823 |
823 |
824 \begin{example}[The bordism $n$-category, ordinary version] |
824 \begin{example}[The bordism $n$-category, ordinary version] |
825 \label{ex:bord-cat} |
825 \label{ex:bord-cat} |
826 \rm |
826 \rm |
827 \label{ex:bordism-category} |
827 \label{ex:bordism-category} |
828 For a $k$-ball $X$, $k<n$, define $\Bord^n(X)$ to be the set of all $k$-dimensional |
828 For a $k$-ball $X$, $k<n$, define $\Bord^n(X)$ to be the set of all $k$-dimensional PL |
829 submanifolds $W$ of $X\times \Real^\infty$ such that the projection $W \to X$ is transverse |
829 submanifolds $W$ of $X\times \Real^\infty$ such that $\bd W$ is |
830 to $\bd X$. |
830 contained in $\bd X \times \Real^\infty$. |
831 For an $n$-ball $X$ define $\Bord^n(X)$ to be homeomorphism classes (rel boundary) of such $n$-dimensional submanifolds; |
831 For an $n$-ball $X$ define $\Bord^n(X)$ to be homeomorphism classes (rel boundary) of such $n$-dimensional submanifolds; |
832 we identify $W$ and $W'$ if $\bd W = \bd W'$ and there is a homeomorphism |
832 we identify $W$ and $W'$ if $\bd W = \bd W'$ and there is a homeomorphism |
833 $W \to W'$ which restricts to the identity on the boundary. |
833 $W \to W'$ which restricts to the identity on the boundary. |
834 \end{example} |
834 \end{example} |
835 |
835 |
894 \begin{example}[The bordism $n$-category, $A_\infty$ version] |
894 \begin{example}[The bordism $n$-category, $A_\infty$ version] |
895 \rm |
895 \rm |
896 \label{ex:bordism-category-ainf} |
896 \label{ex:bordism-category-ainf} |
897 As in Example \ref{ex:bord-cat}, for $X$ a $k$-ball, $k<n$, we define $\Bord^{n,\infty}(X)$ |
897 As in Example \ref{ex:bord-cat}, for $X$ a $k$-ball, $k<n$, we define $\Bord^{n,\infty}(X)$ |
898 to be the set of all $k$-dimensional |
898 to be the set of all $k$-dimensional |
899 submanifolds $W$ of $X\times \Real^\infty$ such that the projection $W \to X$ is transverse |
899 submanifolds $W$ of $X\times \Real^\infty$ such that $\bd W$ is |
900 to $\bd X$. |
900 contained in $\bd X \times \Real^\infty$. |
901 For an $n$-ball $X$ with boundary condition $c$ |
901 For an $n$-ball $X$ with boundary condition $c$ |
902 define $\Bord^{n,\infty}(X; c)$ to be the space of all $k$-dimensional |
902 define $\Bord^{n,\infty}(X; c)$ to be the space of all $k$-dimensional |
903 submanifolds $W$ of $X\times \Real^\infty$ such that |
903 submanifolds $W$ of $X\times \Real^\infty$ such that |
904 $W$ coincides with $c$ at $\bd X \times \Real^\infty$. |
904 $W$ coincides with $c$ at $\bd X \times \Real^\infty$. |
905 (The topology on this space is induced by ambient isotopy rel boundary. |
905 (The topology on this space is induced by ambient isotopy rel boundary. |