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148 \begin{equation*} |
148 \begin{equation*} |
149 \xymatrix{\bc_*^{\cC}(S^1) \ar[r]^{\iso}_{\text{qi}} & HC_*(\cC)} |
149 \xymatrix{\bc_*^{\cC}(S^1) \ar[r]^{\iso}_{\text{qi}} & HC_*(\cC)} |
150 \end{equation*} |
150 \end{equation*} |
151 \end{property} |
151 \end{property} |
152 |
152 |
153 \begin{property}[Evaluation map] |
153 \nn{$HC_*$ or $\rm{Hoch}_*$?} |
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154 |
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155 \begin{property}[$C_*(\Diff(\cdot))$ action] |
154 \label{property:evaluation}% |
156 \label{property:evaluation}% |
155 There is an `evaluation' chain map |
157 There is a chain map |
156 \begin{equation*} |
158 \begin{equation*} |
157 \ev_X: \CD{X} \tensor \bc_*(X) \to \bc_*(X). |
159 \ev_X: \CD{X} \tensor \bc_*(X) \to \bc_*(X). |
158 \end{equation*} |
160 \end{equation*} |
159 (Here $\CD{X}$ is the singular chain complex of the space of diffeomorphisms of $X$, fixed on $\bdy X$.) |
161 (Here $\CD{X}$ is the singular chain complex of the space of diffeomorphisms of $X$, fixed on $\bdy X$.) |
160 |
162 |
168 \ar@/_4ex/[r]_{\ev_{X_1} \otimes \ev_{X_2}} \ar[u]^{\gl^{\Diff}_Y \otimes \gl_Y} & |
170 \ar@/_4ex/[r]_{\ev_{X_1} \otimes \ev_{X_2}} \ar[u]^{\gl^{\Diff}_Y \otimes \gl_Y} & |
169 \bc_*(X_1) \otimes \bc_*(X_2) \ar[u]_{\gl_Y} |
171 \bc_*(X_1) \otimes \bc_*(X_2) \ar[u]_{\gl_Y} |
170 } |
172 } |
171 \end{equation*} |
173 \end{equation*} |
172 \nn{should probably say something about associativity here (or not?)} |
174 \nn{should probably say something about associativity here (or not?)} |
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175 \nn{maybe do self-gluing instead of 2 pieces case} |
173 \end{property} |
176 \end{property} |
174 |
177 |
175 |
178 |
176 \begin{property}[Gluing formula] |
179 \begin{property}[Gluing formula] |
177 \label{property:gluing}% |
180 \label{property:gluing}% |