1418 |
1418 |
1419 \nn{this should probably not be a section by itself. i'm just trying to write down the outline |
1419 \nn{this should probably not be a section by itself. i'm just trying to write down the outline |
1420 while it's still fresh in my mind.} |
1420 while it's still fresh in my mind.} |
1421 |
1421 |
1422 If $C$ is a commutative algebra it |
1422 If $C$ is a commutative algebra it |
1423 can (and will) also be thought of as an $n$-category with trivial $j$-morphisms for |
1423 can (and will) also be thought of as an $n$-category whose $j$-morphisms are trivial for |
1424 $j<n$ and $n$-morphisms are $C$. |
1424 $j<n$ and whose $n$-morphisms are $C$. |
1425 The goal of this \nn{subsection?} is to compute |
1425 The goal of this \nn{subsection?} is to compute |
1426 $\bc_*(M^n, C)$ for various commutative algebras $C$. |
1426 $\bc_*(M^n, C)$ for various commutative algebras $C$. |
1427 |
1427 |
1428 Let $k[t]$ denote the ring of polynomials in $t$ with coefficients in $k$. |
1428 Let $k[t]$ denote the ring of polynomials in $t$ with coefficients in $k$. |
1429 |
1429 |
1430 Let $\Sigma^i(M)$ denote the $i$-th symmetric power of $M$, the configuration space of $i$ |
1430 Let $\Sigma^i(M)$ denote the $i$-th symmetric power of $M$, the configuration space of $i$ |
1431 unlabeled points in $M$. |
1431 unlabeled points in $M$. |
1432 Note that $\Sigma^i(M)$ is a point. |
1432 Note that $\Sigma^0(M)$ is a point. |
1433 Let $\Sigma^\infty(M) = \coprod_{i=0}^\infty \Sigma^i(M)$. |
1433 Let $\Sigma^\infty(M) = \coprod_{i=0}^\infty \Sigma^i(M)$. |
1434 |
1434 |
1435 Let $C_*(X)$ denote the singular chain complex of the space $X$. |
1435 Let $C_*(X)$ denote the singular chain complex of the space $X$. |
1436 |
1436 |
1437 \begin{prop} |
1437 \begin{prop} |
1443 |
1443 |
1444 \begin{lemma} |
1444 \begin{lemma} |
1445 Let $A_*$ and $B_*$ be chain complexes, and assume $A_*$ is equipped with |
1445 Let $A_*$ and $B_*$ be chain complexes, and assume $A_*$ is equipped with |
1446 a basis (e.g.\ blob diagrams or singular simplices). |
1446 a basis (e.g.\ blob diagrams or singular simplices). |
1447 For each basis element $c \in A_*$ assume given a contractible subcomplex $R(c)_* \sub B_*$ |
1447 For each basis element $c \in A_*$ assume given a contractible subcomplex $R(c)_* \sub B_*$ |
1448 such that $R(c') \sub R(c)$ whenever $c'$ is a basis element which is part of $\bd c$. |
1448 such that $R(c')_* \sub R(c)_*$ whenever $c'$ is a basis element which is part of $\bd c$. |
1449 Then the complex of chain maps (and (iterated) homotopies) $f:A_*\to B_*$ such that |
1449 Then the complex of chain maps (and (iterated) homotopies) $f:A_*\to B_*$ such that |
1450 $f(c) \in R(c)_*$ for all $c$ is contractible (and in particular non-empty). |
1450 $f(c) \in R(c)_*$ for all $c$ is contractible (and in particular non-empty). |
1451 \end{lemma} |
1451 \end{lemma} |
1452 |
1452 |
1453 \begin{proof} |
1453 \begin{proof} |
1454 \nn{easy, but should probably write the details eventually} |
1454 \nn{easy, but should probably write the details eventually} |
1455 \end{proof} |
1455 \end{proof} |
1456 |
1456 |
1457 \nn{...} |
1457 Our first task: For each blob diagram $b$ define a subcomplex $R(b)_* \sub C_*(\Sigma^\infty(M))$ |
1458 |
1458 satisfying the conditions of the above lemma. |
|
1459 If $b$ is a 0-blob diagram, then it is just a $k[t]$ field on $M$, which is a |
|
1460 finite unordered collection of points of $M$ with multiplicities, which is |
|
1461 a point in $\Sigma^\infty(M)$. |
|
1462 Define $R(b)_*$ to be the singular chain complex of this point. |
|
1463 If $(B, u, r)$ is an $i$-blob diagram, let $D\sub M$ be its support (the union of the blobs). |
|
1464 The path components of $\Sigma^\infty(D)$ are contractible, and these components are indexed |
|
1465 by the numbers of points in each component of $D$. |
|
1466 We may assume that the blob labels $u$ have homogeneous $t$ degree in $k[t]$, and so |
|
1467 $u$ picks out a component $X \sub \Sigma^\infty(D)$. |
|
1468 The field $r$ on $M\setminus D$ can be thought of as a point in $\Sigma^\infty(M\setminus D)$, |
|
1469 and using this point we can embed $X$ in $\Sigma^\infty(M)$. |
|
1470 Define $R(B, u, r)_*$ to be the singular chain complex of $X$, thought of as a |
|
1471 subspace of $\Sigma^\infty(M)$. |
|
1472 It is easy to see that $R(\cdot)_*$ satisfies the condition on boundaries from the above lemma. |
|
1473 Thus we have defined (up to homotopy) a map from |
|
1474 $\bc_*(M^n, k[t])$ to $C_*(\Sigma^\infty(M))$. |
|
1475 |
|
1476 Next we define, for each simplex $c$ of $C_*(\Sigma^\infty(M))$, a contractible subspace |
|
1477 $R(c)_* \sub \bc_*(M^n, k[t])$. |
|
1478 If $c$ is a 0-simplex we use the identification of the fields $\cC(M)$ and |
|
1479 $\Sigma^\infty(M)$ described above. |
|
1480 Now let $c$ be an $i$-simplex of $\Sigma^j(M)$. |
|
1481 Choose a metric on $M$, which induces a metric on $\Sigma^j(M)$. |
|
1482 We may assume that the diameter of $c$ is small --- that is, $C_*(\Sigma^j(M))$ |
|
1483 is homotopy equivalent to the subcomplex of small simplices. |
|
1484 How small? $(2r)/3j$, where $r$ is the radius of injectivity of the metric. |
|
1485 Let $T\sub M$ be the ``track" of $c$ in $M$. |
|
1486 \nn{do we need to define this precisely?} |
|
1487 Choose a neighborhood $D$ of $T$ which is a disjoint union of balls of small diameter. |
|
1488 \nn{need to say more precisely how small} |
|
1489 Define $R(c)_*$ to be $\bc_*(D, k[t]) \sub \bc_*(M^n, k[t])$. |
|
1490 This is contractible by \ref{bcontract}. |
|
1491 We can arrange that the boundary/inclusion condition is satisfied if we start with |
|
1492 low-dimensional simplices and work our way up. |
|
1493 \nn{need to be more precise} |
|
1494 |
|
1495 \nn{still to do: show indep of choice of metric; show compositions are homotopic to the identity |
|
1496 (for this, might need a lemma that says we can assume that blob diameters are small)} |
1459 \end{proof} |
1497 \end{proof} |
1460 |
|
1461 |
1498 |
1462 |
1499 |
1463 |
1500 |
1464 |
1501 |
1465 |
1502 |