An algebraic chain model of string topology
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Abstract:
A chain complex model for the free loop space of a connected, closed and oriented manifold is presented, and on its homology, the Gerstenhaber and Batalin-Vilkovisky algebra structures are defined and identified with the string topology structures. The gravity algebra on the equivariant homology of the free loop space is also modeled. The construction includes the non-simply connected case, and therefore gives an algebraic and chain level model of Chas-Sullivan’s String Topology.References
- J. F. Adams, On the cobar construction, Proc. Nat. Acad. Sci. U.S.A. 42 (1956), 409–412. MR 79266, DOI 10.1073/pnas.42.7.409
- Alejandro Adem and Yongbin Ruan, Twisted orbifold $K$-theory, Comm. Math. Phys. 237 (2003), no. 3, 533–556. MR 1993337, DOI 10.1007/s00220-003-0849-x
- A. K. Bousfield, On the homology spectral sequence of a cosimplicial space, Amer. J. Math. 109 (1987), no. 2, 361–394. MR 882428, DOI 10.2307/2374579
- Edgar H. Brown Jr., Twisted tensor products. I, Ann. of Math. (2) 69 (1959), 223–246. MR 105687, DOI 10.2307/1970101
- Bohumil Cenkl and Richard Porter, de Rham theorem with cubical forms, Pacific J. Math. 112 (1984), no. 1, 35–48. MR 739139
- Chas, M. and Sullivan, D., String topology, arXiv:math-GT/9911159.
- Moira Chas and Dennis Sullivan, Closed string operators in topology leading to Lie bialgebras and higher string algebra, The legacy of Niels Henrik Abel, Springer, Berlin, 2004, pp. 771–784. MR 2077595
- Kuo Tsai Chen, Iterated path integrals, Bull. Amer. Math. Soc. 83 (1977), no. 5, 831–879. MR 454968, DOI 10.1090/S0002-9904-1977-14320-6
- Ralph L. Cohen, Kathryn Hess, and Alexander A. Voronov, String topology and cyclic homology, Advanced Courses in Mathematics. CRM Barcelona, Birkhäuser Verlag, Basel, 2006. Lectures from the Summer School held in Almería, September 16–20, 2003. MR 2251006
- Ralph L. Cohen and John D. S. Jones, A homotopy theoretic realization of string topology, Math. Ann. 324 (2002), no. 4, 773–798. MR 1942249, DOI 10.1007/s00208-002-0362-0
- Ralph L. Cohen, John R. Klein, and Dennis Sullivan, The homotopy invariance of the string topology loop product and string bracket, J. Topol. 1 (2008), no. 2, 391–408. MR 2399136, DOI 10.1112/jtopol/jtn001
- Alain Connes, Noncommutative differential geometry, Inst. Hautes Études Sci. Publ. Math. 62 (1985), 257–360. MR 823176
- Kevin Costello, Topological conformal field theories and Calabi-Yau categories, Adv. Math. 210 (2007), no. 1, 165–214. MR 2298823, DOI 10.1016/j.aim.2006.06.004
- Yves Félix and Jean-Claude Thomas, Rational BV-algebra in string topology, Bull. Soc. Math. France 136 (2008), no. 2, 311–327 (English, with English and French summaries). MR 2415345, DOI 10.24033/bsmf.2558
- Yves Félix, Jean-Claude Thomas, and Micheline Vigué-Poirrier, Rational string topology, J. Eur. Math. Soc. (JEMS) 9 (2007), no. 1, 123–156. MR 2283106, DOI 10.4171/jems/75
- Murray Gerstenhaber, The cohomology structure of an associative ring, Ann. of Math. (2) 78 (1963), 267–288. MR 161898, DOI 10.2307/1970343
- E. Getzler, Batalin-Vilkovisky algebras and two-dimensional topological field theories, Comm. Math. Phys. 159 (1994), no. 2, 265–285. MR 1256989
- E. Getzler, Two-dimensional topological gravity and equivariant cohomology, Comm. Math. Phys. 163 (1994), no. 3, 473–489. MR 1284793
- Ezra Getzler, John D. S. Jones, and Scott Petrack, Differential forms on loop spaces and the cyclic bar complex, Topology 30 (1991), no. 3, 339–371. MR 1113683, DOI 10.1016/0040-9383(91)90019-Z
- Godin, V., Higher string topology operations, arxiv:0711.4859.
- Hamilton, A. and Lazarev, A., Homotopy algebras and noncommutative geometry, arxiv:math.QA/0403340.
