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?

Improve this question
$\endgroup$
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. $\endgroup$
    – Neil Strickland
    Commented Sep 3, 2011 at 17:10

1 Answer 1

Reset to default
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).

Improve this answer
$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .