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Equivariant K-theory and cohomology of the free loop space of a projective space

by Mia Hauge Dollerup
PhD Dissertations August 2009

The free loop space LX of a space X is the space of continuous maps from S1 to X. The circle group S1 acts on LX by rotation, and we study the space of homotopy orbits, LXhS1=ES1×S1LX, sometimes called the Borel construction. The aim of the thesis is to compute the cohomology and complex K-theory of LXhS1 when X is a projective space (complex or quaternion). The method is Morse theory on LX as developed by Klingenberg.

We obtain generators for the cohomology H(XhS1;Fp) as a module over H(BS1;Fp), when X is a quaternion projective space. Moreover, we find the K-theory of LXhS1 when X is a complex projective space, and this is one of the first such calculations for a non-trivial X. Our result is that K0(XhS1)=K0(BS1), and K1(XhS1) is free abelian. We conjecture that K1(XhS1) could be the completion of some ''representation theory'' type of group, in analogy with the classical result of Atiyah for the classifying space of a compact Lie group.

Format available: PDF (682 KB)
Thesis advisor: Marcel Bökstedt