## Homology, Homotopy and Applications

### The tangent bundle of an almost-complex free loopspace

Jack Morava

#### Abstract

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

#### Article information

Source
Homology Homotopy Appl., Volume 3, Number 2 (2001), 407-415.

Dates
First available in Project Euclid: 13 February 2006