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Let $X, Y, B$ be three smooth manifolds, and $f : X\to B$, $g : Y\to B$ submersions.

Then $X\times_BY$ exists.

(1) If $X, Y, B$ have the homotopy type of a finite CW complex, does $X\times_BY$?

(2) What is $\pi_n(X\times_BY)$ in relation to $\pi_n(X),\pi_n(Y),\pi_n(B)$?

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  • $\begingroup$ I also wanted to know this,. $\endgroup$
    – Praphulla Koushik
    Commented Mar 29, 2019 at 3:30

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Let $X\times_B Y$ denote the homotopy pullback in spaces of maps $X,Y\to B$. (It's not clear whether you mean the ordinary pullback or the homotopy pullback; if $X$ or $Y$ is compact, then $f$ or $g$ (respectively) will be a fibration, by Ehresmann's theorem; therefore, the homotopy pullback will agree with the ordinary pullback.) Then there is a fiber sequence $\Omega B \to X\times_B Y \to X\times Y$. This gives a long exact sequence $$\cdots \to \pi_k \Omega B = \pi_{k+1} B \to \pi_k(X\times_B Y) \to \pi_k(X\times Y) = \pi_k(X) \oplus \pi_k(Y) \to \pi_k B \to \cdots$$ Given $\pi_\ast(B),\pi_\ast(X)$, and $\pi_\ast(Y)$, you can use this to get at $\pi_\ast(X\times_B Y)$.

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    $\begingroup$ @John : Please see math.stackexchange.com/questions/542889/… This is for reference of other users (including me :D).. there is a relation between homotopy pullback denoted by $X\times^h_BY$ and the pullback given in question denoted by $X\times_BY$.. $\endgroup$
    – Praphulla Koushik
    Commented Mar 29, 2019 at 5:43
  • $\begingroup$ Can you please give reference for “the fiber sequence $\Omega B\rightarrow X\times_BY\rightarrow X\times Y$”... I am assuming $\Omega B$ means loop space of $B$.. Just for confirmation, your $X\times_BY$ is the homotopy pullback and not the usual pullback... $\endgroup$
    – Praphulla Koushik
    Commented Mar 29, 2019 at 5:45
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    $\begingroup$ @PraphullaKoushik This is classical. For a reference, see, e.g., Proposition 3.2 here: people.math.binghamton.edu/malkiewich//fibration_sequences.pdf. $\endgroup$
    – skd
    Commented Mar 29, 2019 at 6:44
  • $\begingroup$ Thanks. I read about homotopy pullback only recently. So I do not know what is classic and what is not.,, thanks for link... I will read,,,, :) $\endgroup$
    – Praphulla Koushik
    Commented Mar 29, 2019 at 7:15
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    $\begingroup$ Thanks. I read about homotopy pullback only recently. So I do not know what is classic and what is not.,, thanks for link... I will read,,,, :) I also found some reference. page no 14 of uregina.ca/~franklam/Math527/Math527_0308.pdf $\endgroup$
    – Praphulla Koushik
    Commented Mar 29, 2019 at 8:17

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