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Counting Homotopy Types of Gauge Groups
Author(s) -
Crabb M. C.,
Sutherland W. A.
Publication year - 2000
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/s0024611500012545
Subject(s) - mathematics , homotopy , equivalence (formal languages) , bott periodicity theorem , n connected , homotopy group , whitehead theorem , pure mathematics , regular homotopy , mathematics subject classification , gauge group , gauge theory , mathematical physics
For K a connected finite complex and G a compact connected Lie group, a finiteness result is proved for gauge groups G( P ) of principal G ‐bundles P over K : as P ranges over all principal G ‐bundles with base K , the number of homotopy types of G( P ) is finite; indeed this remains true when these gauge groups are classified by H ‐equivalence, that is, homotopy equivalences which respect multiplication up to homotopy. A case study is given for K = S 4 , G = SU(2): there are eighteen H ‐equivalence classes of gauge group in this case. These questions are studied via fibre homotopy theory of bundles of groups; the calculations in the case study involve K ‐theories and e ‐invariants. 1991 Mathematics Subject Classification : 54C35, 55P15, 55R10.