(Homotopy)-Fibre of H-spaces

Question

By H-space I mean a unital magma in $Ho(\mathsf{Top}_*)$, i.e. the homotopy category of pointed spaces (feel free to use any of your favourite model :) ).

This means that we have a pointed space $X$ and a map $\mu_X:X \times X \to X$ such that $\mu_X \circ j \simeq \nabla$, where $\nabla : X \vee X \to X$ is the fold (or co-diagonal) map, and $j:X \vee X \to X \times X$ is the pushout-product of the map $* \to X$ with itself.

An H-map between H-spaces is a map $f:X \to Y$ of the underlying spaces such that the obvious square involving $f$ and the multiplication maps is homotopy commutative.

Here comes the question: how to prove that the homotopy fibre of $f$ is again an H-space (in a sensible way, of course)?

Homotopy pullbacks of spaces are weak pullbacks in the homotopy category, and that's how you get a multiplication on the fibre compatible with that of $X$. However, I can't seem to find how to check the desired (homotopy) commutativity, as that would seem to require that weak pullback to be a strict one, which is known to be false.

Thanks in advance for any answer/comment!

Answer 1

This is done explicitly by Zabrodsky in his book Hopf Spaces, although I cannot find the exact theorem.

First note that we need a more strict definition of an H-map. An H-map $(f,F):(X,\mu_X)\rightarrow (Y,\mu_Y)$ is a map $f:X\rightarrow Y$ and a homotopy $F:X\times X\rightarrow Y^I$ relative $X\vee X$, satisfying $e_0\circ F=f\circ\mu_x$ and $e_1\circ F=\mu_Y\circ (f\times f)$, where for $a=0,1$, the map $e_a:Y^I\rightarrow Y$ is the evaluation at $a$.

Now the homotopy fibre $F_f$ of $f$ is the (categorical) pullback of $(X\xrightarrow{f} Y\xleftarrow{e_0} PY)$ where $PY=\{l:[0,1]\rightarrow Y\,|\,l(1)=\ast \}$ is the path space over $Y$. That is, $F_f=\{(x,l)\in X\times PY\,|\,f(x)=l(0)\}$. Define $\mu_f:F_f\times F_f\rightarrow F_f$ by

$\mu_f\left((x,l),(y,m)\right)=\left(\mu_X(x,y),F(x,y)+P\mu_Y(l,m)\right)$

It is straightforward to check that this gives a well defined multiplication on $F_f$. Moreover the canonical projection $F_f\rightarrow X$ and the fibre inclusion $\Omega Y\rightarrow F_f$ are H-maps. The multiplication $\mu_f$ depends on the choice of homotopy $F$. If $\mu_X$, $\mu_Y$ have other properties (homotopy associativity, homotopy commutativity, etc) and $f$ preserves these, then often $\mu_f$ too can be shown to inherit these properties too.