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I asked this question on math.stackexchange, but did not get an answer.

Let $f\colon X\rightarrow X'$ be a continuous map between two spaces $X,X'$, which might be arbitrary wild, especially I don't want to work in any convenient category of topological spaces. Let $X=U\cup V$ and $X'=U'\cup V'$ be open covers such that $f(U)\subseteq U'$ and $f(V)\subseteq V'$ holds.

Consider the following claim.

If the three restrictions $f\colon U\rightarrow V$, $f\colon V\rightarrow V'$ and $f\colon U\cap U'\rightarrow V\cap V'$ are weak homotopy equivalences, then so is $f\colon X\rightarrow X'$.

  1. Is this claim true in general? If not, are there mild assumptions on $X$, $X'$ or $X$ and $X'$, such that the claim holds, e.g. does the claim hold if $X$ and $X'$ are Hausdorff spaces?
  2. What about the corresponding claim with homotopy equivalences instead of weak equivalences?
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  • $\begingroup$ @RonnieBrown: Is this related to your work? $\endgroup$
    – Sean Tilson
    Commented Mar 25, 2015 at 10:23

3 Answers 3

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For weak homotopy equivalences this holds always (Theorem 6.7.9 in tom Dieck's Algebraic Topology).

For homotopy equivalences this holds provided the open covers are numerable (Theorem 4.2.7 loc. cit.)

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    $\begingroup$ The original post says that U and V are open, and in this case the pushout is always a homotopy pushout, regardless of numerability. $\endgroup$
    – Dmitri Pavlov
    Commented Mar 23, 2015 at 10:43
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    $\begingroup$ @Dmitri That's true for homotopy pushouts with respect to weak homotopy equivalences and that's what the first part of my answer says. The notion of a homotopy pushout with respect to homotopy equivalences is stronger and here you need numerability. $\endgroup$
    – Karol Szumiło
    Commented Mar 23, 2015 at 11:00
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    $\begingroup$ Indeed, thanks for the clarification. I must say that the usage of “homotopy pushout” in tom Dieck's book without any additional clarifiers to mean “homotopy pushout in topological spaces equipped with homotopy equivalences” is rather confusing. $\endgroup$
    – Dmitri Pavlov
    Commented Mar 23, 2015 at 12:22
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    $\begingroup$ That terminology is indeed non-standard, but it is unfair to say that it is not clarified. The definition is given right above Proposition 4.2.3 which I originally cited. (Tom later changed this to 4.2.7 which does answer his question more directly.) $\endgroup$
    – Karol Szumiło
    Commented Mar 23, 2015 at 12:27
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    $\begingroup$ The way I see it, readers who skip definitions can only blame themselves. $\endgroup$
    – Karol Szumiło
    Commented Mar 23, 2015 at 12:35
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Let me offer sufficient conditions in both cases. They follow from the existence of the following two left proper model structures on the category of topological spaces, and the well-known gluing lemma holding in such categories:

  1. Weak equivalences = weak homotopy equivalences, cofibrations = retracts of relative CW-complexes, fibrations = Serre fibrations [Quillen].

  2. Weak equivalences = homotopy equivalences, cofibrations = closed immersion with the homotopy extension property, fibrations = Hurewicz fibrations [Strom].

In either case, it is enough to assume that $U\cap V$ contains a deformation retract $A\subset U\cap V$ such that $A\subset U$ or $A\subset V$ is a cofibration, and similarly fo $X'$.

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The claim about weak equivalences follows as soon as one proves that the cocartesian squares generated by U←U∩V→V and U'←U'∩V'→V' are also homotopy cocartesian.

To this end one can use Lurie's Seifert-van Kampen theorem (Theorem A.3.1 in Higher Algebra) to establish that these squares are always homotopy cocartesian: in Lurie's notation, take C={1←0→2} and the functor χ sends C to {U←U∩V→V}. The fact that {U,V} is an open cover of X establishes the required property (*).

(Of course, this particular fact had been established long before Lurie's book came out and can be found in many older, less accessible references.)

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  • $\begingroup$ Do you mean, for example, the work of @RonnieBrown? $\endgroup$
    – Sean Tilson
    Commented Mar 24, 2015 at 11:30
  • $\begingroup$ @SeanTilson: If you are referring to the last paragraph, the relevant statement is almost certainly much older and can probably be dug out from the papers by Borsuk, Leray, Weil, and McCord. $\endgroup$
    – Dmitri Pavlov
    Commented Mar 24, 2015 at 21:24

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