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# 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

#### 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