Affiliation:
1. Department of Mathematics, and Institute of Pure and Applied Mathematics, Jeonbuk National University, 567 Baekje-daero, Deokjin-gu , Jeonju-si , Jeollabuk-do 54896 , Republic of Korea
Abstract
Abstract
For each positive integer
n
n
, we let
φ
n
:
Σ
C
P
∞
→
Σ
C
P
∞
{\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
[
φ
m
,
φ
n
]
:
Σ
C
P
∞
→
Σ
C
P
∞
\left[{\varphi }_{m},{\varphi }_{n}]:\Sigma {\mathbb{C}}{P}^{\infty }\to \Sigma {\mathbb{C}}{P}^{\infty }
be a commutator of self-maps
φ
m
{\varphi }_{m}
and
φ
n
{\varphi }_{n}
for any positive integers
m
m
and
n
n
. In the current study, we show that the image of the homomorphism
[
φ
ˆ
m
,
φ
ˆ
n
]
∗
{\left[{\hat{\varphi }}_{m},{\hat{\varphi }}_{n}]}_{\ast }
in homology induced by the adjoint
[
φ
ˆ
m
,
φ
ˆ
n
]
:
C
P
∞
→
Ω
Σ
C
P
∞
\left[{\hat{\varphi }}_{m},{\hat{\varphi }}_{n}]:{\mathbb{C}}{P}^{\infty }\to \Omega \Sigma {\mathbb{C}}{P}^{\infty }
of the commutator
[
φ
m
,
φ
n
]
\left[{\varphi }_{m},{\varphi }_{n}]
is both primitive and decomposable. As a further support of the above statement, we provide an example.