arXiv subjects
Publications and source records attributed to Pavel Hajek.
We prove that on 2-connected closed oriented manifolds, the analytic and algebraic constructions of an IBL$_\infty$ structure associated to a closed oriented manifold coincide. The corresponding structure is invariant under orientation preserving homotopy equivalences and induces on homology the involutive Lie bialgebra structure of Chas and Sullivan.
We extend a CDGA $V$ with a perfect pairing of degree $n$ on cohomology to a CDGA $\hat V$ with a pairing of degree $n$ on chain level such that $\hat V$ admits a Hodge decomposition and retracts onto $V$ preserving the pairing on cohomology; here we suppose that $V$ is either 1-connected, or that $V$ is connected, of finite type, and $n$ is odd. We show that a Hodge decomposition of $\hat V$ induces a differential Poincaré duality model of $V$ in a natural way. Assuming that $H(V)$ is 1-connected, we apply our extension to a Sullivan model of $V$ in the proof of the existence and "uniqueness" of a 1-connected differential Poincaré duality model of $V$ by Lambrechts & Stanley; we eliminate their extra assumptions in the uniqueness statement, including $H^2(V)=0$ if $n$ is odd.
Following Cieliebak, Fukaya, Latschev and Volkov, we construct an IBL-infinity chain model for equivariant string topology on cyclic Hochschild cochains of de Rham cohomology. We study its properties and perform explicit computations.
We study the application of IBL-infinity-algebras to string topology and explicitly compute the case of spheres. This involves finding a Green kernel and computing integrals associated to trivalent ribbon graphs, which are similar to integrals from perturbative Chern-Simons theory. We generalize the computation and show that the twisted IBL-infinity-structure in question is in many cases trivial.