# HG changeset patch
# User Kevin Walker <kevin@canyon23.net>
# Date 1313089146 21600
# Node ID 937214896458d311aeaf5d52d50e2c1008facdca
# Parent  651d1612699971106260ae9de14739ccdec28aa1
add referees request for comment on hh_*(loops)

diff -r 651d16126999 -r 937214896458 RefereeReport.pdf
Binary file RefereeReport.pdf has changed
diff -r 651d16126999 -r 937214896458 blob to-do
--- a/blob to-do	Thu Aug 11 12:08:38 2011 -0600
+++ b/blob to-do	Thu Aug 11 12:59:06 2011 -0600
@@ -3,8 +3,10 @@
 
 * better discussion of systems of fields from disk-like n-cats
     (Is this done by now?)
+    
+* ?? say clearly that certain lemmas don't work for TOP; we're only claiming DIFF and PL (requires small changes in many places)
 
-* need to fix fam-o-homeo argument per discussion with Rob
+* need to fix fam-o-homeo argument per discussion with Rob (or just remove it)
 
 * need to change module axioms to follow changes in n-cat axioms; search for and destroy all the "Homeo_\bd"'s, add a v-cone axiom
 
diff -r 651d16126999 -r 937214896458 text/a_inf_blob.tex
--- a/text/a_inf_blob.tex	Thu Aug 11 12:08:38 2011 -0600
+++ b/text/a_inf_blob.tex	Thu Aug 11 12:59:06 2011 -0600
@@ -418,14 +418,31 @@
 \begin{rem}
 Lurie has shown in \cite[Theorem 3.8.6]{0911.0018} that the topological chiral homology 
 of an $n$-manifold $M$ with coefficients in a certain $E_n$ algebra constructed from $T$ recovers 
-the same space of singular chains on maps from $M$ to $T$, with the additional hypothesis that $T$ is $n-1$-connected.
+the same space of singular chains on maps from $M$ to $T$, with the additional hypothesis that $T$ is $n{-}1$-connected.
 This extra hypothesis is not surprising, in view of the idea described in Example \ref{ex:e-n-alg} 
 that an $E_n$ algebra is roughly equivalent data to an $A_\infty$ $n$-category which 
 is trivial at levels 0 through $n-1$.
 Ricardo Andrade also told us about a similar result.
+
+Specializing still further, Theorem \ref{thm:map-recon} is related to the classical result that for connected spaces $T$
+we have $HH_*(C_*(\Omega T)) \cong H_*(LT)$, that is, the Hochschild homology of based loops in $T$ is isomorphic
+to the homology of the free loop space of $T$ (see \cite{MR793184} and \cite{MR842427}).
+Theorem \ref{thm:map-recon} says that for any space $T$ (connected or not) we have
+$\bc_*(S^1; C_*(\pi^\infty_{\le 1}(T))) \simeq C_*(LT)$.
+Here $C_*(\pi^\infty_{\le 1}(T))$ denotes the singular chain version of the fundamental infinity-groupoid of $T$, 
+whose objects are points in $T$ and morphism chain complexes are $C_*(\paths(t_1 \to t_2))$ for $t_1, t_2 \in T$.
+If $T$ is connected then the $A_\infty$ 1-category $C_*(\pi^\infty_{\le 1}(T))$ is Morita equivalent to the
+$A_\infty$ algebra $C_*(\Omega T)$; 
+the bimodule for the equivalence is the singular chains of the space of paths which start at the base point of $T$.
+Theorem \ref{thm:hochschild} holds for $A_\infty$ 1-categories (though we do not prove that in this paper),
+which then implies that
+\[
+	Hoch_*(C_*(\Omega T)) \simeq Hoch_*(C_*(\pi^\infty_{\le 1}(T)))
+			\simeq \bc_*(S^1; C_*(\pi^\infty_{\le 1}(T))) \simeq C_*(LT) .
+\]
 \end{rem}
 
-\begin{proof}
+\begin{proof}[Proof of Theorem \ref{thm:map-recon}]
 The proof is again similar to that of Theorem \ref{thm:product}.
 
 We begin by constructing a chain map $\psi: \cB^\cT(M) \to C_*(\Maps(M\to T))$.
diff -r 651d16126999 -r 937214896458 text/kw_macros.tex
--- a/text/kw_macros.tex	Thu Aug 11 12:08:38 2011 -0600
+++ b/text/kw_macros.tex	Thu Aug 11 12:59:06 2011 -0600
@@ -64,7 +64,7 @@
 
 % \DeclareMathOperator{\pr}{pr} etc.
 \def\declaremathop#1{\expandafter\DeclareMathOperator\csname #1\endcsname{#1}}
-\applytolist{declaremathop}{im}{gl}{ev}{coinv}{tr}{rot}{Eq}{obj}{mor}{ob}{Rep}{Tet}{cat}{Maps}{Diff}{Homeo}{sign}{supp}{Nbd}{res}{rad}{Compat}{Coll}{Cone}{pr};
+\applytolist{declaremathop}{im}{gl}{ev}{coinv}{tr}{rot}{Eq}{obj}{mor}{ob}{Rep}{Tet}{cat}{Maps}{Diff}{Homeo}{sign}{supp}{Nbd}{res}{rad}{Compat}{Coll}{Cone}{pr}{paths};
 
 \DeclareMathOperator*{\colim}{colim}
 \DeclareMathOperator*{\hocolim}{hocolim}