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
In the following exercises 
be a connected Poincaré complex of formal dimension 
.
Exercise 0.1.
Let 
be a spherical fibration with 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 of .
Here are some hints for this problem: Tangent bundles of bundles (Ex), [Wall1966a], [Chazin1975]
