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650 |
650 |
651 \nn{should also include example of ncats coming from TQFTs, or refer ahead to where we discuss that example} |
651 \nn{should also include example of ncats coming from TQFTs, or refer ahead to where we discuss that example} |
652 |
652 |
653 \newcommand{\Bord}{\operatorname{Bord}} |
653 \newcommand{\Bord}{\operatorname{Bord}} |
654 \begin{example}[The bordism $n$-category, plain version] |
654 \begin{example}[The bordism $n$-category, plain version] |
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655 \label{ex:bord-cat} |
655 \rm |
656 \rm |
656 \label{ex:bordism-category} |
657 \label{ex:bordism-category} |
657 For a $k$-ball $X$, $k<n$, define $\Bord^n(X)$ to be the set of all $k$-dimensional |
658 For a $k$-ball $X$, $k<n$, define $\Bord^n(X)$ to be the set of all $k$-dimensional |
658 submanifolds $W$ of $X\times \Real^\infty$ such that the projection $W \to X$ is transverse |
659 submanifolds $W$ of $X\times \Real^\infty$ such that the projection $W \to X$ is transverse |
659 to $\bd X$. |
660 to $\bd X$. |
717 There's no way for the blob complex to magically recover all the data of $\pi^\infty_{\leq 0}(T) \iso C_* T$. |
718 There's no way for the blob complex to magically recover all the data of $\pi^\infty_{\leq 0}(T) \iso C_* T$. |
718 |
719 |
719 \begin{example}[The bordism $n$-category, $A_\infty$ version] |
720 \begin{example}[The bordism $n$-category, $A_\infty$ version] |
720 \rm |
721 \rm |
721 \label{ex:bordism-category-ainf} |
722 \label{ex:bordism-category-ainf} |
722 blah blah \nn{to do...} |
723 As in Example \ref{ex:bord-cat}, for $X$ a $k$-ball, $k<n$, we define $\Bord^{n,\infty}(X)$ |
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724 to be the set of all $k$-dimensional |
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725 submanifolds $W$ of $X\times \Real^\infty$ such that the projection $W \to X$ is transverse |
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726 to $\bd X$. |
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727 For an $n$-ball $X$ with boundary condition $c$ |
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728 define $\Bord^{n,\infty}(X; c)$ to be the space of all $k$-dimensional |
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729 submanifolds $W$ of $X\times \Real^\infty$ such that |
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730 $W$ coincides with $c$ at $\bd X \times \Real^\infty$. |
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731 (The topology on this space is induced by ambient isotopy rel boundary. |
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732 This is homotopy equivalent to a disjoint union of copies $\mathrm{B}\!\Homeo(W')$, where |
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733 $W'$ runs though representatives of homeomorphism types of such manifolds.) |
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734 \nn{check this} |
723 \end{example} |
735 \end{example} |
724 |
736 |
725 |
737 |
726 |
738 |
727 Let $\cE\cB_n$ be the operad of smooth embeddings of $k$ (little) |
739 Let $\cE\cB_n$ be the operad of smooth embeddings of $k$ (little) |
762 --- composition and $\Diff(X\to X')$ action --- |
774 --- composition and $\Diff(X\to X')$ action --- |
763 also comes from the $\cE\cB_n$ action on $A$. |
775 also comes from the $\cE\cB_n$ action on $A$. |
764 \nn{should we spell this out?} |
776 \nn{should we spell this out?} |
765 |
777 |
766 \nn{Should remark that this is just Lurie's topological chiral homology construction |
778 \nn{Should remark that this is just Lurie's topological chiral homology construction |
767 applied to $n$-balls (check this).} |
779 applied to $n$-balls (check this). |
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780 Hmmm... Does Lurie do both framed and unframed cases?} |
768 \end{example} |
781 \end{example} |
769 |
782 |
770 |
783 |
771 |
784 |
772 |
785 |