arXiv · math/0309038
A De Rham model for string topology based on the theory of iterated integrals is presented.
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S. A. Merkulov. 2004-05-25. De Rham model for string topology. https://arxiv.org/abs/math/0309038
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We give a continuous version of the comparison theorem for free simplicial resolutions in the categories of prounipotent groups over a field of characteristic zero and of pro-$p$ groups. The resolutions carry free bases compatible with degeneracies; their ranks may be infinite. In characteristic zero the lifting step follows from continuous linear splittings and the universal property of completed free Lie algebras. In the pro-$p$ case it follows from the classical projectivity of free pro-$p$ groups. A relative lifting argument and an explicit simplicial cylinder establish homotopy uniqueness. We formulate the free bases using actual surjections in the simplex category, so degeneracy words related by simplicial identities are identified from the outset, and we spell out the full matching objects used in the lifting argument. The question arose naturally in our preceding work on Bousfield--Kan completions of subcontractible presentations.
We study Bousfield--Kan completions through the interaction of free simplicial resolutions, their filtration spectral sequences, and a noncommutative arithmetic square. For every free discrete simplicial group of finite type, we express its integral pronilpotent completion as the homotopy pullback of its rational prounipotent completion and the product of its pro-$p$ completions over an explicit adelic simplicial group. The adelic entry is formed by taking restricted products at finite nilpotent stages and then their inverse limit; no nilpotency assumption on the original group of components is required. Finite subpresentations of contractible presentations provide an explicit application of this construction. Independence of the specified relators makes the positive-degree terms of the rational and mod-$p$ filtration spectral sequences vanish, with convergence verified on the quotient towers. Continuous comparison of free simplicial resolutions then realizes, in characteristic zero, the equivalence with a constant free prounipotent group by morphisms and homotopies in that category. For the corresponding presentation complex $K$ we obtain $R_\infty K\simeq K(F_R(Z),1)$ for $R=\mathbb Q,\mathbb F_p,\mathbb Z$, where $Z$ indexes a complementary basis and $F_R(Z)$ denotes, respectively, the rational points of a free prounipotent group, a free pro-$p$ group, or a free pronilpotent group. Compatible contractions at the nilpotent stages identify all four entries of the arithmetic square in this case.
We consider the space of embeddings of finitely many circles that bound disks in non-positively curved surfaces. We index the connected components of this space with finite rooted trees and show that the connected components are classifying spaces of the ``braided" automorphism groups of the associated trees. An intermediate step to proving these results is to construct a strong deformation retract onto the subspace of geometric circles; moreover, this strong deformation retraction is equivariant with respect to transformations of the surface.