Primitive and decomposable elements in homology of ΩΣℂP∞

Abstract

For each positive integer nn, we let φn:ΣCP∞→ΣCP∞{\varphi }_{n}:\Sigma {\mathbb{C}}{P}^{\infty }\to \Sigma {\mathbb{C}}{P}^{\infty } be the self-maps of the suspension of the infinite complex projective space, or the localization of this space at a set of primes which may be an empty set. Furthermore, let [φm,φn]:ΣCP∞→ΣCP∞\left[{\varphi }_{m},{\varphi }_{n}]:\Sigma {\mathbb{C}}{P}^{\infty }\to \Sigma {\mathbb{C}}{P}^{\infty } be a commutator of self-maps φm{\varphi }_{m} and φn{\varphi }_{n} for any positive integers mm and nn. In the current study, we show that the image of the homomorphism [φˆm,φˆn]∗{\left[{\hat{\varphi }}_{m},{\hat{\varphi }}_{n}]}_{\ast } in homology induced by the adjoint [φˆm,φˆn]:CP∞→ΩΣCP∞\left[{\hat{\varphi }}_{m},{\hat{\varphi }}_{n}]:{\mathbb{C}}{P}^{\infty }\to \Omega \Sigma {\mathbb{C}}{P}^{\infty } of the commutator [φm,φn]\left[{\varphi }_{m},{\varphi }_{n}] is both primitive and decomposable. As a further support of the above statement, we provide an example.

Keywords

• commutator
• localization
• leray-serre spectral sequence
• 55s37
• 55p10
• 17b01