I recently read the original paper by ChasSullivan 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)BatalinVilkovisky 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 ChasSullivan 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 ChasSullivan operations in a way that more or less follows the original ideas (e.g. without using any homotopy theory).

2$\begingroup$ Could you explain exactly what you don't like about the CohenJones set up? I've always thought it very constructive! It's all about loop spaces and explicit pullback 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. $\endgroup$ – Loop Space Mar 1 '10 at 20:49

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. Lin Mar 1 '10 at 21:13

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 ChasSullivan might help validate, e.g., Fukaya's observation that the fundamental chain for the moduli space of leastarea pseudoholomorphic discs in a symp manifold with Lagrangian b.c. should satisfy the MaurerCartan equation with respect to the string bracket in the Lagrangian. $\endgroup$ – Tim Perutz Mar 1 '10 at 21:40

$\begingroup$ @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. $\endgroup$ – Loop Space Mar 2 '10 at 8:42

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 ChasSullivan could be made rigorous. $\endgroup$ – Kevin H. Lin Mar 2 '10 at 18:19
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.hausdorffresearchinstitute.unibonn.de/files/kreckDA.pdf). I'm sure he will send you an electronic version of his thesis: lennart@meierbielefeld.de.
Matthias Kreck

3$\begingroup$ Meier's thesis is available online: math.unibonn.de/people/lmeier/… $\endgroup$ – Dmitri Pavlov Mar 3 '10 at 8:36
I think this was one of the main motivations for the following paper of McClure.
math/0410450 On the chainlevel intersection pairing for PL manifolds. J. E. McClure. Geom. Topol. 10 (2006) 13911424 and Geom. Topol. 13 (2009) 17751777. math.QA (math.GT).