Homology, Homotopy and Applications

Volume 3 (2001)

Number 2

Volume of a Workshop at Stanford University

The tangent bundle of an almost-complex free loopspace

Pages: 407 – 415

DOI: http://dx.doi.org/10.4310/HHA.2001.v3.n2.a7

Author

Jack Morava (Department of Mathematics, Johns Hopkins University, Baltimore, Maryland, U.S.A.)

Abstract

The space $LV$ of free loops on a manifold $V$ inherits an action of the circle group ${\mathbb T}$. When $V$ has an almost-complex structure, the tangent bundle of the free loopspace, pulled back to a certain infinite cyclic cover ${\widetilde{LV}}$, has an equivariant decomposition as a completion of ${\bf T} V \otimes (\oplus \, {\mathbb C}(k))$, where ${\bf T} V$ is an equivariant bundle on the cover, nonequivariantly isomorphic to the pullback of $TV$ along evaluation at the basepoint (and $\oplus \, {\mathbb C}(k)$ denotes an algebra of Laurent polynomials). On a flat manifold, this analogue of Fourier analysis is classical.

Keywords

free loopspace, circle action, holonomy, polarization

2010 Mathematics Subject Classification

53C29, 55P91, 58Dxx

Full Text (PDF format)

An erratum to this article is available as HHA 5(1) pp. 71-71.