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I have been reading the discussion from Pushforward and pullback..

I understand that it is quite straight forward to construct a pullback of a vector bundle. In the discussion it is clear that if we have a map $f \colon M \to N$ betweeen manifolds $M$ and $N$ and a vector bundle $\pi_E \colon E \to M$ over $M$ then we can construct a pushforward $f_*E$ over $N$ provided that $f$ is a diffeomorphism. We can get these pullback vector bundles from the cartesian lift. In the case of a opcartesian lift which is related to the pushfoward, is it possible if get this notion provided that $M$ and $N$ are topological spaces?

It is not clear in the discussion that we can get such a pushforward $f_*E$ over $N$ where $N$ and $M$ are topological spaces, in case where $f$ is not necessarily a diffeomorphism. Can $f$ be a homeomorphism in this case? I does not seem possible to me, however I want to understand this rather rigourously.

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    $\begingroup$ The pushforward is always defined as a sheaf. Are you asking for a criterion for it to be a vector bundle? $\endgroup$
    – Vik78
    Commented Sep 28, 2023 at 20:34
  • $\begingroup$ Yes, precisely when $M$ and $N$ are topological spaces in general rather than smooth manifolds. $\endgroup$
    – Siya
    Commented Sep 28, 2023 at 20:37
  • $\begingroup$ Smaller question: what can we say when $N$ is a single point? $\endgroup$
    – Vik78
    Commented Sep 28, 2023 at 21:43
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    $\begingroup$ If $f$ is a homeomorphism and $\xi$ is a bundle on the domain, then the pullback of $\xi$ via $f^{-1}$ is the pushforward of $\xi$ via $f$. $\endgroup$
    – Igor Belegradek
    Commented Sep 28, 2023 at 21:48
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    $\begingroup$ Was that a question to my comment? Yes, the pullback of a vector bundle is a vector bundle, so the pushforward $f_*\xi$ of $\xi$ via $f$ is a vector bundle. On the other hand, some extra structure can be lost, just like for the pullback. For example, if $\xi$ has smooth coordinate charts, and $f$ is a homeomorphism of smooth manifolds, then the obvious coordinate charts for $f_*\xi$ need not be smooth. You never specified what you mean by a vector bundle. $\endgroup$
    – Igor Belegradek
    Commented Sep 29, 2023 at 10:40

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