66 research outputs found
Models for classifying spaces and derived deformation theory
Using the theory of extensions of L-infinity algebras, we construct rational
homotopy models for classifying spaces of fibrations, giving answers in terms
of classical homological functors, namely the Chevalley-Eilenberg and Harrison
cohomology. We also investigate the algebraic structure of the
Chevalley-Eilenberg complexes of L-infinity algebras and show that they
possess, along with the Gerstenhaber bracket, an L-infinity structure that is
homotopy abelian.Comment: 23 pages. This version contains minor technical corrections and a new
section with a list of open problems. To appear in Proceedings of the LM
The Stasheff model of a simply-connected manifold and the string bracket
We revisit Stasheff's construction of a minimal Lie-Quillen model of a
simply-connected closed manifold using the language of infinity-algebras.
This model is then used to construct a graded Lie bracket on the equivariant
homology of the free loop space of minus a point similar to the
Chas-Sullivan string bracket.Comment: 9 page
L-infinity maps and twistings
We give a construction of an L-infinity map from any L-infinity algebra into
its truncated Chevalley-Eilenberg complex as well as its cyclic and A-infinity
analogues. This map fits with the inclusion into the full Chevalley-Eilenberg
complex (or its respective analogues) to form a homotopy fiber sequence of
L-infinity-algebras. Application to deformation theory and graph homology are
given. We employ the machinery of Maurer-Cartan functors in L-infinity and
A-infinity algebras and associated twistings which should be of independent
interest.Comment: 16 pages, to appear in Homology, Homotopy and Applications. This
version contains many corrections of technical nature and minor improvement
Unimodular homotopy algebras and Chern-Simons theory
Quantum Chern-Simons invariants of differentiable manifolds are analyzed from
the point of view of homological algebra. Given a manifold M and a Lie (or,
more generally, an L-infinity) algebra g, the vector space H^*(M) \otimes g has
the structure of an L-infinity algebra whose homotopy type is a homotopy
invariant of M. We formulate necessary and sufficient conditions for this
L-infinity algebra to have a quantum lift. We also obtain structural results on
unimodular L-infinity algebras and introduce a doubling construction which
links unimodular and cyclic L-infinity algebras.Comment: 37 pages, expanded introduction and made minor correction
Dual Feynman transform for modular operads
We introduce and study the notion of a dual Feynman transform of a modular
operad. This generalizes and gives a conceptual explanation of Kontsevich's
dual construction producing graph cohomology classes from a contractible
differential graded Frobenius algebra. The dual Feynman transform of a modular
operad is indeed linear dual to the Feynman transform introduced by Getzler and
Kapranov when evaluated on vacuum graphs. In marked contrast to the Feynman
transform, the dual notion admits an extremely simple presentation via
generators and relations; this leads to an explicit and easy description of its
algebras. We discuss a further generalization of the dual Feynman transform
whose algebras are not necessarily contractible. This naturally gives rise to a
two-colored graph complex analogous to the Boardman-Vogt topological tree
complex.Comment: 27 pages. A few conceptual changes in the last section; in particular
we prove that the two-colored graph complex is a resolution of the
corresponding modular operad. It is now called 'BV-resolution' as suggested
by Sasha Vorono
Disconnected rational homotopy theory
We construct two algebraic versions of homotopy theory of rational
disconnected topological spaces, one based on differential graded commutative
associative algebras and the other one on complete differential graded Lie
algebras. As an application of the developed technology we obtain results on
the structure of Maurer-Cartan spaces of complete differential graded Lie
algebras.Comment: 50 pages, a couple of typos corrected and references adde
Curved infinity-algebras and their characteristic classes
In this paper we study a natural extension of Kontsevich's characteristic
class construction for A-infinity and L-infinity algebras to the case of curved
algebras. These define homology classes on a variant of his graph homology
which allows vertices of valence >0. We compute this graph homology, which is
governed by star-shaped graphs with odd-valence vertices. We also classify
nontrivially curved cyclic A-infinity and L-infinity algebras over a field up
to gauge equivalence, and show that these are essentially reduced to algebras
of dimension at most two with only even-ary operations. We apply the reasoning
to compute stability maps for the homology of Lie algebras of formal vector
fields. Finally, we explain a generalization of these results to other types of
algebras, using the language of operads.Comment: Final version, to appear in J. Topology. 28 page
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