On vector bundles for a Morse decomposition of $L\mathbb{C}\mathrm{P}^n$


  • Iver Ottosen




We give a description of the negative bundles for the energy integral on the free loop space $L\mathbb{C}\mathrm{P}^n$ in terms of circle vector bundles over projective Stiefel manifolds. We compute the mod $p$ Chern classes of the associated homotopy orbit bundles.


Atiyah, M. F., $K$-theory, Lecture notes by D. W. Anderson, W. A. Benjamin, Inc., New York-Amsterdam, 1967.

Bökstedt, M., Hsiang, W. C., and Madsen, I., The cyclotomic trace and algebraic $K$-theory of spaces, Invent. Math. 111 (1993), no. 3, 465–539. https://doi.org/10.1007/BF01231296

tom Dieck, T., Transformation groups, De Gruyter Studies in Mathematics, vol. 8, Walter de Gruyter & Co., Berlin, 1987. https://doi.org/10.1515/9783110858372.312

Gallot, S., Hulin, D., and Lafontaine, J., Riemannian geometry, second ed., Universitext, Springer-Verlag, Berlin, 1990. https://doi.org/10.1007/978-3-642-97242-3

Gromoll, D. and Meyer, W., Periodic geodesics on compact Riemannian manifolds, J. Differential Geometry 3 (1969), 493–510.

Hatcher, A., Vector bundles and $K$-theory, http://www.math.cornell.edu/~hatcher/VBKT/VBpage.html, 2009.

Hingston, N., On the growth of the number of closed geodesics on the two-sphere, Internat. Math. Res. Notices (1993), no. 9, 253–262. https://doi.org/10.1155/S1073792893000285

Husemoller, D., Fibre bundles, third ed., Graduate Texts in Mathematics, vol. 20, Springer-Verlag, New York, 1994. https://doi.org/10.1007/978-1-4757-2261-1

Klingenberg, W., The space of closed curves on the sphere, Topology 7 (1968), 395–415. https://doi.org/10.1016/0040-9383(68)90015-3

Klingenberg, W., The space of closed curves on a projective space, Quart. J. Math. Oxford Ser. (2) 20 (1969), 11–31. https://doi.org/10.1093/qmath/20.1.11

Klingenberg, W., Lectures on closed geodesics, Springer-Verlag, Berlin-New York, 1978, Grundlehren der Mathematischen Wissenschaften, Vol. 230.

Kobayashi, S. and Nomizu, K., Foundations of differential geometry. Vol. II, Interscience Tracts in Pure and Applied Mathematics, no. 15, John Wiley & Sons, Inc., New York-London-Sydney, 1969.

Madsen, I. and Tornehave, J., From calculus to cohomology, Cambridge University Press, Cambridge, 1997.

Ndombol, B. and El Haouari, M., The free loop space equivariant cohomology algebra of some formal spaces, Math. Z. 266 (2010), no. 4, 863–875. https://doi.org/10.1007/s00209-009-0602-z

Ottosen, I. and Bökstedt, M., String cohomology groups of complex projective spaces, Algebr. Geom. Topol. 7 (2007), 2165–2238. https://doi.org/10.2140/agt.2007.7.2165

Vigué-Poirrier, M. and Sullivan, D., The homology theory of the closed geodesic problem, J. Differential Geometry 11 (1976), no. 4, 633–644.

Ziller, W., The free loop space of globally symmetric spaces, Invent. Math. 41 (1977), no. 1, 1–22. https://doi.org/10.1007/BF01390161



How to Cite

Ottosen, I. (2017). On vector bundles for a Morse decomposition of $L\mathbb{C}\mathrm{P}^n$. MATHEMATICA SCANDINAVICA, 121(2), 186–218. https://doi.org/10.7146/math.scand.a-96622