I recently read the original paper by Chas-Sullivan on string topology, in which they introduce some operations on homology of free loopspace LM, where M is a compact oriented manifold, giving it the structure of a (Gerstenhaber-)Batalin-Vilkovisky algebra. However, the arguments in this paper rely on some transversality assumptions, and I'm not sure whether these assumptions are justified. I know that the Chas-Sullivan operations have been constructed via homotopy theoretic methods by Cohen, Jones, Voronov (hopefully I'm not missing any names here), but I am wondering whether anybody has managed to construct the Chas-Sullivan operations in a way that more or less follows the original ideas (e.g. without using any homotopy theory).
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asked Mar 1, 2010 at 20:30
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2$\begingroup$ Could you explain exactly what you don't like about the Cohen-Jones set up? I've always thought it very constructive! It's all about loop spaces and explicit pull-back maps. I know it works in cohomology rather than homology, is that your issue? Incidentally, you should also look up work by Veronique Godin on this. $\endgroup$– Andrew StaceyCommented Mar 1, 2010 at 20:49
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1$\begingroup$ I don't have any problems with it. I am just wondering whether it's possible to make the original idea rigorous, that's all. $\endgroup$– Kevin H. LinCommented Mar 1, 2010 at 21:13
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4$\begingroup$ @Andrew. Intersection theory on ordinary manifolds can be understood cohomologically (etc. etc.), but that doesn't vitiate the usefulness of transversality theory. A rigorous version of Chas-Sullivan might help validate, e.g., Fukaya's observation that the fundamental chain for the moduli space of least-area pseudo-holomorphic discs in a symp manifold with Lagrangian b.c. should satisfy the Maurer-Cartan equation with respect to the string bracket in the Lagrangian. $\endgroup$– Tim PerutzCommented Mar 1, 2010 at 21:40
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$\begingroup$ @Tim: That wasn't my point. I was reacting to the slightly dismissive tone of "via homotopy theoretic methods" and "without using any homotopy theory". Transversality arguments need homotopy theory! It seemed that the question was "I don't like homotopy theory, can string topology be done geometrically?" and I wished to point out that the "homotopy theory" method of Cohen and Jones is actually very geometric and "hands on". There's no "up to homotopy" in the actual construction. $\endgroup$– Andrew StaceyCommented Mar 2, 2010 at 8:42
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2$\begingroup$ Andrew: I am not dismissive of homotopy theory at all, nor do I dislike homotopy theory. It was definitely not my intention to sound that way. I just wanted to know if the arguments of Chas-Sullivan could be made rigorous. $\endgroup$– Kevin H. LinCommented Mar 2, 2010 at 18:19
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2 Answers
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I would like to point at the Diploma thesis of my student Lennart Meier, who has given various elementary descriptions of the Chaas Sullivan product (for example using my description of singular homology in terms of bordism groups of stratifolds, see: http://www.hausdorff-research-institute.uni-bonn.de/files/kreck-DA.pdf). I'm sure he will send you an electronic version of his thesis: [email protected].
Matthias Kreck
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3$\begingroup$ Meier's thesis is available online: math.uni-bonn.de/people/lmeier/… $\endgroup$ Commented Mar 3, 2010 at 8:36
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$\begingroup$ The link above is not active anymore. Use this webspace.science.uu.nl/~meier007/… $\endgroup$ Commented Jul 24 at 14:25
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I think this was one of the main motivations for the following paper of McClure.
math/0410450 On the chain-level intersection pairing for PL manifolds. J. E. McClure. Geom. Topol. 10 (2006) 1391-1424 and Geom. Topol. 13 (2009) 1775-1777. math.QA (math.GT).
