# HG changeset patch # User Kevin Walker # Date 1319311613 21600 # Node ID 75c1e11d0f25361c0b6149c7d1efd57a928ad615 # Parent c43f9f8fb3959f89534650a4888c1698ce552479 add remarks about the missing TOP case; searched for all occurrances of "topological" and "continuous" to make sure all other mentions of TOP have been expunged; other minor changes diff -r c43f9f8fb395 -r 75c1e11d0f25 blob to-do --- a/blob to-do Fri Oct 14 08:35:15 2011 -0700 +++ b/blob to-do Sat Oct 22 13:26:53 2011 -0600 @@ -1,10 +1,6 @@ ====== big ====== -* double-check for mentions of TOP - -* (maybe) say somewhere what's missing for TOP - * 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 * probably should go through and refer to new splitting axiom when we need to choose refinements etc. diff -r c43f9f8fb395 -r 75c1e11d0f25 text/appendixes/famodiff.tex --- a/text/appendixes/famodiff.tex Fri Oct 14 08:35:15 2011 -0700 +++ b/text/appendixes/famodiff.tex Sat Oct 22 13:26:53 2011 -0600 @@ -372,6 +372,13 @@ \medskip +Topological (merely continuous) homeomorphisms are conspicuously absent from the +list of classes of maps for which the above lemma hold. +The $k=1$ case of Lemma \ref{basic_adaptation_lemma} for plain, continuous homeomorphisms +is more or less equivalent to Corollary 1.3 of \cite{MR0283802}. +We suspect that the proof found in \cite{MR0283802} of that corollary can be adapted to many-parameter families of +homeomorphisms, but so far the details have alluded us. + %%%%%% Lo, \noop{...} \noop{ diff -r c43f9f8fb395 -r 75c1e11d0f25 text/intro.tex --- a/text/intro.tex Fri Oct 14 08:35:15 2011 -0700 +++ b/text/intro.tex Sat Oct 22 13:26:53 2011 -0600 @@ -496,7 +496,7 @@ Much more could be said about other types of manifolds, in particular oriented, $\operatorname{Spin}$ and $\operatorname{Pin}^{\pm}$ manifolds, where boundary issues become more complicated. (We'd recommend thinking about boundaries as germs, rather than just codimension $1$ manifolds.) -We've also take the path of least resistance by considering $\operatorname{PL}$ manifolds; +We've also take the path of least resistance by concentrating on PL manifolds; there may be some differences for topological manifolds and smooth manifolds. The paper ``Skein homology'' \cite{MR1624157} has similar motivations, and it may be diff -r c43f9f8fb395 -r 75c1e11d0f25 text/ncat.tex --- a/text/ncat.tex Fri Oct 14 08:35:15 2011 -0700 +++ b/text/ncat.tex Sat Oct 22 13:26:53 2011 -0600 @@ -19,7 +19,7 @@ and avoid combinatorial questions about, for example, finding a minimal sufficient collection of generalized associativity axioms; we prefer maximal sets of axioms to minimal sets. It is easy to show that examples of topological origin -(e.g.\ categories whose morphisms are maps into spaces or decorated balls, or bordism categories), +(e.g.\ categories whose morphisms are maps into spaces or decorated balls, or bordism categories) satisfy our axioms. To show that examples of a more purely algebraic origin satisfy our axioms, one would typically need the combinatorial @@ -42,7 +42,7 @@ Strictly speaking, before we can state the axioms for $k$-morphisms we need all the axioms for $k{-}1$-morphisms. Readers who prefer things to be presented in a strictly logical order should read this -subsection $n+1$ times, first setting $k=0$, then $k=1$, and so on until they reach $k=n$. +subsection $n{+}1$ times, first setting $k=0$, then $k=1$, and so on until they reach $k=n$. \medskip @@ -91,6 +91,11 @@ For each flavor of manifold there is a corresponding flavor of $n$-category. For simplicity, we will concentrate on the case of PL unoriented manifolds. +(An interesting open question is whether the techniques of this paper can be adapted to topological +manifolds and plain, merely continuous homeomorphisms. +The main obstacles are proving a version of Lemma \ref{basic_adaptation_lemma} and adapting the +transversality arguments used in Lemma \ref{lem:colim-injective}.) + An ambitious reader may want to keep in mind two other classes of balls. The first is balls equipped with a map to some other space $Y$ (c.f. \cite{MR2079378}). This will be used below (see the end of \S \ref{ss:product-formula}) to describe the blob complex of a fiber bundle with @@ -660,7 +665,6 @@ The revised axiom is -%\addtocounter{axiom}{-1} \begin{axiom}[Extended isotopy invariance in dimension $n$] \label{axiom:extended-isotopies} Let $X$ be an $n$-ball, $b \in \cC(X)$, and $f: X\to X$ be a homeomorphism which @@ -675,7 +679,7 @@ We need one additional axiom. It says, roughly, that given a $k$-ball $X$, $k