Book Review
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MathSciNet review:
2721043
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Book Information:
Authors:
Ralph L. Cohen,
Kathryn Hess and
Alexander A. Voronov
Title:
String topology and cyclic homology
Additional book information:
Advanced Courses in Mathematics, CRM Barcelona,
Birkh{\"a}user Verlag,
Basel,
2006,
viii+163 pp.,
ISBN 3-7643-2182-2,
softcover
Alberto Abbondandolo and Matthias Schwarz, On the Floer homology of cotangent bundles, Comm. Pure Appl. Math. 59 (2006), no. 2, 254–316. MR 2190223, DOI 10.1002/cpa.20090
A. Abbondandolo and M. Schwarz, Floer homology of cotangent bundles and the loop product, arXiv:0810.1995.
A. J. Berrick, Algebraic $K$-theory and algebraic topology, Contemporary developments in algebraic $K$-theory, ICTP Lect. Notes, XV, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2004, pp. 97–190. MR 2175641
M. Bökstedt, W. C. Hsiang, and I. Madsen, The cyclotomic trace and algebraic $K$-theory of spaces, Invent. Math. 111 (1993), no. 3, 465–539. MR 1202133, DOI 10.1007/BF01231296
D. Burghelea and Z. Fiedorowicz, Cyclic homology and algebraic $K$-theory of spaces. II, Topology 25 (1986), no. 3, 303–317. MR 842427, DOI 10.1016/0040-9383(86)90046-7
Alberto S. Cattaneo, Jürg Fröhlich, and Bill Pedrini, Topological field theory interpretation of string topology, Comm. Math. Phys. 240 (2003), no. 3, 397–421. MR 2005849, DOI 10.1007/s00220-003-0917-2
M. Chas and D. Sullivan, String topology, preprint math.GT/9911159 (1999).
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
David Chataur, A bordism approach to string topology, Int. Math. Res. Not. 46 (2005), 2829–2875. MR 2180465, DOI 10.1155/IMRN.2005.2829
K. Cieliebak and J. Latschev, The role of string topology in symplectic field theory, arXiv:0706:3284.
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
Alain Connes, Cohomologie cyclique et foncteurs $\textrm {Ext}^n$, C. R. Acad. Sci. Paris Sér. I Math. 296 (1983), no. 23, 953–958 (French, with English summary). MR 777584
Alain Connes, Noncommutative differential geometry, Inst. Hautes Études Sci. Publ. Math. 62 (1985), 257–360. MR 823176
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
Kenji Fukaya, Application of Floer homology of Langrangian submanifolds to symplectic topology, Morse theoretic methods in nonlinear analysis and in symplectic topology, NATO Sci. Ser. II Math. Phys. Chem., vol. 217, Springer, Dordrecht, 2006, pp. 231–276. MR 2276953, DOI 10.1007/1-4020-4266-3_{0}6
K. Fukaya, Y.-G. Oh, H. Ohta and K. Ono, Lagrangian intersection Floer homology—anomaly and obstruction, 2008.
Murray Gerstenhaber, The cohomology structure of an associative ring, Ann. of Math. (2) 78 (1963), 267–288. MR 161898, DOI 10.2307/1970343
V. Godin, Higher string topology operations, arxiv:0711.4859.
Thomas G. Goodwillie, Cyclic homology, derivations, and the free loopspace, Topology 24 (1985), no. 2, 187–215. MR 793184, DOI 10.1016/0040-9383(85)90055-2
Detlef Gromoll and Wolfgang Meyer, Periodic geodesics on compact riemannian manifolds, J. Differential Geometry 3 (1969), 493–510. MR 264551
John D. S. Jones, Cyclic homology and equivariant homology, Invent. Math. 87 (1987), no. 2, 403–423. MR 870737, DOI 10.1007/BF01389424
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
Ib Madsen, Algebraic $K$-theory and traces, Current developments in mathematics, 1995 (Cambridge, MA), Int. Press, Cambridge, MA, 1994, pp. 191–321. MR 1474979
Martin Markl, Steve Shnider, and Jim Stasheff, Operads in algebra, topology and physics, Mathematical Surveys and Monographs, vol. 96, American Mathematical Society, Providence, RI, 2002. MR 1898414, DOI 10.1090/surv/096
S. A. Merkulov, De Rham model for string topology, Int. Math. Res. Not. 55 (2004), 2955–2981. MR 2099178, DOI 10.1155/S1073792804132662
A. Ramirez, Open-closed string topology via fat graphs, math/0606512.
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
D. Sullivan and M. Sullivan, in preparation.
Micheline Vigué-Poirrier and Dennis Sullivan, The homology theory of the closed geodesic problem, J. Differential Geometry 11 (1976), no. 4, 633–644. MR 455028
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
References
- A. Abbondandolo and M. Schwarz, On the Floer homology of cotangent bundles, Comm. Pure Appl. Math. 59 (2006), no. 2, 254–316. MR 2190223 (2006m:53137)
- A. Abbondandolo and M. Schwarz, Floer homology of cotangent bundles and the loop product, arXiv:0810.1995.
