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String topology background and present state. (English) Zbl 1171.55003

Jerison, David (ed.) et al., Current developments in mathematics, 2005. Somerville, MA: International Press (ISBN 978-1-57146-166-7/hbk). 41-88 (2007).
This paper is a survey concerning the recent development of string topology; it consists of 3 parts with an appendix concerning the homotopy theory of the master equation.
Part 1. In part 1, the author explains the results of string topology treated in the following two papers: [M. Chas and D. Sullivan, “String topology”, preprint (ArXiv: math. GT/9911159). M. Chas and D. Sullivan, The legacy of Niels Henrik Abel. Papers from the Abel bicentennial conference, University of Oslo, Oslo, Norway, June 3–8, 2002. Berlin: Springer. 771–784 (2004; Zbl 1068.55009).
First, the author recalls the intersection product \(m:\mathbb H_i\otimes \mathbb H_j\to \mathbb H_{i+j}\) which is a generalization of the usual intersection product, where \(\mathbb H_k=H_{k+d}(LM^d)\) (and \(LM^d:\) the free loop space of an orientable \(d\) dimensional manifold \(M^d\)). Then he studies its chain level generalization and the Batalin-Vilkovisky algebra (BV-algebra) structure on it. To understand this, he uses the \(S^1\) equivariant homology \(\mathbb H_i^{S^1}\) of \(LM\) and its related long exact sequence \[ \cdots @>{M}>>\mathbb H_{i+2} @>{E}>>\mathbb H_{i+2}^{S^1} @>{\cap c}>>\mathbb H_i^{S^1} @>{M}>>\mathbb H_{i+1} @>{E}>>\mathbb H_{i+1}^{S^1}\to\cdots \]
Then he shows that \(\mathbb H_*^{S^1},m,(\Delta)\) form a BV-algebra, where \(\Delta =M\circ E\).
Next, after explaining the string diagrams for closed strings, dessins d’enfants, the combinatorial model and the general constructions for the equivariant loop space, zipping up the noncomposition type A boundary of the combinatorial moduli space etc., finally he obtains the following:
Theorem (open strings). The top chain in each moduli space yields an operation from strings to surfaces such that the total sum \(X\) satisfies a master equation
\[ dX+X*X+\delta_1X+\delta_2X+\cdots =0 \]
where \(*\) denotes the sum over all inputs, outputs, and gluings, \(\delta_1\) refers to the operation inverse to erasing an output boundary puncture, \(\delta_2\) refers to the operation of gluing which is inverse to cutting along the small arc,…. The \(\delta\) operations involve capping with Poincaré dual cocycles.
Part 2. In part II, the author explains the history of string topology and its background with related works. The sections are named Thurston’s work, Wolpert’s formula, Goldman’s brackets, Turaev’s question, Chas’ conjecture on embedded conjugacy classes and group theory equivalent of the Poincaré conjecture, Algebra perspective on string topology, Homotopy theory or algebraic topology perspective on string topology, Symplectic topology perspective on string topology and Riemannian geometry perspective on string topology.
Part 3. In part III, the author explains the diffusion intersection, the short geodesic construction of string topology and states his main results. More precisely, to state these, let \(\mathbb L_*^{S^1}(k)\) denote the “diffuse equivariant” chains for the mapping space of \(k\)-labelled circles with the structure group the \(k\)-torus acting by rotations on the \(k\)-labelled domain circles. Then his main results are stated as follows:
Theorem. (i) On the reduced equivariant chains of the free loop space \(\mathbb L^{S^1}_*\), the “diffuse intersection” string topology construction produces an involutive Lie bialgebra structure up to homotopy, and it has the structure of a graded differential Lie algebra of degree \(-d+2\).
(ii) Workhouse theorem. (Because the statement is too long, details are omitted.)
(iii) The chain level string bracket construction, together with the moduli space chain homotopies, yields a coderivation differential \(d=d_2+d_3+\cdots \) of degree \(-1\) on the free graded algebra generated by the equivariant homology \(\mathbb H_*^{S^1}\) shifted by \(-d+3\), where the differential \(d\) is well defined up to isomorphism homotopic to the identity.
This survey report explains the geometrical ideas nicely and intuitively and is suitable for experts as well.
For the entire collection see [Zbl 1149.00019].

MSC:

55N45 Products and intersections in homology and cohomology
55P35 Loop spaces
57R56 Topological quantum field theories (aspects of differential topology)
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
81T45 Topological field theories in quantum mechanics
55U35 Abstract and axiomatic homotopy theory in algebraic topology
55U15 Chain complexes in algebraic topology

Citations:

Zbl 1068.55009
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