2
$\begingroup$

Suppose we have a simplicial group $G$. What do we need from $G$ to get first countable $BG$, where $BG$ is a geometric realization of $G$?

$\endgroup$
1
$\begingroup$

A CW-complex is first countable if and only if it's locally finite, and the geometric realization of a simplicial set is locally finite if and only if the original simplicial set was, in that only finitely many non-degenerate simplices may meet any other non-degenerate simplex. I don't see much improvement in this result from starting with a simplicial group.

$\endgroup$
  • $\begingroup$ Thanks, Kevin. I will think about that $\endgroup$ – Fat ninja Nov 30 '18 at 7:59
  • 1
    $\begingroup$ Not so much improvement but a shortcut - for a group, it should suffice to check the condition only for simplices containing one fixed 0-simplex. $\endgroup$ – მამუკა ჯიბლაძე Dec 30 '18 at 6:20

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Not the answer you're looking for? Browse other questions tagged or ask your own question.