Loop homology of spheres and complex projective spaces

In his Inventiones paper, Ziller (Invent. Math: 1-22, 1977) computed the integral homology as a graded abelian group of the free loop space of compact, globally symmetric spaces of rank 1. Chas and Sullivan (String Topology, 1999)showed that the homology of the free loop space of a compact closed orientable manifold can be equipped with a loop product and a BV-operator making it a Batalin-Vilkovisky algebra. Cohen, Jones and Yan (The loop homology algebra of spheres and projective spaces, 2004) developed a spectral sequence which converges to the loop homology as a spectral sequence of algebras. They computed the algebra structure of the loop homology of spheres and complex projective spaces by using Ziller's results and the method of Brown-Shih (Ann. of Math. 69:223-246, 1959, Publ. Math. Inst. Hautes \'Etudes Sci. 3: 93-176, 1962). In this note we compute the loop homology algebra by using only spectral sequences and the technique of universal examples. We therefore not only obtain Zillers' and Brown-Shihs' results in an elementary way, we also replace the roundabout computations of Cohen, Jones and Yan (The loop homology algebra of spheres and projective spaces, 2004) making them independent of Ziller's and Brown-Shihs' work. Moreover we offer an elementary technique which we expect can easily be generalized and applied to a wider family of spaces, not only the globally symmetric ones.Comment: 10 pages, 8 figure

String topology for spheres

Let M be a compact oriented d-dimensional smooth manifold. Chas and Sullivan have defined a structure of Batalin-Vilkovisky algebra on H*(LM). Extending work of Cohen, Jones and Yan, we compute this Batalin-Vilkovisky algebra structure when M is a sphere Sd, d ≥ 1. In particular, we show that H*(LS2;{F}2) and the Hochschild cohomology HH*(H*(S2);H*(S2)) are surprisingly not isomorphic as Batalin-Vilkovisky algebras, although we prove that, as expected, the underlying Gerstenhaber algebras are isomorphic. The proof requires the knowledge of the Batalin-Vilkovisky algebra H*(Ω2S3;{F}2) that we compute in the Appendix

Batalin-Vilkovisky algebra structures on Hochschild Cohomology

Soit M une variété lisse orientée compact simplement connexe de dimension d. Soit F un corps commutatif quelconque. Nous montrons que la structure d\u27algèbre de Gerstenhaber sur la cohomologie de Hochschild des cochaînes singulières de M, HH* (S* (M), S*(M)), s\u27étend en une algèbre de Batalin-Vilkovisky. L\u27existence d\u27une telle algèbre de Batalin-Vilkovisky était conjecturée. Il est prévu qu\u27une telle algèbre soit isomorphe à l\u27algèbre de Batalin-Vilkovisky sur l\u27homologie des lacets libres sur M, H*+d(LM), introduite par Chas and Sullivan. Nous montrons aussi que la cohomologie cyclique négative HC―*(S*(M)) possède un crochet de Lie. Ce crochet de Lie devrait coincider avec le crochet des cordes de Chas et Sullivan sur l\u27homologie équivariante HS1*(LM)

String topology of classifying spaces

Let G be a finite group or a compact connected Lie group and let BG be its classifying space. Let ℒBG ≔ map(S1, BG) be the free loop space of BG, i.e. the space of continuous maps from the circle S1 to BG. The purpose of this paper is to study the singular homology H*(ℒBG) of this loop space. We prove that when taken with coefficients in a field the homology of ℒBG is a homological conformal field theory. As a byproduct of our Main Theorem, we get a Batalin–Vilkovisky algebra structure on the cohomology H*(ℒBG). We also prove an algebraic version of this result by showing that the Hochschild cohomology HH*(S*(G), S*(G)) of the singular chains of G is a Batalin–Vilkovisky algebra.Comments (0

