Homology, Homotopy and Applications

The tangent bundle of an almost-complex free loopspace

Jack Morava

Full-text: Open access


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

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

First available in Project Euclid: 13 February 2006

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 58B25: Group structures and generalizations on infinite-dimensional manifolds [See also 22E65, 58D05]
Secondary: 53C29: Issues of holonomy 55P91: Equivariant homotopy theory [See also 19L47]


Morava, Jack. The tangent bundle of an almost-complex free loopspace. Homology Homotopy Appl. 3 (2001), no. 2, 407--415. https://projecteuclid.org/euclid.hha/1139840261

Export citation