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Let $f: X \to Y$ be a fibration of pointed Kan complexes, and let $F$ be the fiber.

Question: How do you prove that the following diagram of homotopy groups commutes?:

$\pi_n(Y) \to \pi_{n-1}(\Omega Y)$

$\ \ \downarrow \ \ \ \ \ \ \ \ \ \ \ \ \ \downarrow$

$\pi_{n-1}(F) \to \pi_{n-2}(\Omega F)$

Admittedly, I don't know for certain that it commutes, but it looks like it should.

All the arrows are boundary maps ($\delta$) from long exact sequences of homotopy groups for a fibration. The horizontal maps are isomorphisms.

The definition of $\delta$ that I'm using is that for for a fibration $X \to Y$, $\alpha \in \pi_n(Y)$, $\delta ([\alpha]) = [\beta d^0]$, where $\beta:\Delta^n \to X$ fits into the following diagram:

$ \Lambda^n_0 \to^* X$

$\ \downarrow \ \ \ \ \ \downarrow$

$\Delta^n \to^\alpha Y$

So far, I've attempted to chase elements around this diagram and use prismatic arguments, but I haven't found one that works.

$\Omega F \to \Omega X \to \Omega Y$

$\ \ \downarrow\ \ \ \ \ \ \downarrow\ \ \ \ \ \ \ \downarrow$

$PF \to PX \to PY$

$\ \ \downarrow\ \ \ \ \ \ \downarrow\ \ \ \ \ \ \ \downarrow$

$\ \ F \to \ \ X \to \ \ Y$

It feels like a useful fact that $PX \to PY\times_Y X$ is a fibration, which is true since pointed simplicial sets form a simplicial model category.

Edit: The path and loop spaces I'm using are defined by the following: For pointed simplicial sets X and Y, define the simplicial set $hom_*(X,Y)$ to have n- simplices $hom_{sSet_*}(X \wedge \Delta^n_+,Y)$, where $\Delta^n_+$ is the standard n-simplex with a disjoint basepoint.

Then I'm using $PX=hom_*(\Delta^1 , X)$

and $\Omega X=hom_*(\Delta^1/\partial \Delta^1,X)$

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You haven't said what model for the loop space you're using... –  Dan Ramras Sep 14 '10 at 23:55
    
Thanks, I've added that now. –  user9109 Sep 15 '10 at 1:55
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1 Answer

up vote 1 down vote accepted

The boundary map of homotopy groups is induced by a map $\partial : \Omega Y \to F$ of spaces; then the commutativity follows from the naturality of the isomorphism $\pi_n \circ \Omega \cong \pi_{n+1}$.

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Thanks, I found maps of spaces $\Omega Y \to PB\times_B E ^{\simeq\atop \leftarrow} F$, which cut my diagram in half and makes both parts commute. –  user9109 Sep 15 '10 at 14:03
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