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The Stasheff model of a simply-connected manifold and the string bracket
Author:
A. Lazarev
Journal:
Proc. Amer. Math. Soc. 136 (2008), 735-745
MSC (2000):
Primary 55P62; Secondary 13D03, 57T30
Posted:
October 24, 2007
MathSciNet review:
2358516
Full-text PDF
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Additional Information
Abstract: We revisit Stasheff's construction of a minimal Lie-Quillen model of a simply-connected closed manifold  using the language of infinity-algebras. This model is then used to construct a graded Lie bracket on the equivariant homology of the free loop space of  minus a point similar to the Chas-Sullivan string bracket.
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- 1.
- A. K. Bousfield, V. K. A. M. Gugenheim. On
 de Rham theory and rational homotopy type. Mem. Amer. Math. Soc. 8 (1976), no. 179. MR 0425956 (54:13906)
- 2.
- M. Aubry, S. Halperin, J.-M. Lemaire. Poincaré duality models, preprint.
- 3.
- H. L. Baues, J.-M. Lemaire. Minimal models in homotopy theory. Math. Ann. 225 (1977), no. 3, 219-242. MR 0431172 (55:4174)
- 4.
- M. Chas, D. Sullivan. String topology. arXiv:math.GT/9911159.
- 5.
- K. Costello. Topological conformal field theories and Calabi-Yau categories. Adv. in Math., Vol. 210, 1, 165-214, 2007.
- 6.
- E. Getzler, J.D.S. Jones. Operads, Homotopy Algebra, and Iterated Integrals for double Loop Spaces. arXiv:hep-th/9403055.
- 7.
- J.D.S. Jones, Cyclic homology and equivariant homology. Inventiones Mathematicae, 87, 1987, 403-423. MR 870737 (88f:18016)
- 8.
- A. Hamilton, A. Lazarev. Homotopy algebras and noncommutative geometry. arXiv:math. QA/0410621.
- 9.
- M. Kontsevich, Formal Noncommutative Symplectic Geometry. The Gelfand Mathematical Seminars, 1990-1992, pp. 173-187, Birkhäuser Boston, Boston, MA, 1993. MR 1247289 (94i:58212)
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- M. Kontsevich, Feynman Diagrams and Low-Dimensional Topology. First European Congress of Mathematics, Vol. 2 (Paris, 1992), pp. 97-121, Progr. Math., Vol. 120, Birkhäuser, Basel, 1994. MR 1341841 (96h:57027)
- 11.
- A. Lazarev. Hochschild cohomology and moduli spaces of strongly homotopy associative algebras. Homology Homotopy Appl. 5(2003), no. 1, 73-100. MR 1989615 (2004k:16018)
- 12.
- M. Kontsevich, Y. Soibelman. Notes on A-infinity algebras, A-infinity categories and non-commutative geometry I. arXiv:math.RA/0606241.
- 13.
- J.-L. Loday, Cyclic Homology. Grundlehren der mathematischen Wissenschaften 301, second edition, Springer, 1998. MR 1600246 (98h:16014)
- 14.
- J. Neisendorfer. Lie algebras, coalgebras and rational homotopy theory for nilpotent spaces. Pacific J. Math. 74 (1978), no. 2, 429-460. MR 494641 (80b:55010)
- 15.
- J. Stasheff. Rational Poincaré duality spaces. Illinois J. Math. 27 (1983), no. 1, 104-109. MR 684544 (85c:55012)
- 16.
- R. Umble. Homotopy conditions that determine rational homotopy type. J. Pure Appl. Algebra 60 (1989), no. 2, 205-217. MR 1020716 (90i:55021)
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Additional Information
A. Lazarev
Affiliation:
Department of Mathematics, University of Leicester, Leicester LE1 7RH, England
Email:
al179@le.ac.uk
DOI:
http://dx.doi.org/10.1090/S0002-9939-07-09040-5
PII:
S 0002-9939(07)09040-5
Received by editor(s):
December 30, 2005
Received by editor(s) in revised form:
December 2, 2006
Posted:
October 24, 2007
Additional Notes:
This research was partially supported by the EPSRC grant No. GR/SO7148/01
Communicated by:
Paul Goerss
Article copyright:
© Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
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