615 \begin{axiom}[\textup{\textbf{[$A_\infty$ version]}} Families of homeomorphisms act in dimension $n$.] |
615 \begin{axiom}[\textup{\textbf{[$A_\infty$ version]}} Families of homeomorphisms act in dimension $n$.] |
616 For each $n$-ball $X$ and each $c\in \cl{\cC}(\bd X)$ we have a map of chain complexes |
616 For each $n$-ball $X$ and each $c\in \cl{\cC}(\bd X)$ we have a map of chain complexes |
617 \[ |
617 \[ |
618 C_*(\Homeo_\bd(X))\ot \cC(X; c) \to \cC(X; c) . |
618 C_*(\Homeo_\bd(X))\ot \cC(X; c) \to \cC(X; c) . |
619 \] |
619 \] |
620 These action maps are required to be associative up to homotopy |
620 These action maps are required to be associative up to homotopy, |
621 \nn{iterated homotopy?}, and also compatible with composition (gluing) in the sense that |
621 %\nn{iterated homotopy?} |
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622 and also compatible with composition (gluing) in the sense that |
622 a diagram like the one in Theorem \ref{thm:CH} commutes. |
623 a diagram like the one in Theorem \ref{thm:CH} commutes. |
623 \nn{repeat diagram here?} |
624 %\nn{repeat diagram here?} |
624 \nn{restate this with $\Homeo(X\to X')$? what about boundary fixing property?} |
625 %\nn{restate this with $\Homeo(X\to X')$? what about boundary fixing property?} |
625 \end{axiom} |
626 \end{axiom} |
626 |
627 |
627 We should strengthen the above axiom to apply to families of collar maps. |
628 We should strengthen the above axiom to apply to families of collar maps. |
628 To do this we need to explain how collar maps form a topological space. |
629 To do this we need to explain how collar maps form a topological space. |
629 Roughly, the set of collared $n{-}1$-balls in the boundary of an $n$-ball has a natural topology, |
630 Roughly, the set of collared $n{-}1$-balls in the boundary of an $n$-ball has a natural topology, |
822 This example will be used in Theorem \ref{thm:product} below, which allows us to compute the blob complex of a product. |
823 This example will be used in Theorem \ref{thm:product} below, which allows us to compute the blob complex of a product. |
823 Notice that with $F$ a point, the above example is a construction turning a topological |
824 Notice that with $F$ a point, the above example is a construction turning a topological |
824 $n$-category $\cC$ into an $A_\infty$ $n$-category. |
825 $n$-category $\cC$ into an $A_\infty$ $n$-category. |
825 We think of this as providing a ``free resolution" |
826 We think of this as providing a ``free resolution" |
826 of the topological $n$-category. |
827 of the topological $n$-category. |
827 \nn{say something about cofibrant replacements?} |
828 %\nn{say something about cofibrant replacements?} |
828 In fact, there is also a trivial, but mostly uninteresting, way to do this: |
829 In fact, there is also a trivial, but mostly uninteresting, way to do this: |
829 we can think of each vector space associated to an $n$-ball as a chain complex concentrated in degree $0$, |
830 we can think of each vector space associated to an $n$-ball as a chain complex concentrated in degree $0$, |
830 and take $\CD{B}$ to act trivially. |
831 and take $\CD{B}$ to act trivially. |
831 |
832 |
832 Be careful that the ``free resolution" of the topological $n$-category $\pi_{\leq n}(T)$ is not the $A_\infty$ $n$-category $\pi^\infty_{\leq n}(T)$. |
833 Be careful that the ``free resolution" of the topological $n$-category $\pi_{\leq n}(T)$ is not the $A_\infty$ $n$-category $\pi^\infty_{\leq n}(T)$. |
846 submanifolds $W$ of $X\times \Real^\infty$ such that |
847 submanifolds $W$ of $X\times \Real^\infty$ such that |
847 $W$ coincides with $c$ at $\bd X \times \Real^\infty$. |
848 $W$ coincides with $c$ at $\bd X \times \Real^\infty$. |
848 (The topology on this space is induced by ambient isotopy rel boundary. |
849 (The topology on this space is induced by ambient isotopy rel boundary. |
849 This is homotopy equivalent to a disjoint union of copies $\mathrm{B}\!\Homeo(W')$, where |
850 This is homotopy equivalent to a disjoint union of copies $\mathrm{B}\!\Homeo(W')$, where |
850 $W'$ runs though representatives of homeomorphism types of such manifolds.) |
851 $W'$ runs though representatives of homeomorphism types of such manifolds.) |
851 \nn{check this} |
|
852 \end{example} |
852 \end{example} |
853 |
853 |
854 |
854 |
855 |
855 |
856 Let $\cE\cB_n$ be the operad of smooth embeddings of $k$ (little) |
856 Let $\cE\cB_n$ be the operad of smooth embeddings of $k$ (little) |
857 copies of the standard $n$-ball $B^n$ into another (big) copy of $B^n$. |