- John D. S. Jones, Cyclic homology and equivariant homology, Invent. Math. 87 (1987), no. 2, 403–423. MR 870737, DOI 10.1007/BF01389424
- Bernhard Keller, Hochschild cohomology and derived Picard groups, J. Pure Appl. Algebra 190 (2004), no. 1-3, 177–196. MR 2043327, DOI 10.1016/j.jpaa.2003.10.030
- Ralph M. Kaufmann, A proof of a cyclic version of Deligne’s conjecture via cacti, Math. Res. Lett. 15 (2008), no. 5, 901–921. MR 2443991, DOI 10.4310/MRL.2008.v15.n5.a7
- Jean-Louis Loday, Cyclic homology, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 301, Springer-Verlag, Berlin, 1998. Appendix E by María O. Ronco; Chapter 13 by the author in collaboration with Teimuraz Pirashvili. MR 1600246, DOI 10.1007/978-3-662-11389-9
- Ernesto Lupercio, Bernardo Uribe, and Miguel A. Xicotencatl, Orbifold string topology, Geom. Topol. 12 (2008), no. 4, 2203–2247. MR 2431019, DOI 10.2140/gt.2008.12.2203
- John McCleary, Homotopy theory and closed geodesics, Homotopy theory and related topics (Kinosaki, 1988) Lecture Notes in Math., vol. 1418, Springer, Berlin, 1990, pp. 86–94. MR 1048178, DOI 10.1007/BFb0083695
- John McCleary, A user’s guide to spectral sequences, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 58, Cambridge University Press, Cambridge, 2001. MR 1793722
- J. E. McClure, On the chain-level intersection pairing for PL manifolds, Geom. Topol. 10 (2006), 1391–1424. MR 2255502, DOI 10.2140/gt.2006.10.1391
- Luc Menichi, Batalin-Vilkovisky algebras and cyclic cohomology of Hopf algebras, $K$-Theory 32 (2004), no. 3, 231–251. MR 2114167, DOI 10.1007/s10977-004-0480-4
- Luc Menichi, Batalin-Vilkovisky algebra structures on Hochschild cohomology, Bull. Soc. Math. France 137 (2009), no. 2, 277–295 (English, with English and French summaries). MR 2543477, DOI 10.24033/bsmf.2576
- S. A. Merkulov, De Rham model for string topology, Int. Math. Res. Not. 55 (2004), 2955–2981. MR 2099178, DOI 10.1155/S1073792804132662
- Daniel Quillen, Rational homotopy theory, Ann. of Math. (2) 90 (1969), 205–295. MR 258031, DOI 10.2307/1970725
- David L. Rector, Steenrod operations in the Eilenberg-Moore spectral sequence, Comment. Math. Helv. 45 (1970), 540–552. MR 278310, DOI 10.1007/BF02567352
- Dennis Sullivan, Open and closed string field theory interpreted in classical algebraic topology, Topology, geometry and quantum field theory, London Math. Soc. Lecture Note Ser., vol. 308, Cambridge Univ. Press, Cambridge, 2004, pp. 344–357. MR 2079379, DOI 10.1017/CBO9780511526398.014
- Dennis Sullivan, String topology background and present state, Current developments in mathematics, 2005, Int. Press, Somerville, MA, 2007, pp. 41–88. MR 2459297
- Tradler, T., Two BV-structures identified: The Hochschild cohomology and the homology of the free loop space. CUNY Ph.D. Thesis, 2002.
- Alexander A. Voronov, Homotopy Gerstenhaber algebras, Conférence Moshé Flato 1999, Vol. II (Dijon), Math. Phys. Stud., vol. 22, Kluwer Acad. Publ., Dordrecht, 2000, pp. 307–331. MR 1805923
- Alexander A. Voronov, Notes on universal algebra, Graphs and patterns in mathematics and theoretical physics, Proc. Sympos. Pure Math., vol. 73, Amer. Math. Soc., Providence, RI, 2005, pp. 81–103. MR 2131012, DOI 10.1090/pspum/073/2131012
- A. A. Voronov and M. Gerstenkhaber, Higher-order operations on the Hochschild complex, Funktsional. Anal. i Prilozhen. 29 (1995), no. 1, 1–6, 96 (Russian, with Russian summary); English transl., Funct. Anal. Appl. 29 (1995), no. 1, 1–5. MR 1328534, DOI 10.1007/BF01077036
- Craig Westerland, Equivariant operads, string topology, and Tate cohomology, Math. Ann. 340 (2008), no. 1, 97–142. MR 2349769, DOI 10.1007/s00208-007-0140-0
- J. H. C. Whitehead, On $C^1$-complexes, Ann. of Math. (2) 41 (1940), 809–824. MR 2545, DOI 10.2307/1968861
- Wilson, S., Partial algebras over operads of complexes and applications, arXiv:math.AT/0410405.
Additional Information
- Xiaojun Chen
- Affiliation: School of Mathematics, Sichuan University, Chengdu 610064, People’s Republic of China – and – Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, Michigan 48104
- Email: xch@umich.edu
- Received by editor(s): February 25, 2010
- Received by editor(s) in revised form: October 16, 2010, and November 13, 2010
- Published electronically: November 9, 2011
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 364 (2012), 2749-2781
- MSC (2010): Primary 55P50, 55P35
- DOI: https://doi.org/10.1090/S0002-9947-2011-05518-2
- MathSciNet review: 2888227