|
1 %!TEX root = ../blob1.tex |
|
2 |
|
3 \section{Commutative algebras as \texorpdfstring{$n$}{n}-categories} |
|
4 \label{sec:comm_alg} |
|
5 |
|
6 If $C$ is a commutative algebra it |
|
7 can also be thought of as an $n$-category whose $j$-morphisms are trivial for |
|
8 $j<n$ and whose $n$-morphisms are $C$. |
|
9 The goal of this appendix is to compute |
|
10 $\bc_*(M^n, C)$ for various commutative algebras $C$. |
|
11 |
|
12 Moreover, we conjecture that the blob complex $\bc_*(M^n, $C$)$, for $C$ a commutative |
|
13 algebra is homotopy equivalent to the higher Hochschild complex for $M^n$ with |
|
14 coefficients in $C$ (see \cite{MR0339132, MR1755114, MR2383113}). |
|
15 This possibility was suggested to us by Thomas Tradler. |
|
16 |
|
17 |
|
18 \medskip |
|
19 |
|
20 Let $k[t]$ denote the ring of polynomials in $t$ with coefficients in $k$. |
|
21 |
|
22 Let $\Sigma^i(M)$ denote the $i$-th symmetric power of $M$, the configuration space of $i$ |
|
23 unlabeled points in $M$. |
|
24 Note that $\Sigma^0(M)$ is a point. |
|
25 Let $\Sigma^\infty(M) = \coprod_{i=0}^\infty \Sigma^i(M)$. |
|
26 |
|
27 Let $C_*(X, k)$ denote the singular chain complex of the space $X$ with coefficients in $k$. |
|
28 |
|
29 \begin{prop} \label{sympowerprop} |
|
30 $\bc_*(M, k[t])$ is homotopy equivalent to $C_*(\Sigma^\infty(M), k)$. |
|
31 \end{prop} |
|
32 |
|
33 \begin{proof} |
|
34 We will use acyclic models (\S \ref{sec:moam}). |
|
35 Our first task: For each blob diagram $b$ define a subcomplex $R(b)_* \sub C_*(\Sigma^\infty(M))$ |
|
36 satisfying the conditions of Theorem \ref{moam-thm}. |
|
37 If $b$ is a 0-blob diagram, then it is just a $k[t]$ field on $M$, which is a |
|
38 finite unordered collection of points of $M$ with multiplicities, which is |
|
39 a point in $\Sigma^\infty(M)$. |
|
40 Define $R(b)_*$ to be the singular chain complex of this point. |
|
41 If $(B, u, r)$ is an $i$-blob diagram, let $D\sub M$ be its support (the union of the blobs). |
|
42 The path components of $\Sigma^\infty(D)$ are contractible, and these components are indexed |
|
43 by the numbers of points in each component of $D$. |
|
44 We may assume that the blob labels $u$ have homogeneous $t$ degree in $k[t]$, and so |
|
45 $u$ picks out a component $X \sub \Sigma^\infty(D)$. |
|
46 The field $r$ on $M\setminus D$ can be thought of as a point in $\Sigma^\infty(M\setminus D)$, |
|
47 and using this point we can embed $X$ in $\Sigma^\infty(M)$. |
|
48 Define $R(B, u, r)_*$ to be the singular chain complex of $X$, thought of as a |
|
49 subspace of $\Sigma^\infty(M)$. |
|
50 It is easy to see that $R(\cdot)_*$ satisfies the condition on boundaries from |
|
51 Theorem \ref{moam-thm}. |
|
52 Thus we have defined (up to homotopy) a map from |
|
53 $\bc_*(M, k[t])$ to $C_*(\Sigma^\infty(M))$. |
|
54 |
|
55 Next we define a map going the other direction. |
|
56 First we replace $C_*(\Sigma^\infty(M))$ with a homotopy equivalent |
|
57 subcomplex $S_*$ of small simplices. |
|
58 Roughly, we define $c\in C_*(\Sigma^\infty(M))$ to be small if the |
|
59 corresponding track of points in $M$ |
|
60 is contained in a disjoint union of balls. |
|
61 Because there could be different, inequivalent choices of such balls, we must a bit more careful. |
|
62 \nn{this runs into the same issues as in defining evmap. |
|
63 either refer there for details, or use the simp-space-ish version of the blob complex, |