857 copies of the standard $n$-ball $B^n$ into another (big) copy of $B^n$. |
858 (We require that the interiors of the little balls be disjoint, but their |
858 (We require that the interiors of the little balls be disjoint, but their |
859 boundaries are allowed to meet. |
859 boundaries are allowed to meet. |
860 Note in particular that the space for $k=1$ contains a copy of $\Diff(B^n)$, namely |
860 Note in particular that the space for $k=1$ contains a copy of $\Diff(B^n)$, namely |
861 the embeddings of a ``little" ball with image all of the big ball $B^n$. |
861 the embeddings of a ``little" ball with image all of the big ball $B^n$. |
862 \nn{should we warn that the inclusion of this copy of $\Diff(B^n)$ is not a homotopy equivalence?}) |
862 (But note also that this inclusion is not |
|
863 necessarily a homotopy equivalence.) |
863 The operad $\cE\cB_n$ is homotopy equivalent to the standard framed little $n$-ball operad: |
864 The operad $\cE\cB_n$ is homotopy equivalent to the standard framed little $n$-ball operad: |
864 by shrinking the little balls (precomposing them with dilations), |
865 by shrinking the little balls (precomposing them with dilations), |
865 we see that both operads are homotopic to the space of $k$ framed points |
866 we see that both operads are homotopic to the space of $k$ framed points |
866 in $B^n$. |
867 in $B^n$. |
867 It is easy to see that $n$-fold loop spaces $\Omega^n(T)$ have |
868 It is easy to see that $n$-fold loop spaces $\Omega^n(T)$ have |
868 an action of $\cE\cB_n$. |
869 an action of $\cE\cB_n$. |
869 \nn{add citation for this operad if we can find one} |
870 %\nn{add citation for this operad if we can find one} |
870 |
871 |
871 \begin{example}[$E_n$ algebras] |
872 \begin{example}[$E_n$ algebras] |
872 \rm |
873 \rm |
873 \label{ex:e-n-alg} |
874 \label{ex:e-n-alg} |
874 |
875 |
891 --- composition and $\Diff(X\to X')$ action --- |
892 --- composition and $\Diff(X\to X')$ action --- |
892 also comes from the $\cE\cB_n$ action on $A$. |
893 also comes from the $\cE\cB_n$ action on $A$. |
893 \nn{should we spell this out?} |
894 \nn{should we spell this out?} |
894 |
895 |
895 \nn{Should remark that the associated hocolim for manifolds |
896 \nn{Should remark that the associated hocolim for manifolds |
896 is agrees with Lurie's topological chiral homology construction; maybe wait |
897 agrees with Lurie's topological chiral homology construction; maybe wait |
897 until next subsection to say that?} |
898 until next subsection to say that?} |
898 |
899 |
899 Conversely, one can show that a topological $A_\infty$ $n$-category $\cC$, where the $k$-morphisms |
900 Conversely, one can show that a topological $A_\infty$ $n$-category $\cC$, where the $k$-morphisms |
900 $\cC(X)$ are trivial (single point) for $k<n$, gives rise to |
901 $\cC(X)$ are trivial (single point) for $k<n$, gives rise to |
901 an $\cE\cB_n$-algebra. |
902 an $\cE\cB_n$-algebra. |
916 In the case of plain $n$-categories, this construction factors into a construction of a |
917 In the case of plain $n$-categories, this construction factors into a construction of a |
917 system of fields and local relations, followed by the usual TQFT definition of a |
918 system of fields and local relations, followed by the usual TQFT definition of a |
918 vector space invariant of manifolds given as Definition \ref{defn:TQFT-invariant}. |
919 vector space invariant of manifolds given as Definition \ref{defn:TQFT-invariant}. |
919 For an $A_\infty$ $n$-category, $\cl{\cC}$ is defined using a homotopy colimit instead. |
920 For an $A_\infty$ $n$-category, $\cl{\cC}$ is defined using a homotopy colimit instead. |
920 Recall that we can take a plain $n$-category $\cC$ and pass to the ``free resolution", |
921 Recall that we can take a plain $n$-category $\cC$ and pass to the ``free resolution", |
921 an $A_\infty$ $n$-category $\bc_*(\cC)$, by computing the blob complex of balls (recall Example \ref{ex:blob-complexes-of-balls} above). |
922 an $A_\infty$ $n$-category $\bc_*(\cC)$, by computing the blob complex of balls |
|
923 (recall Example \ref{ex:blob-complexes-of-balls} above). |
922 We will show in Corollary \ref{cor:new-old} below that the homotopy colimit invariant |
924 We will show in Corollary \ref{cor:new-old} below that the homotopy colimit invariant |
923 for a manifold $M$ associated to this $A_\infty$ $n$-category is actually the same as the original blob complex for $M$ with coefficients in $\cC$. |
925 for a manifold $M$ associated to this $A_\infty$ $n$-category is actually the |
924 |
926 same as the original blob complex for $M$ with coefficients in $\cC$. |
