From 82db3620d25000bf1828910f057b088f557909d5 Mon Sep 17 00:00:00 2001 From: Joshua Moerman Date: Tue, 13 Jan 2015 17:16:00 +0100 Subject: [PATCH] Adds group iso (stub) --- thesis/notes/A_K_Quillen_Pair.tex | 19 ++++++++++++++++--- 1 file changed, 16 insertions(+), 3 deletions(-) diff --git a/thesis/notes/A_K_Quillen_Pair.tex b/thesis/notes/A_K_Quillen_Pair.tex index 59759be..67bbd57 100644 --- a/thesis/notes/A_K_Quillen_Pair.tex +++ b/thesis/notes/A_K_Quillen_Pair.tex @@ -52,10 +52,10 @@ The induced adjunction in the previous corollary is given by $LA(X) = A(X)$ for \section{Homotopy groups} -The homotopy groups of cdga's are precisely the dual of the homotopy groups of their associated spaces. +The homotopy groups of augmented cdga's are precisely the dual of the homotopy groups of their associated spaces. \Theorem{cdga-dual-homotopy-groups}{ - Let $X$ be a cofibrant augmented cdga, then + Let $X$ be a cofibrant augmented cdga, then there is a natural bijection $$ \pi_n(KX) \iso \pi^n(X)^\ast. $$ } \Proof{ @@ -71,7 +71,20 @@ The homotopy groups of cdga's are precisely the dual of the homotopy groups of t \todo{Group structure?} } -We get a particularly nice result for minimal cdga's, because the functor $Q$ is the left inverse of the functor $\Lambda$ and the differential is decomposable. +\Theorem{cdga-dual-homotopy-groups-iso}{ + Let $X$ be a $1$-connected cofibrant augmented cdga, then there is a natural group isomorphism $ \pi_n(KX) \iso \pi^n(X)^\ast $. +} +\Proof{ + We will prove that the map in the previous theorem preserves the group structure. We will prove this by endowing a certain cdga with a coalgebra structure, which induces the multiplication in both $\pi_n(KX)$ and $\pi^n(X)^\ast$. + + Since every $1$-connected cdga admits a minimal model, we will assume that $X$ is a minimal model, generated by $V$ (filtered by degree). We first observe that $\pi^n(X) \iso \pi^n(\Lambda V(n))$, since elements of degree $n+1$ or higher do not influence the homology of $QX$. + + Now consider the cofibration $i: \Lambda V(n-1) \cof \Lambda V(n)$ and its associated long exact sequence (\CorollaryRef{long-exact-cdga-homotopy}). It follows that $\pi^n(\Lambda V(n)) \iso \pi^n(\coker(i))$. Now $\coker(i)$ is generated by elements of degree $n$ only (as algebra), i.e. $\coker(i) = (\Lambda W, 0)$ for some vector space $W = W^n$. Define a comultiplication on generators $w \in W$: + \[ \Delta : \Lambda W \to \Lambda W \tensor \Lambda W : w \mapsto 1 \tensor w + w \tensor 1. \] + Since the differential is trivial, this defines a map of cdga's. \todo{show the induced maps in $\pi$} +} + +We get a particularly nice result for minimal cdga's, because the functor $Q$ is the left inverse of the functor $\Lambda$ and the differential is decomposable. \todo{this remark has already been made earlier?} \Corollary{minimal-cdga-homotopy-groups}{ For a minimal cdga $X = \Lambda V$ we get