Non-reducible Spivak Normal Fibrations (Ex)

Let $[\iota_2, \iota_3] \in \pi_4(S^2 \vee S^3)$$<wikitex>; Let $[\iota_2, \iota_3] \in \pi_4(S^2 \vee S^3)$ be the Whitehead product of the inclusion of the two factors: $[\iota_2, \iota_3]$ is the attaching map for the top cell of $S^2 \times S^3$. Let $X \simeq S^2 \vee S^3 \cup_\phi E^5$ be the space obtained by attaching a $-cell as indicated where the map $$ \phi \colon S^4 \to S^2 \vee S^3$$ is given by $\phi = [\iota_2, \iota_3] + \iota_2 \circ \eta^2_2$ and here $\iota_i \colon S^i \to S^2 \vee S^3$ is the obvious inclusion, $[\iota_2, \iota_3]$ is the Whitehead product and $\eta^2_2 \colon S^4 \to S^2$ is essential. {{beginthm|Exercise|c.f. {{citeD|Madsen&Milgram1979|2.5}}}} # Show that $X$ is a Poincaré complex. # Find a self-homotopy equivalence $f \colon S^2 \times S^2 \simeq S^2 \times S^2$ such that there is a homotopy equivalence $$ X \simeq (S^2 \times D^3) \cup_f (S^2 \times D^3).$$# Show that the Spivak normal fibration of $X$ has no vector bundle reduction. {{endthm}} </wikitex> == References == {{#RefList:}} [[Category:Exercises]] [[Category:Exercises without solution]][\iota_2, \iota_3] \in \pi_4(S^2 \vee S^3)$ be the Whitehead product of the inclusion of the two factors: $[\iota_2, \iota_3]$$[\iota_2, \iota_3]$ is the attaching map for the top cell of $S^2 \times S^3$$S^2 \times S^3$. Let $X \simeq S^2 \vee S^3 \cup_\phi E^5$$X \simeq S^2 \vee S^3 \cup_\phi E^5$ be the space obtained by attaching a $5$$5$-cell as indicated where the map

$\displaystyle \phi \colon S^4 \to S^2 \vee S^3$ $$\displaystyle \phi \colon S^4 \to S^2 \vee S^3$$

is given by $\phi = [\iota_2, \iota_3] + \iota_2 \circ \eta^2_2$$\phi = [\iota_2, \iota_3] + \iota_2 \circ \eta^2_2$ and here $\iota_i \colon S^i \to S^2 \vee S^3$$\iota_i \colon S^i \to S^2 \vee S^3$ is the obvious inclusion, $[\iota_2, \iota_3]$$[\iota_2, \iota_3]$ is the Whitehead product and $\eta^2_2 \colon S^4 \to S^2$$\eta^2_2 \colon S^4 \to S^2$ is essential.

Exercise 0.1 c.f. [Madsen&Milgram1979, 2.5].

1. Show that $X$$X$ is a Poincaré complex.
2. Find a self-homotopy equivalence $f \colon S^2 \times S^2 \simeq S^2 \times S^2$$f \colon S^2 \times S^2 \simeq S^2 \times S^2$ such that there

is a homotopy equivalence

$\displaystyle X \simeq (S^2 \times D^3) \cup_f (S^2 \times D^3).$ $$\displaystyle X \simeq (S^2 \times D^3) \cup_f (S^2 \times D^3).$$

# Show that the Spivak normal fibration of $X$$X$ has no vector bundle reduction.