925 We will first define the ``cell-decomposition" poset $\cell(W)$ for any $k$-manifold $W$, for $1 \leq k \leq n$. |
927 |
|
928 We will first define the ``decomposition" poset $\cell(W)$ for any $k$-manifold $W$, for $1 \leq k \leq n$. |
926 An $n$-category $\cC$ provides a functor from this poset to the category of sets, |
929 An $n$-category $\cC$ provides a functor from this poset to the category of sets, |
927 and we will define $\cl{\cC}(W)$ as a suitable colimit |
930 and we will define $\cl{\cC}(W)$ as a suitable colimit |
928 (or homotopy colimit in the $A_\infty$ case) of this functor. |
931 (or homotopy colimit in the $A_\infty$ case) of this functor. |
929 We'll later give a more explicit description of this colimit. |
932 We'll later give a more explicit description of this colimit. |
930 In the case that the $n$-category $\cC$ is enriched (e.g. associates vector spaces or chain complexes to $n$-balls with boundary data), |
933 In the case that the $n$-category $\cC$ is enriched (e.g. associates vector spaces or chain complexes to $n$-balls with boundary data), |
931 then the resulting colimit is also enriched, that is, the set associated to $W$ splits into subsets according to boundary data, and each of these subsets has the appropriate structure (e.g. a vector space or chain complex). |
934 then the resulting colimit is also enriched, that is, the set associated to $W$ splits into subsets according to boundary data, and each of these subsets has the appropriate structure (e.g. a vector space or chain complex). |
932 |
935 |
933 Define a {\it permissible decomposition} of $W$ to be a cell decomposition |
936 Recall (Definition \ref{defn:gluing-decomposition}) that a {\it ball decomposition} of $W$ is a |
934 \[ |
937 sequence of gluings $M_0\to M_1\to\cdots\to M_m = W$ such that $M_0$ is a disjoint union of balls |
935 W = \bigcup_a X_a , |
938 $\du_a X_a$. |
936 \] |
939 Abusing notation, we let $X_a$ denote both the ball (component of $M_0$) and |
937 where each closed top-dimensional cell $X_a$ is an embedded $k$-ball. |
940 its image in $W$ (which is not necessarily a ball --- parts of $\bd X_a$ may have been glued together). |
938 \nn{need to define this more carefully} |
941 Define a {\it permissible decomposition} of $W$ to be a map |
939 Given permissible decompositions $x$ and $y$, we say that $x$ is a refinement |
942 \[ |
940 of $y$, or write $x \le y$, if each $k$-ball of $y$ is a union of $k$-balls of $x$. |
943 \coprod_a X_a \to W, |
|
944 \] |
|
945 which can be completed to a ball decomposition $\du_a X_a = M_0\to\cdots\to M_m = W$. |
|
946 Roughly, a permissible decomposition is like a ball decomposition where we don't care in which order the balls |
|
947 are glued up to yield $W$, so long as there is some (non-pathological) way to glue them. |
|
948 |
|
949 Given permissible decompositions $x = \{X_a\}$ and $y = \{Y_b\}$ or $W$, we say that $x$ is a refinement |
|
950 of $y$, or write $x \le y$, if there is a ball decomposition $\du_a X_a = M_0\to\cdots\to M_m = W$ |
|
951 with $\du_b Y_b = M_i$ for some $i$. |
941 |
952 |
942 \begin{defn} |
953 \begin{defn} |
943 The category (poset) $\cell(W)$ has objects the permissible decompositions of $W$, |
954 The category (poset) $\cell(W)$ has objects the permissible decompositions of $W$, |
944 and a unique morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$. |
955 and a unique morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$. |
945 See Figure \ref{partofJfig} for an example. |
956 See Figure \ref{partofJfig} for an example. |
1226 (For $k=n$, see below.)} |
1237 (For $k=n$, see below.)} |
1227 \end{module-axiom} |
1238 \end{module-axiom} |
1228 |
1239 |
1229 \begin{module-axiom}[Strict associativity] |
1240 \begin{module-axiom}[Strict associativity] |
1230 The composition and action maps above are strictly associative. |
1241 The composition and action maps above are strictly associative. |
|
1242 Given any decomposition of a large marked ball into smaller marked and unmarked balls |
|
1243 any sequence of pairwise gluings yields (via composition and action maps) the same result. |
1231 \end{module-axiom} |
1244 \end{module-axiom} |
1232 |
|
1233 \nn{should say that this is multifold, not just 3-fold} |
|
1234 |
1245 |
1235 Note that the above associativity axiom applies to mixtures of module composition, |
1246 Note that the above associativity axiom applies to mixtures of module composition, |
1236 action maps and $n$-category composition. |
1247 action maps and $n$-category composition. |
1237 See Figure \ref{zzz1b}. |
1248 See Figure \ref{zzz1b}. |
1238 |
1249 |