# pullback and fiber sequence

Let $$A\rightarrow D\leftarrow C$$ a diagram of connected pointed toplogical space where $$A\rightarrow D$$ is a fibration. Denote $$P=A\times_{D}C$$. We obtain a homotopy fiber sequence $$\Omega D\rightarrow P\rightarrow A\times C$$

If we suppose that $$D=\Omega X$$ for some pointed topological space $$X$$. Do we obtain a homotopy fiber sequence $$P\rightarrow A\times C\rightarrow D ?$$ where the map $$A\times C\rightarrow D$$ is obtained as a composition $$A\times C\rightarrow D\times D\rightarrow D$$ (the second map is a concatenation of loops)

• The hofiber of $A\times C\to D$ has a model consisting of points $(a,c)$ together with a homotopy $f(a)g(c)^{-1} \sim *$, so points $(a,c)$ together with a homotopy $f(a)\sim g(c)$, so it amounts to a point in a model of the homotopy pullback $A\times^h_D C$, but $P$ is precisely that homotopy pullback, because $A\to D$ is a fibration. – Max Jan 18 at 13:42
• @Max if i understand correctly your comment then the answer is YES ? – lab Jan 18 at 13:49
• This is true if one of the loops is inverted before the concatenation in the definition of the map $D \times D \to D$ and the chosen point of $D$ is the trivial loop. – Valery Isaev Jan 18 at 14:24
• @ValeryIsaev Now I see thanks! – lab Jan 18 at 14:32

1. The space $$P$$ sits in homotopy pullback diagram $$\require{AMScd}$$ $$\begin{CD} P @>>> D \\ @VVV@VVV \\ A\times C @>>> D\times D \end{CD}$$ where the the right vertical map is the diagonal. In fact, one can see this by replacing the latter map with the free path fibration $$D^I \to D\times D$$. After this replacement the diagram becomes a pullback and a pullback with one of the terminal maps a fibration is always a homotopy pullback.
2. When $$D = \Omega X$$, there the diagonal is induced by the map $$m:D\times D \to D$$ given by $$(\gamma_1,\gamma_2) \mapsto \gamma_1 \cdot \bar\gamma_2$$, where the bar means loop inversion. This means that there is a commutative homotopy pullback $$\begin{CD} D @>\text{diag} >> D\times D \\ @VVV @VV m V \\ * @>>> D \end{CD}$$ where $$*$$ is some contractible space. (Perhaps the easiest way to see this is to note that the diagonal $$X\to X \times X$$ has homotopy fiber $$D$$.)
3. The base change of a map which is induced in also induced: this means that there is a commutative homotopy pullback $$\begin{CD} P @>>> A \times C \\ @VVV @VVV \\ * @>>> D \end{CD}$$ where the bottom map is the same as in the previous diagram.