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# Homology, Homotopy and Applications

## Volume 20 (2018)

### Number 1

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

Pages: 5 – 36

#### Author

#### Abstract

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

Received 27 January 2017

Received revised 14 April 2017

Published 19 December 2017