|
64 which makes things easier here.} |
|
65 |
|
66 \nn{...} |
|
67 |
|
68 |
|
69 We will define, for each simplex $c$ of $S_*$, a contractible subspace |
|
70 $R(c)_* \sub \bc_*(M, k[t])$. |
|
71 If $c$ is a 0-simplex we use the identification of the fields $\cC(M)$ and |
|
72 $\Sigma^\infty(M)$ described above. |
|
73 Now let $c$ be an $i$-simplex of $S_*$. |
|
74 Choose a metric on $M$, which induces a metric on $\Sigma^j(M)$. |
|
75 We may assume that the diameter of $c$ is small --- that is, $C_*(\Sigma^j(M))$ |
|
76 is homotopy equivalent to the subcomplex of small simplices. |
|
77 How small? $(2r)/3j$, where $r$ is the radius of injectivity of the metric. |
|
78 Let $T\sub M$ be the ``track" of $c$ in $M$. |
|
79 \nn{do we need to define this precisely?} |
|
80 Choose a neighborhood $D$ of $T$ which is a disjoint union of balls of small diameter. |
|
81 \nn{need to say more precisely how small} |
|
82 Define $R(c)_*$ to be $\bc_*(D; k[t]) \sub \bc_*(M; k[t])$. |
|
83 This is contractible by Proposition \ref{bcontract}. |
|
84 We can arrange that the boundary/inclusion condition is satisfied if we start with |
|
85 low-dimensional simplices and work our way up. |
|
86 \nn{need to be more precise} |
|
87 |
|
88 \nn{still to do: show indep of choice of metric; show compositions are homotopic to the identity |
|
89 (for this, might need a lemma that says we can assume that blob diameters are small)} |
|
90 \end{proof} |
|
91 |
|
92 |
|
93 \begin{prop} \label{ktchprop} |
|
94 The above maps are compatible with the evaluation map actions of $C_*(\Homeo(M))$. |
|
95 \end{prop} |
|
96 |
|
97 \begin{proof} |
|
98 The actions agree in degree 0, and both are compatible with gluing. |
|
99 (cf. uniqueness statement in Theorem \ref{thm:CH}.) |
|
100 \nn{if Theorem \ref{thm:CH} is rewritten/rearranged, make sure uniqueness discussion is properly referenced from here} |
|
101 \end{proof} |
|
102 |
|
103 \medskip |
|
104 |
|
105 In view of Theorem \ref{thm:hochschild}, we have proved that $HH_*(k[t]) \cong C_*(\Sigma^\infty(S^1), k)$, |
|
106 and that the cyclic homology of $k[t]$ is related to the action of rotations |
|
107 on $C_*(\Sigma^\infty(S^1), k)$. |
|
108 \nn{probably should put a more precise statement about cyclic homology and $S^1$ actions in the Hochschild section} |
|
109 Let us check this directly. |
|
110 |
|
111 The algebra $k[t]$ has a resolution |
|
112 $k[t] \tensor k[t] \xrightarrow{t\tensor 1 - 1 \tensor t} k[t] \tensor k[t]$, |
|
113 which has coinvariants $k[t] \xrightarrow{0} k[t]$. |
|
114 So we have $HH_i(k[t]) \cong k[t]$ for $i=0,1$ and zero for $i\ge 2$. |
|
115 (See also \cite[3.2.2]{MR1600246}.) This computation also tells us the $t$-gradings: |
|
116 $HH_0(k[t]) \iso k[t]$ is in the usual grading, and $HH_1(k[t]) \iso k[t]$ is shifted up by one. |
|
117 |
|
118 We can define a flow on $\Sigma^j(S^1)$ by having the points repel each other. |
|
119 The fixed points of this flow are the equally spaced configurations. |
|
120 This defines a deformation retraction from $\Sigma^j(S^1)$ to $S^1/j$ ($S^1$ modulo a $2\pi/j$ rotation). |
|
121 The fiber of this map is $\Delta^{j-1}$, the $(j-1)$-simplex, |
|
122 and the holonomy of the $\Delta^{j-1}$ bundle |
|
123 over $S^1/j$ is induced by the cyclic permutation of its $j$ vertices. |
|
124 |
|
