# On the real cohomology of spaces of free loops on manifolds

Fundamenta Mathematicae (1996)

- Volume: 150, Issue: 2, page 173-188
- ISSN: 0016-2736

## Access Full Article

top## Abstract

top## How to cite

topKuribayashi, Katsuhiko. "On the real cohomology of spaces of free loops on manifolds." Fundamenta Mathematicae 150.2 (1996): 173-188. <http://eudml.org/doc/212168>.

@article{Kuribayashi1996,

abstract = {Let LX be the space of free loops on a simply connected manifold X. When the real cohomology of X is a tensor product of algebras generated by a single element, we determine the algebra structure of the real cohomology of LX by using the cyclic bar complex of the de Rham complex Ω(X) of X. In consequence, the algebra generators of the real cohomology of LX can be represented by differential forms on LX through Chen’s iterated integral map. Let $\mathbb \{T\}$ be the circle group. The $\mathbb \{T\}$-equivariant cohomology of LX is also studied in terms of the cyclic homology of Ω(X).},

author = {Kuribayashi, Katsuhiko},

journal = {Fundamenta Mathematicae},

keywords = {space of free loops; simply connected manifold; real cohomology; cyclic bar complex; de Rham complex; iterated integral map},

language = {eng},

number = {2},

pages = {173-188},

title = {On the real cohomology of spaces of free loops on manifolds},

url = {http://eudml.org/doc/212168},

volume = {150},

year = {1996},

}

TY - JOUR

AU - Kuribayashi, Katsuhiko

TI - On the real cohomology of spaces of free loops on manifolds

JO - Fundamenta Mathematicae

PY - 1996

VL - 150

IS - 2

SP - 173

EP - 188

AB - Let LX be the space of free loops on a simply connected manifold X. When the real cohomology of X is a tensor product of algebras generated by a single element, we determine the algebra structure of the real cohomology of LX by using the cyclic bar complex of the de Rham complex Ω(X) of X. In consequence, the algebra generators of the real cohomology of LX can be represented by differential forms on LX through Chen’s iterated integral map. Let $\mathbb {T}$ be the circle group. The $\mathbb {T}$-equivariant cohomology of LX is also studied in terms of the cyclic homology of Ω(X).

LA - eng

KW - space of free loops; simply connected manifold; real cohomology; cyclic bar complex; de Rham complex; iterated integral map

UR - http://eudml.org/doc/212168

ER -

## References

top- [1] W. Andrzejewski and A. Tralle, Cohomology of some graded differential algebras, Fund. Math. 145 (1994), 181-204. Zbl0847.55007
- [2] M. F. Atiyah and R. Bott, The moment map and equivariant cohomology, Topology 23 (1984), 1-28. Zbl0521.58025
- [3] E. J. Beggs, The de Rham complex on infinite dimensional manifolds, Quart. J. Math. 38 (1987), 131-154. Zbl0636.58004
- [4] R. Bott and L. Tu, Differential Forms in Algebraic Topology, Grad. Texts in Math. 82, Springer, New York, 1982. Zbl0496.55001
- [5] D. Burghelea and M. Vigué-Poirrier, Cyclic homology of commutative algebras I, in: Lecture Notes in Math. 1318, Springer, 1988, 51-72.
- [6] H. Cartan and S. Eilenberg, Homological Algebra, Princeton University Press, Princeton, 1956.
- [7] K. T. Chen, Iterated integrals of differential forms and loop space homology, Ann. of Math. 97 (1973), 217-246. Zbl0227.58003
- [8] E. Getzler and J. D. S. Jones, ${A}_{\infty}$-algebras and cyclic bar complex, Illinois J. Math. 34 (1990), 256-283. Zbl0701.55009
- [9] E. Getzler, J. D. S. Jones and S. Petrack, Differential form on loop spaces and the cyclic bar complex, Topology 30 (1991), 339-371. Zbl0729.58004
- [10] T. Goodwillie, Cyclic homology, derivations, and the free loop space, Topology 24 (1985), 187-215. Zbl0569.16021
- [11] C. Hood and J. D. S. Jones, Some algebraic properties of cyclic homology groups, K-Theory 1 (1987), 361-384. Zbl0636.18005
- [12] J. D. S. Jones, Cyclic homology and equivariant homology, Invent. Math. 87 (1987), 403-423. Zbl0644.55005
- [13] D. Kraines and C. Schochet, Differentials in the Eilenberg-Moore spectral sequence, J. Pure Appl. Algebra 2 (1972), 131-148. Zbl0237.55016
- [14] KK. Kuribayashi, On the mod p cohomology of spaces of free loops on the Grassmann and Stiefel manifolds, J. Math. Soc. Japan 43 (1991), 331-346. Zbl0729.55010
- [15] K. Kuribayashi and T. Yamaguchi, On additive K-theory with the Loday-Quillen *-product, preprint. Zbl0968.19001
- [16] J. L. Loday and D. G. Quillen, Cyclic homology and the Lie algebra homology of matrices, Comment. Math. Helv. 59 (1984), 565-591. Zbl0565.17006
- [17] V. Mathai and D. G. Quillen, Superconnections, equivariant differential forms and the Thom class, Topology 25 (1986), 85-110. Zbl0592.55015
- [18] MJ. P. May, The cohomology of restricted Lie algebras and of Hopf algebras, J. Algebra 3 (1966), 123-146.
- [19] J. McCleary, User's Guide to Spectral Sequences, Publish or Perish, Wilmington, 1985. Zbl0577.55001
- [20] L. Smith, Homological algebra and the Eilenberg-Moore spectral sequence, Trans. Amer. Math. Soc. 129 (1967), 58-93. Zbl0177.51402
- [21] L. Smith, On the characteristic zero cohomology of the free loop space, Amer. J. Math. 103 (1981), 887-910. Zbl0475.55004
- [22] M. Vigué-Poirrier, Réalisation de morphismes donnés en cohomologie et site spectrale d'Eilenberg-Moore, Trans. Amer. Math. Soc. 265 (1981), 447-484. Zbl0474.55009
- [23] M. Vigué-Poirrier, Sur l'algèbre de cohomologie cyclique d'un espace 1-connexe, Illinois J. Math. 32 (1988), 40-52.
- [24] M. Vigué-Poirrier and D. Burghelea, A model for cyclic homology and algebraic K-theory of 1-connected topological spaces, J. Differential Geom. 22 (1985), 243-253. Zbl0595.55009

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.