Non-reducible Spivak Normal Fibrations (Ex)
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− | + | Let $[\iota_2, \iota_3] \in \pi_4(S^2 \vee S^3)$ be the Whitehead product of the inclusion of the two factors: $[\iota_2, \iota_3]$ is the attaching map for the top cell of $S^2 \times S^3$. | |
Let $X \simeq S^2 \vee S^3 \cup_\phi E^5$ be the space obtained by attaching a $5$-cell as indicated where the map | Let $X \simeq S^2 \vee S^3 \cup_\phi E^5$ be the space obtained by attaching a $5$-cell as indicated where the map | ||
$$ \phi \colon S^4 \to S^2 \vee S^3$$ | $$ \phi \colon S^4 \to S^2 \vee S^3$$ | ||
− | is given by $\phi = [\iota_2, \iota_3] + \iota_2 \circ \eta^2_2$ and here $\iota_i \colon S^i \to S^2 \vee S^3$ is the obvious inclusion, $[\iota_2, \iota_3]$ is the Whitehead product and $\eta^2_2 \colon S^4 \to S^2$ is essential. Show that $X$ is a Poincaré complex. | + | is given by $\phi = [\iota_2, \iota_3] + \iota_2 \circ \eta^2_2$ and here $\iota_i \colon S^i \to S^2 \vee S^3$ is the obvious inclusion, $[\iota_2, \iota_3]$ is the Whitehead product and $\eta^2_2 \colon S^4 \to S^2$ is essential. |
+ | {{beginthm|Exercise|c.f. {{citeD|Madsen&Milgram1979|2.5}}}} | ||
+ | # Show that $X$ is a Poincaré complex. | ||
+ | # Find a self-homotopy equivalence $f \colon S^2 \times S^2 \simeq S^2 \times S^2$ such that there | ||
+ | is a homotopy equivalence | ||
+ | $$ X \simeq (S^2 \times D^3) \cup_f (S^2 \times D^3).$$ | ||
+ | # Show that the Spivak normal fibration of $X$ has no vector bundle reduction. | ||
{{endthm}} | {{endthm}} | ||
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== References == | == References == |
Revision as of 23:35, 25 August 2013
Let be the Whitehead product of the inclusion of the two factors: is the attaching map for the top cell of . Let be the space obtained by attaching a -cell as indicated where the map
is given by and here is the obvious inclusion, is the Whitehead product and is essential.
Exercise 0.1 c.f. [Madsen&Milgram1979, 2.5].
- Show that is a Poincaré complex.
- Find a self-homotopy equivalence such that there
is a homotopy equivalence
- Show that the Spivak normal fibration of has no vector bundle reduction.