arXiv subjects
Publications and source records attributed to Andrey Lazarev.
We give a conceptual formulation of Kontsevich's `dual construction' producing graph cohomology classes from a differential graded Frobenius algebra with an odd scalar product. Our construction -- whilst equivalent to the original one -- is combinatorics-free and is based on the Batalin-Vilkovisky formalism, from which its gauge-independence is immediate.
We study the structure of the formal groups associated to the Morava $K$-theories of integral Eilenberg-Mac Lane spaces. The main result is that every formal group in the collection $\{K(n)^*K({\mathbb Z}, q), q=2,3,...\}$ for a fixed $n$ enters in it together with its Serre dual, an analogue of a principal polarization on an abelian variety. We also identify the isogeny class of each of these formal groups over an algebraically closed field. These results are obtained with the help of the Dieudonné correspondence between bicommutative Hopf algebras and Dieudonné modules. We extend P. Goerss's results on the bilinear products of such Hopf algebras and corresponding Dieudonné modules.
A standard combinatorial construction, due to Kontsevich, associates to any A-infinity algebra with an invariant inner product, an inhomogeneous class in the cohomology of the moduli spaces of Riemann surfaces with marked points. We describe an alternative version of this construction based on noncommutative geometry and use it to prove that homotopy equivalent algebras give rise to the same cohomology classes. Along the way we re-prove Kontsevich's theorem relating graph homology to the homology of certain infinite-dimensional Lie algebras. An application to topological conformal field theories is given.
We revisit Stasheff's construction of a minimal Lie-Quillen model of a simply-connected closed manifold $M$ 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 $M$ minus a point similar to the Chas-Sullivan string bracket.
We study cohomology theories of strongly homotopy algebras, namely $A_\infty, C_\infty$ and $L_\infty$-algebras and establish the Hodge decomposition of Hochschild and cyclic cohomology of $C_\infty$-algebras thus generalising previous work by Loday and Gerstenhaber-Schack. These results are then used to show that a $C_\infty$-algebra with an invariant inner product on its cohomology can be uniquely extended to a symplectic $C_\infty$-algebra (an $\infty$-generalisation of a commutative Frobenius algebra introduced by Kontsevich). As another application, we show that the `string topology' operations (the loop product, the loop bracket and the string bracket) are homotopy invariant and can be defined on the homology or equivariant homology of an arbitrary Poincare duality space.
We explore two constructions in homotopy category with algebraic precursors in the theory of noncommutative rings and homological algebra, namely the Hochschild cohomology of ring spectra and Morita theory. The present paper provides an extension of the algebraic theory to include the case when $M$ is not necessarily a progenerator. Our approach is complementary to recent work of Dwyer and Greenlees and of Schwede and Shipley. A central notion of noncommutative ring theory related to Morita equivalence is that of central separable or Azumaya algebras. For such an Azumaya algebra A, its Hochschild cohomology HH^*(A,A) is concentrated in degree 0 and is equal to the center of A. We introduce a notion of topological Azumaya algebra and show that in the case when the ground S-algebra R is an Eilenberg-Mac Lane spectrum of a commutative ring this notion specializes to classical Azumaya algebras. A canonical example of a topological Azumaya R-algebra is the endomorphism R-algebra F_R(M,M) of a finite cell R-module. We show that the spectrum of mod 2 topological K-theory KU/2 is a nontrivial topological Azumaya algebra over the 2-adic completion of the K-theory spectrum widehat{KU}_2. This leads to the determination of THH(KU/2,KU/2), the topological Hochschild cohomology of KU/2. As far as we know this is the first calculation of THH(A,A) for a noncommutative S-algebra A.
We develop a simple theory of André-Quillen cohomology for commutative differential graded algebras over a field of characteristic zero. We then relate it to the homotopy groups of function spaces and spaces of homotopy self-equivalences of rational nilpotent $CW$-complexes. This puts certain results of Sullivan in a more conceptual framework.
Motivated by ideas from stable homotopy theory we study the space of strongly homotopy associative multiplications on a two-cell chain complex. In the simplest case this moduli space is isomorphic to the set of orbits of a group of invertible power series acting on a certain space. The Hochschild cohomology rings of resulting A-infinity algebras have an interpretation as totally ramified extensions of discrete valuation rings. All A-infinity algebras are supposed to be unital and we give a detailed analysis of unital structures which is of independent interest.
This is a survey paper on cohomology theories for $A_\infty$ and $E_\infty$ ring spectra. Different constructions and main properties of topological André-Quillen cohomology and of topological derivations are described. We give sample calculations of these cohomology theories and outline applications to the existence of $A_\infty$ and $E_\infty$ structures on various spectra. We also explain the relationship between topological derivations, spaces of multiplicative maps and moduli spaces of multiplicative structures.
We discuss the Adams Spectral Sequence for R-modules based on commutative localized regular quotient ring spectra over a commutative S-algebra R in the sense of Elmendorf, Kriz, Mandell, May and Strickland. The formulation of this spectral sequence is similar to the classical case and the calculation of its E_2-term involves the cohomology of certain `brave new Hopf algebroids' E^R_*E. In working out the details we resurrect Adams' original approach to Universal Coefficient Spectral Sequences for modules over an R ring spectrum. We show that the Adams Spectral Sequence for S_R based on a commutative localized regular quotient R ring spectrum E=R/I[X^{-1}] converges to the homotopy of the E-nilpotent completion pi_*hat{L}^R_ES_R=R_*[X^{-1}]^hat_{I_*}. We also show that when the generating regular sequence of I_* is finite, hatL^R_ES_R is equivalent to L^R_ES_R, the Bousfield localization of S_R with respect to E-theory. The spectral sequence here collapses at its E_2-term but it does not have a vanishing line because of the presence of polynomial generators of positive cohomological degree. Thus only one of Bousfield's two standard convergence criteria applies here even though we have this equivalence. The details involve the construction of an I-adic tower R/I <-- R/I^2 <-- ... <-- R/I^s <-- R/I^{s+1} <-- ... whose homotopy limit is hatL^R_ES_R. We describe some examples for the motivating case R=MU.