10
$\begingroup$

Given a space $X$, what conditions on $X$ can you give to ensure that the diagonal map $X\to X\times X$ is a Hurewicz cofibration? (I am happy to assume that $X$ is compactly generated weak Hausdorff, or even just Hausdorf.)

More generally, given a map $f:X\to Y$, when is $X\to X\times_YX$ a Hurewicz cofibration?

$\endgroup$
1
  • 4
    $\begingroup$ This condition is called "equi-locally connected" or "locally equiconnected". If you search for those terms you'll find some results, but I don't think that the literature is extensive. I think Peter May and his collaborators (Elmendorf, Sigurdsson?) have used this condition in various places in connection with parametrised spectra etc. $\endgroup$ Sep 3, 2011 at 17:10

1 Answer 1

9
$\begingroup$

The diagonal condition was used crucially in Milnor's classical paper. Milnor, John. On spaces having the homotopy type of a CW-complex. Trans. Amer. Math. Soc. 90 1959 272–280. He gives earlier references to Fox and Serre. In the parametrized generalization, there is a general fiberwise NDR pair characterization of cofibrations that applies (e.g. Lemma 5.2.4 in May and Sigurdsson, Parametrized homotopy theory).

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

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