# Smoothing the Kervaire manifold (Ex)

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Let $M_K$$; Let M_K be the PL Kervaire (4k+2)-manifold and let \tau \colon M_K \to BPL and let B\pi \colon BPL \to B(PL/O) be the canonical map. In addition let \Sigma_K \in \Theta_{4k+1} be the Kervaire sphere and let \Psi \colon \Theta_{4k+1} \cong \pi_{4k+1}(PL/O) be the bijection defined taking the standard sphere as the base-point for smooth structures on S^{4k+1}. {{beginthm|Exercise}} # Show that the Spivak normal fibration of M_K admits a vector bundle reduction. # Determine the homotopy class of B \pi \circ \tau \colon M_K \to B(PL/O) in terms of \Psi(\Sigma_K) \in \pi_{4k+1}(PL/O). {{endthm}} == References == {{#RefList:}} [[Category:Exercises]] [[Category:Exercises without solution]]M_K$ be the $PL$$PL$ Kervaire $(4k+2)$$(4k+2)$-manifold and let $\tau \colon M_K \to BPL$$\tau \colon M_K \to BPL$ and let $B\pi \colon BPL \to B(PL/O)$$B\pi \colon BPL \to B(PL/O)$ be the canonical map. In addition let $\Sigma_K \in \Theta_{4k+1}$$\Sigma_K \in \Theta_{4k+1}$ be the Kervaire sphere and let
$\displaystyle \Psi \colon \Theta_{4k+1} \cong \pi_{4k+1}(PL/O)$
be the bijection defined taking the standard sphere as the base-point for smooth structures on $S^{4k+1}$$S^{4k+1}$.

Exercise 0.1.

1. Show that the Spivak normal fibration of $M_K$$M_K$ admits a vector bundle reduction.
2. Determine the homotopy class of $B \pi \circ \tau \colon M_K \to B(PL/O)$$B \pi \circ \tau \colon M_K \to B(PL/O)$ in terms of $\Psi(\Sigma_K) \in \pi_{4k+1}(PL/O)$$\Psi(\Sigma_K) \in \pi_{4k+1}(PL/O)$.