan Swaminathan Sep 7 '17 at 12:21
  • $\begingroup$ Good question. It's probably in Bott and Tu but I can't be certain. $\endgroup$ – Dan Rust Sep 7 '17 at 12:21

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.