Algebraic surgery X (Ex)

Latest revision as of 14:16, 1 June 2012

Let $<wikitex>; Let $(f,b) \colon M \rightarrow X$ be a degree one normal map of $n$-GPC. Denote by $\nu_M$, $\nu_X$ the respective SNFs. We form the $(n+1)$-dimensional geometric (normal,~Poincar\'e) pair $$ \big( (W,M \sqcup X), (\nu_W,\nu_{M \sqcup X}), (\rho_W,\rho_{M \sqcup X}) \big) $$ with $W = \textup{cyl} (f)$. The symbol $\nu_W$ denotes the $k$-spherical fibration over $W$ induced by $b$ and $$ (\rho_W,\rho_{M \sqcup X}) \colon (D^{n+1+k},S^{n+k}) \rightarrow (\textup{Th} (\nu_W), \textup{Th} (\nu_M \sqcup \nu_X)) $$ is the map induced by $\rho_M$ and $\rho_X$ and denote $j \colon M \sqcup X \hookrightarrow W$. Let $C'$ be the underlying chain complex obtained by algebraic surgery on the $(n+1)$-dimensional symmetric pair $$ (j_\ast \colon C(M) \oplus C(X) \rightarrow C(W),(\delta \varphi,\varphi)). $$ Show that it is homotopy equivalent to the mapping cone $\mathcal{C} (f^{!})$ of the 'Umkehr' map associated to $(f,b)$. </wikitex> == References == {{#RefList:}} [[Category:Exercises]] [[Category:Exercises without solution]](f,b) \colon M \rightarrow X$ be a degree one normal map of $n$-GPC. Denote by $\nu_M$, $\nu_X$ the respective SNFs. We form the $(n+1)$-dimensional geometric (normal, Poincaré) pair

$$\displaystyle \big( (W,M \sqcup X), (\nu_W,\nu_{M \sqcup X}), (\rho_W,\rho_{M \sqcup X}) \big)$$

with $W = \textup{cyl} (f)$. The symbol $\nu_W$ denotes the $k$-spherical fibration over $W$ induced by $b$ and

$$\displaystyle (\rho_W,\rho_{M \sqcup X}) \colon (D^{n+1+k},S^{n+k}) \rightarrow (\textup{Th} (\nu_W), \textup{Th} (\nu_M \sqcup \nu_X))$$

is the map induced by $\rho_M$ and $\rho_X$ and denote $j \colon M \sqcup X \hookrightarrow W$.

Let $C'$ be the underlying chain complex obtained by algebraic surgery on the $(n+1)$-dimensional symmetric pair

$$\displaystyle (j_\ast \colon C(M) \oplus C(X) \rightarrow C(W),(\delta \varphi,\varphi)).$$

Show that it is homotopy equivalent to the mapping cone $\mathcal{C} (f^{!})$ of the 'Umkehr' map associated to $(f,b)$.

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