- A. J. Berrick, Algebraic $K$-theory and algebraic topology, in Contemporary Developments in Algebraic $K$-theory, ICTP Lect. Notes, XV, ICTP, Trieste, 2004, 97–190. MR 2175641 (2006m:19002)
- M. Bökstedt, W. C. Hsiang and I. Madsen, The cyclotomic trace and algebraic $K$-theory of spaces, Invent. Math. 111 (1993), no. 3, 465–539. MR 1202133 (94g:55011)
- D. Burghelea and Z. Fiedorowicz, Cyclic homology and algebraic $K$-theory of spaces. II Topology 25 (1986), no. 3, 303–317. MR 842427 (88i:18009b)
- A. Cattaneo, J. Fröhlich and B. Pedrini, Topological field theory interpretation of string topology, Comm. Math. Phys. 240 (2003), no. 3, 397–421. MR 2005849 (2005a:81204)
- M. Chas and D. Sullivan, String topology, preprint math.GT/9911159 (1999).
- M. Chas and D. Sullivan, Closed string operators in topology leading to Lie bialgebras and higher string algebra, in The Legacy of Niels Henrik Abel, Springer, Berlin, 2004, 771–784. MR 2077595 (2005f:55007)
- D. Chataur, A bordism approach to string topology, Int. Math. Res. Not. 2005, no. 46, 2829–2875. MR 2180465 (2007b:55009)
- K. Cieliebak and J. Latschev, The role of string topology in symplectic field theory, arXiv:0706:3284.
- R. Cohen and J. D. S. Jones, A homotopy theoretic realization of string topology, Math. Ann. 324, 2002, 773–798. MR 1942249 (2004c:55019)
- A. Connes, Cohomologie cyclique et foncteurs $\textrm {Ext}^ n$, C. R. Acad. Sci. Paris Sér. I Math. 296 (1983), no. 23, 953–958. MR 777584 (86d:18007)
- A. Connes, Non-commutative differential geometry, Publ. Math. IHES 62, 1985, 257–360. MR 823176 (87i:58162)
- Y. Félix and J.-C. Thomas, Rational BV-algebra in string topology, Bull. Soc. Math. France 136 (2008), no. 2, 311–327. MR 2415345 (2009c:55015)
- K. Fukaya, Application of Floer homology of Lagrangian submanifolds to symplectic topology, in Morse Theoretic Methods in Nonlinear Analysis and in Symplectic Topology, ed. by P. Biran, O. Cornea and F. Lalonde, 231–276. MR 2276953 (2007j:53112)
- K. Fukaya, Y.-G. Oh, H. Ohta and K. Ono, Lagrangian intersection Floer homology—anomaly and obstruction, 2008.
- M. Gerstenhaber, The cohomology structure of an associative ring, Ann. of Math. (2) 78 1963 267–288. MR 0161898 (28:5102)
- V. Godin, Higher string topology operations, arxiv:0711.4859.
- T. G. Goodwillie, Cyclic homology, derivations, and the free loopspace, Topology 24 (1985), no. 2, 187–215. MR 793184 (87c:18009)
- D. Gromoll and W. Meyer, Periodic geodesics on compact Riemannian manifolds, J. Diff. Geom. 3, 1969, 493–510. MR 0264551 (41:9143)
- J. D. S. Jones, Cyclic homology and equivariant homology, Invent. Math. 87 (1987), no. 2, 403–423. MR 870737 (88f:18016)
- J.-L. Loday. Cyclic Homology, 2nd edition, Grundlehren der mathematischen Wissenschaften 301, Springer, Berlin, 1998. MR 1600246 (98h:16014)
- I. Madsen, Algebraic $K$-theory and traces, Current Developments in Mathematics, 1995 (Cambridge, MA), 191–321, Int. Press, Cambridge, MA, 1994. MR 1474979 (98g:19004)
- M. Markl, S. Shnider and J. Stasheff, Operads in algebra, topology and physics, Mathematical Surveys and Monographs, 96. American Mathematical Society, Providence, RI, 2002. x+349 pp., ISBN: 0-8218-2134-2. MR 1898414 (2003f:18011)
- S. A. Merkulov, De Rham model for string topology, Int. Math. Res. Not. 2004, no. 55, 2955–2981. MR 2099178 (2005g:57055)
- A. Ramirez, Open-closed string topology via fat graphs, math/0606512.
- D. Sullivan, Open and closed string field theory interpreted in classical algebraic topology, in Topology, Geometry and Quantum Field Theory, 344–357, London Math. Soc. Lecture Note Ser., 308, Cambridge Univ. Press, Cambridge, 2004. MR 2079379 (2005g:81289)
- D. Sullivan, String topology: background and present state, Current Developments in Mathematics (2005), 41–88, Int. Press, Somerville, MA, 2007; arxiv:0710.4141. MR 2459297
- D. Sullivan and M. Sullivan, in preparation.
- D. Sullivan and M. Vigué-Poirrier, The homology theory of the closed geodesic problem, J. Diff. Geom. 11, 1976, 633–644. MR 0455028 (56:13269)
- A. A. Voronov, Notes on universal algebra, in Graphs and Patterns in Mathematics and Theoretical Physics, 81–103, Proc. Sympos. Pure Math., 73, Amer. Math. Soc., Providence, RI, 2005. MR 2131012 (2005k:18017)
Review Information:
Reviewer:
Janko Latschev
Affiliation:
Departement Mathematik, ETH Zürich
Email:
janko@math.ethz.ch
Journal:
Bull. Amer. Math. Soc.
47 (2010), 705-712
DOI:
https://doi.org/10.1090/S0273-0979-09-01265-8
Published electronically:
June 23, 2009
Review copyright:
© Copyright 2009
American Mathematical Society
