equal
deleted
inserted
replaced
25 |
25 |
26 Suppose that for all $c \in \cC(\bd B^n)$ |
26 Suppose that for all $c \in \cC(\bd B^n)$ |
27 we have a splitting $s: H_0(\bc_*(B^n, c)) \to \bc_0(B^n; c)$ |
27 we have a splitting $s: H_0(\bc_*(B^n, c)) \to \bc_0(B^n; c)$ |
28 of the quotient map |
28 of the quotient map |
29 $p: \bc_0(B^n; c) \to H_0(\bc_*(B^n, c))$. |
29 $p: \bc_0(B^n; c) \to H_0(\bc_*(B^n, c))$. |
30 For example, this is always the case if you coefficient ring is a field. |
30 For example, this is always the case if the coefficient ring is a field. |
31 Then |
31 Then |
32 \begin{prop} \label{bcontract} |
32 \begin{prop} \label{bcontract} |
33 For all $c \in \cC(\bd B^n)$ the natural map $p: \bc_*(B^n, c) \to H_0(\bc_*(B^n, c))$ |
33 For all $c \in \cC(\bd B^n)$ the natural map $p: \bc_*(B^n, c) \to H_0(\bc_*(B^n, c))$ |
34 is a chain homotopy equivalence |
34 is a chain homotopy equivalence |
35 with inverse $s: H_0(\bc_*(B^n, c)) \to \bc_*(B^n; c)$. |
35 with inverse $s: H_0(\bc_*(B^n, c)) \to \bc_*(B^n; c)$. |
64 \end{prop} |
64 \end{prop} |
65 |
65 |
66 |
66 |
67 \begin{prop} |
67 \begin{prop} |
68 For fixed fields ($n$-cat), $\bc_*$ is a functor from the category |
68 For fixed fields ($n$-cat), $\bc_*$ is a functor from the category |
69 of $n$-manifolds and diffeomorphisms to the category of chain complexes and |
69 of $n$-manifolds and homeomorphisms to the category of chain complexes and |
70 (chain map) isomorphisms. |
70 (chain map) isomorphisms. |
71 \qed |
71 \qed |
72 \end{prop} |
72 \end{prop} |
73 |
73 |
74 In particular, |
74 In particular, |
75 \begin{prop} \label{diff0prop} |
75 \begin{prop} \label{diff0prop} |
76 There is an action of $\Diff(X)$ on $\bc_*(X)$. |
76 There is an action of $\Homeo(X)$ on $\bc_*(X)$. |
77 \qed |
77 \qed |
78 \end{prop} |
78 \end{prop} |
79 |
79 |
80 The above will be greatly strengthened in Section \ref{sec:evaluation}. |
80 The above will be greatly strengthened in Section \ref{sec:evaluation}. |
81 |
81 |
104 \end{prop} |
104 \end{prop} |
105 |
105 |
106 The above map is very far from being an isomorphism, even on homology. |
106 The above map is very far from being an isomorphism, even on homology. |
107 This will be fixed in Section \ref{sec:gluing} below. |
107 This will be fixed in Section \ref{sec:gluing} below. |
108 |
108 |
109 \nn{Next para not need, since we already use bullet = gluing notation above(?)} |
109 %\nn{Next para not needed, since we already use bullet = gluing notation above(?)} |
110 |
110 |
111 An instance of gluing we will encounter frequently below is where $X = X_1 \du X_2$ |
111 %An instance of gluing we will encounter frequently below is where $X = X_1 \du X_2$ |
112 and $X\sgl = X_1 \cup_Y X_2$. |
112 %and $X\sgl = X_1 \cup_Y X_2$. |
113 (Typically one of $X_1$ or $X_2$ is a disjoint union of balls.) |
113 %(Typically one of $X_1$ or $X_2$ is a disjoint union of balls.) |
114 For $x_i \in \bc_*(X_i)$, we introduce the notation |
114 %For $x_i \in \bc_*(X_i)$, we introduce the notation |
115 \eq{ |
115 %\eq{ |
116 x_1 \bullet x_2 \deq \gl(x_1 \otimes x_2) . |
116 % x_1 \bullet x_2 \deq \gl(x_1 \otimes x_2) . |
117 } |
117 %} |
118 Note that we have resumed our habit of omitting boundary labels from the notation. |
118 %Note that we have resumed our habit of omitting boundary labels from the notation. |
119 |
119 |
120 |
120 |
121 |
121 |