Spivak normal fibration (Ex)
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In the following exercises $X$ be a connected Poincaré complex of formal dimension $n$. | In the following exercises $X$ be a connected Poincaré complex of formal dimension $n$. | ||
{{beginthm|Exercise}} | {{beginthm|Exercise}} | ||
− | Let $\xi \colon E \to X$ be a spherical fibration $X$ with homotopy fibre $S^k$. Show that $E$ is a Poincaré complex of formal dimension $n + k$. | + | Let $\xi \colon E \to X$ be a spherical fibration $X$ with homotopy fibre $S^k$. Show that $E$ is homotopy equivalent to a Poincaré complex of formal dimension $n + k$. |
{{endthm}} | {{endthm}} | ||
Here is an interesting problem we now confront | Here is an interesting problem we now confront |
Revision as of 12:31, 27 March 2012
Tex syntax errorbe a connected Poincaré complex of formal dimension
Tex syntax error.
Exercise 0.1.
Let be a spherical fibrationTex syntax errorwith homotopy fibre . Show that is homotopy equivalent to a Poincaré complex of formal dimension .
Here is an interesting problem we now confront
Problem 0.2.
Determine the Spivak normal fibration of above in terms of and the Spivak normal fibration ofTex syntax error.
Here are some hints for this problem: Tangent bundles of bundles (Ex), [Wall1966a], [Chazin1975]