The homotopy types of $U(n)$-gauge groups over $S^4$ and $\mathbb{C}P^2$

The homotopy types of $U(n)$-gauge groups over the two most fundamental 4-manifolds $S^4$ and $\mathbb{C}P^2$ are studied. We give homotopy decompositions of the $U(n)$-gauge groups over $S^4$ in terms of certain $SU(n) $- and $PU(n) $-gauge groups and use these decompositions to enumerate the homotopy types of the $U(2)$-, $U(3)$- and $U(5)$-gauge groups. Over $\mathbb{C}P^2$ we provide bounding results on the number of homotopy types of $U(n)$-gauge groups, provide $p$-local decompositions and give homotopy decompositions of certain $U(n)$-gauge groups in terms of certain $SU(n)$-gauge groups. Applications are then given to count the number of homotopy types of $U(2)$-gauge groups over $\mathbb{C}P^2$.

Keywords

gauge group, homotopy type, homotopy decomposition, function space

2010 Mathematics Subject Classification

54C35, 55P15

Full Text (PDF format)

Received 27 January 2017

Received revised 14 April 2017

Published 19 December 2017