62 we are considering. |
62 we are considering. |
63 They could be unoriented or oriented or Spin or $\mbox{Pin}_\pm$. |
63 They could be unoriented or oriented or Spin or $\mbox{Pin}_\pm$. |
64 They could be topological or PL or smooth. |
64 They could be topological or PL or smooth. |
65 %\nn{need to check whether this makes much difference} |
65 %\nn{need to check whether this makes much difference} |
66 (If smooth, ``homeomorphism" should be read ``diffeomorphism", and we would need |
66 (If smooth, ``homeomorphism" should be read ``diffeomorphism", and we would need |
67 to be fussier about corners.) |
67 to be fussier about corners and boundaries.) |
68 For each flavor of manifold there is a corresponding flavor of $n$-category. |
68 For each flavor of manifold there is a corresponding flavor of $n$-category. |
69 We will concentrate on the case of PL unoriented manifolds. |
69 We will concentrate on the case of PL unoriented manifolds. |
70 |
70 |
71 (The ambitious reader may want to keep in mind two other classes of balls. |
71 (The ambitious reader may want to keep in mind two other classes of balls. |
72 The first is balls equipped with a map to some other space $Y$ (c.f. \cite{MR2079378}). |
72 The first is balls equipped with a map to some other space $Y$ (c.f. \cite{MR2079378}). |
1520 More specifically, $D\to D'$ is an antirefinement if $D'$ is obtained from $D$ by |
1520 More specifically, $D\to D'$ is an antirefinement if $D'$ is obtained from $D$ by |
1521 gluing subintervals together and/or omitting some of the rightmost subintervals. |
1521 gluing subintervals together and/or omitting some of the rightmost subintervals. |
1522 (See Figure \ref{fig:lmar}.) |
1522 (See Figure \ref{fig:lmar}.) |
1523 \begin{figure}[t]$$ |
1523 \begin{figure}[t]$$ |
1524 \definecolor{arcolor}{rgb}{.75,.4,.1} |
1524 \definecolor{arcolor}{rgb}{.75,.4,.1} |
1525 \begin{tikzpicture} |
1525 \begin{tikzpicture}[line width=1pt] |
1526 \pgfsetlinewidth{1pt} |
|
1527 \fill (0,0) circle (.1); |
1526 \fill (0,0) circle (.1); |
1528 \draw (0,0) -- (2,0); |
1527 \draw (0,0) -- (2,0); |
1529 \draw (1,0.1) -- (1,-0.1); |
1528 \draw (1,0.1) -- (1,-0.1); |
1530 |
1529 |
1531 \draw [->, arcolor] (1,0.25) -- (1,0.75); |
1530 \draw [->, arcolor] (1,0.25) -- (1,0.75); |
1532 |
1531 |
1533 \fill (0,1) circle (.1); |
1532 \fill (0,1) circle (.1); |
1534 \draw (0,1) -- (2,1); |
1533 \draw (0,1) -- (2,1); |
1535 \end{tikzpicture} |
1534 \end{tikzpicture} |
1536 \qquad |
1535 \qquad |
1537 \begin{tikzpicture} |
1536 \begin{tikzpicture}[line width=1pt] |
1538 \pgfsetlinewidth{1pt} |
|
1539 \fill (0,0) circle (.1); |
1537 \fill (0,0) circle (.1); |
1540 \draw (0,0) -- (2,0); |
1538 \draw (0,0) -- (2,0); |
1541 \draw (1,0.1) -- (1,-0.1); |
1539 \draw (1,0.1) -- (1,-0.1); |
1542 |
1540 |
1543 \draw [->, arcolor] (1,0.25) -- (1,0.75); |
1541 \draw [->, arcolor] (1,0.25) -- (1,0.75); |
1544 |
1542 |
1545 \fill (0,1) circle (.1); |
1543 \fill (0,1) circle (.1); |
1546 \draw (0,1) -- (1,1); |
1544 \draw (0,1) -- (1,1); |
1547 \end{tikzpicture} |
1545 \end{tikzpicture} |
1548 \qquad |
1546 \qquad |
1549 \begin{tikzpicture} |
1547 \begin{tikzpicture}[line width=1pt] |
1550 \pgfsetlinewidth{1pt} |
|
1551 \fill (0,0) circle (.1); |
1548 \fill (0,0) circle (.1); |
1552 \draw (0,0) -- (3,0); |
1549 \draw (0,0) -- (3,0); |
1553 \foreach \x in {0.5, 1.0, 1.25, 1.5, 2.0, 2.5} { |
1550 \foreach \x in {0.5, 1.0, 1.25, 1.5, 2.0, 2.5} { |
1554 \draw (\x,0.1) -- (\x,-0.1); |
1551 \draw (\x,0.1) -- (\x,-0.1); |
1555 } |
1552 } |
1588 \] |
1585 \] |
1589 For fixed $D_0$ and $D_1$, let $\cbar = \cbar'\ot\cbar''$, |
1586 For fixed $D_0$ and $D_1$, let $\cbar = \cbar'\ot\cbar''$, |
1590 where $\cbar'$ corresponds to the subintervals of $D_0$ which map to $D_1$ and |
1587 where $\cbar'$ corresponds to the subintervals of $D_0$ which map to $D_1$ and |
1591 $\cbar''$ corresponds to the subintervals |
1588 $\cbar''$ corresponds to the subintervals |
1592 which are dropped off the right side. |
1589 which are dropped off the right side. |
1593 (Either $\cbar'$ or $\cbar''$ might be empty.) |
1590 (If no such subintervals are dropped, then $\cbar''$ is empty.) |
1594 \nn{surely $\cbar'$ can't be empy: we don't allow $D_1$ to be empty.} |
|
1595 Translating from the boundary map for $(\cM_\cC\ot {_\cC\cN})^*$ appearing in Equation \eqref{eq:tensor-product-boundary}, |
1591 Translating from the boundary map for $(\cM_\cC\ot {_\cC\cN})^*$ appearing in Equation \eqref{eq:tensor-product-boundary}, |
1596 we have |
1592 we have |
1597 \begin{eqnarray*} |
1593 \begin{eqnarray*} |
1598 (\bd g)(\olD\ot x \ot \cbar) &=& \bd(g(\olD\ot x \ot \cbar)) + g(\olD\ot\bd(x\ot\cbar)) + \\ |
1594 (\bd g)(\olD\ot x \ot \cbar) &=& \bd(g(\olD\ot x \ot \cbar)) + g(\olD\ot\bd(x\ot\cbar)) + \\ |
1599 & & \;\; g((\bd_+\olD)\ot x\ot\cbar) + \gl''(g((\bd_0\olD)\ot \gl'(x\ot\cbar'))\ot\cbar'') . |
1595 & & \;\; g((\bd_+\olD)\ot x\ot\cbar) + \gl''(g((\bd_0\olD)\ot \gl'(x\ot\cbar'))\ot\cbar'') . |