On the same 𝑁-type conjecture for the suspension of the infinite complex projective space

Lee, Dae-Woong

doi:10.1090/S0002-9939-08-09666-4

On the same $N$-type conjecture for the suspension of the infinite complex projective space
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Proc. Amer. Math. Soc. 137 (2009), 1161-1168 Request permission

Abstract:

Let $[\varphi _{i_{k}},[\varphi _{i_{k-1}},\cdots ,[\varphi _{i_{1}}, \varphi _{i_{2}}],\cdots ]]$ be an iterated commutator of self-maps $\varphi _{i_{j}}$ on the suspension of the infinite complex projective space. In this paper, we produce useful self-maps of the form $I + [\varphi _{i_{k}},[\varphi _{i_{k-1}},\cdots , [\varphi _{i_{1}}, \varphi _{i_{2}}],\cdots ]]$, where $+$ means the addition of maps on the suspension structure of $\Sigma {\mathbb {C}}P^{\infty }$. We then give the answer to the conjecture saying that the set of all the same homotopy $n$-types of the suspension of the infinite complex projective space is the one element set consisting of a single homotopy type.

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