125 In particular, $\Sigma^j(S^1)$ is homotopy equivalent to a circle for $j>0$, and |
|
126 of course $\Sigma^0(S^1)$ is a point. |
|
127 Thus the singular homology $H_i(\Sigma^\infty(S^1))$ has infinitely many generators for $i=0,1$ |
|
128 and is zero for $i\ge 2$. |
|
129 Note that the $j$-grading here matches with the $t$-grading on the algebraic side. |
|
130 |
|
131 By Proposition \ref{ktchprop}, |
|
132 the cyclic homology of $k[t]$ is the $S^1$-equivariant homology of $\Sigma^\infty(S^1)$. |
|
133 Up to homotopy, $S^1$ acts by $j$-fold rotation on $\Sigma^j(S^1) \simeq S^1/j$. |
|
134 If $k = \z$, $\Sigma^j(S^1)$ contributes the homology of an infinite lens space: $\z$ in degree |
|
135 0, $\z/j \z$ in odd degrees, and 0 in positive even degrees. |
|
136 The point $\Sigma^0(S^1)$ contributes the homology of $BS^1$ which is $\z$ in even |
|
137 degrees and 0 in odd degrees. |
|
138 This agrees with the calculation in \cite[\S 3.1.7]{MR1600246}. |
|
139 |
|
140 \medskip |
|
141 |
|
142 Next we consider the case $C = k[t_1, \ldots, t_m]$, commutative polynomials in $m$ variables. |
|
143 Let $\Sigma_m^\infty(M)$ be the $m$-colored infinite symmetric power of $M$, that is, configurations |
|
144 of points on $M$ which can have any of $m$ distinct colors but are otherwise indistinguishable. |
|
145 The components of $\Sigma_m^\infty(M)$ are indexed by $m$-tuples of natural numbers |
|
146 corresponding to the number of points of each color of a configuration. |
|
147 A proof similar to that of \ref{sympowerprop} shows that |
|
148 |
|
149 \begin{prop} |
|
150 $\bc_*(M, k[t_1, \ldots, t_m])$ is homotopy equivalent to $C_*(\Sigma_m^\infty(M), k)$. |
|
151 \end{prop} |
|
152 |
|
153 According to \cite[3.2.2]{MR1600246}, |
|
154 \[ |
|
155 HH_n(k[t_1, \ldots, t_m]) \cong \Lambda^n(k^m) \otimes k[t_1, \ldots, t_m] . |
|
156 \] |
|
157 Let us check that this is also the singular homology of $\Sigma_m^\infty(S^1)$. |
|
158 We will content ourselves with the case $k = \z$. |
|
159 One can define a flow on $\Sigma_m^\infty(S^1)$ where points of the |
|
160 same color repel each other and points of different colors do not interact. |
|
161 This shows that a component $X$ of $\Sigma_m^\infty(S^1)$ is homotopy equivalent |
|
162 to the torus $(S^1)^l$, where $l$ is the number of non-zero entries in the $m$-tuple |
|
163 corresponding to $X$. |
|
164 The homology calculation we desire follows easily from this. |
|
165 |
|
166 %\nn{say something about cyclic homology in this case? probably not necessary.} |
|
167 |
|
168 \medskip |
|
169 |
|
170 Next we consider the case $C$ is the truncated polynomial |
|
171 algebra $k[t]/t^l$ --- polynomials in $t$ with $t^l = 0$. |
|
172 Define $\Delta_l \sub \Sigma^\infty(M)$ to be configurations of points in $M$ with $l$ or |
|
173 more of the points coinciding. |
|
174 |
|
175 \begin{prop} |
|
176 $\bc_*(M, k[t]/t^l)$ is homotopy equivalent to $C_*(\Sigma^\infty(M), \Delta_l, k)$ |
|
177 (relative singular chains with coefficients in $k$). |
|
178 \end{prop} |
|
179 |
|
180 \begin{proof} |
|
181 \nn{...} |
|
182 \end{proof} |
|
183 |
|
184 \medskip |
|
185 \hrule |
|
186 \medskip |
|
187 |
|
188 Still to do: |
|
189 \begin{itemize} |
|
190 \item compare the topological computation for truncated polynomial algebra with \cite{MR1600246} |
|
191 \item multivariable truncated polynomial algebras (at least mention them) |
|
192 \end{itemize} |
|
193 |