K-invariant for G/PL (Ex)

Latest revision as of 20:32, 28 May 2012

In talk 16 the $<wikitex>; In [[2012 SBR Program#The surgery exact sequence for TOP and PL|talk 16]] the $-local homotopy type of $G/PL$ was described as being $$(G/PL)[2]\cong X^4 \times \prod_{n> 1} K(\mathbb{Z},4n)\times K(\mathbb{Z}_2,4n-2)~,$$ where $X^4$ is a two stage Postnikov system with nontrivial homotopy groups $\pi_2(X)=\mathbb{Z}_2$, $\pi_4(X)=\mathbb{Z}[2]$ and (non-trivial) Postnikov invariant $k^4\in H^5(K(\mathbb{Z},2),\mathbb{Z}[2])$. The goal of this exercise is to show that $k^4$ is nonzero. {{beginthm|Exercise}} Let $\sigma_4:\pi_4(G/PL)\to \mathbb{Z}$ be the surgery obstruction map, and $\sigma_4[2]$ its factorization through the localized Hurewicz map $h_{MSO[2]}$, so that $\sigma_4=\sigma_4[2]\circ h_{MSO[2]}$. Show that if $\sigma_4[2]$ is surjective, then $k^4$ is nonzero. {{endthm}} {{beginthm|Exercise}} Show that $\sigma_4[2]$ is surjective by constructing a $PL$-manifold $M^4$ and a map $f:M\to G/PL$ with surgery invariant 1. {{endthm}} {{beginrem|Hint}} Try to construct a degree one normal map $g:\mathbb{C}P^2\to\mathbb{C}P^2 \,\sharp \,8 \overline{\mathbb{C}P^2}$, where $\sharp$ denotes connected sum and $\overline{\mathbb{C}P^2}$ denotes $\mathbb{C}P^2$ with the conjugate complex struture. {{endrem}} </wikitex> == References == {{#RefList:}} [[Category:Exercises]]2$-local homotopy type of $G/PL$ was described as being

$$\displaystyle (G/PL)[2]\cong X^4 \times \prod_{n> 1} K(\mathbb{Z},4n)\times K(\mathbb{Z}_2,4n-2)~,$$

where $X^4$ is a two stage Postnikov system with nontrivial homotopy groups $\pi_2(X)=\mathbb{Z}_2$, $\pi_4(X)=\mathbb{Z}[2]$ and (non-trivial) Postnikov invariant $k^4\in H^5(K(\mathbb{Z},2),\mathbb{Z}[2])$. The goal of this exercise is to show that $k^4$ is nonzero.