Van den Bergh isomorphisms in string topology

Let M be a path-connected closed oriented d-dimensional smooth manifold and let k be a principal ideal domain. By Chas and Sullivan, the shifted free loop space homology of M, H∗+d(LM) is a Batalin–Vilkovisky algebra. Let G be a topological group such that M is a classifying space of G. Denote by S∗(G) the (normalized) singular chains on G. Suppose that G is discrete or path-connected. We show that there is a Van Den Bergh type isomorphism                                           HH−p(S∗(G),S∗(G))≅HHp+d(S∗(G),S∗(G)). Therefore, the Gerstenhaber algebra HH∗(S∗(G),S∗(G)) is a Batalin–Vilkovisky algebra and we have a linear isomorphism                                          HH∗(S∗(G),S∗(G))≅H∗+d(LM). This linear isomorphism is expected to be an isomorphism of Batalin–Vilkovisky algebras. We also give a new characterization of Batalin–Vilkovisky algebra in terms of the derived bracket

BV-structures on the homology of the framed long knot space

We introduce BV-algebra structures on the homology of the space of framed long knots in $\mathbb{R}^n$ in two ways. The first one is given in a similar fashion to Chas-Sullivan's string topology. The second one is defined on the Hochschild homology associated with a cyclic, multiplicative operad of graded modules. The latter can be applied to Bousfield-Salvatore spectral sequence converging to the homology of the space of framed long knots. Conjecturally these two structures coincide with each other.Comment: 13 pages, 3 figures, to appear in Journal of Homotopy and Related Structure

Symplectic cohomology and q-intersection numbers

Given a symplectic cohomology class of degree 1, we define the notion of an equivariant Lagrangian submanifold. The Floer cohomology of equivariant Lagrangian submanifolds has a natural endomorphism, which induces a grading by generalized eigenspaces. Taking Euler characteristics with respect to the induced grading yields a deformation of the intersection number. Dehn twists act naturally on equivariant Lagrangians. Cotangent bundles and Lefschetz fibrations give fully computable examples. A key step in computations is to impose the "dilation" condition stipulating that the BV operator applied to the symplectic cohomology class gives the identity. Equivariant Lagrangians mirror equivariant objects of the derived category of coherent sheaves.Comment: 32 pages, 9 figures, expanded introduction, added details of example 7.5, added discussion of sign

The YqfN protein of Bacillus subtilis is the tRNA: m1A22 methyltransferase (TrmK)

N1-methylation of adenosine to m1A occurs in several different positions in tRNAs from various organisms. A methyl group at position N1 prevents Watson–Crick-type base pairing by adenosine and is therefore important for regulation of structure and stability of tRNA molecules. Thus far, only one family of genes encoding enzymes responsible for m1A methylation at position 58 has been identified, while other m1A methyltransferases (MTases) remain elusive. Here, we show that Bacillus subtilis open reading frame yqfN is necessary and sufficient for N1-adenosine methylation at position 22 of bacterial tRNA. Thus, we propose to rename YqfN as TrmK, according to the traditional nomenclature for bacterial tRNA MTases, or TrMet(m1A22) according to the nomenclature from the MODOMICS database of RNA modification enzymes. tRNAs purified from a ΔtrmK strain are a good substrate in vitro for the recombinant TrmK protein, which is sufficient for m1A methylation at position 22 as are tRNAs from Escherichia coli, which natively lacks m1A22. TrmK is conserved in Gram-positive bacteria and present in some Gram-negative bacteria, but its orthologs are apparently absent from archaea and eukaryota. Protein structure prediction indicates that the active site of TrmK does not resemble the active site of the m1A58 MTase TrmI, suggesting that these two enzymatic activities evolved independently

Gerstenhaber duality in Hochschild cohomology

Let C be a differential graded chain coalgebra, &UOmega; C the reduced cobar construction on C and C-V the dual algebra. We prove that for a large class of coalgebras C there is a natural isomorphism of Gerstenhaber algebras between the Hochschild cohomologies, HH*(C-V, C-V) and HH*(&UOmega; C; &UOmega; C). This result yields to a Hodge decomposition of the loop space homology of a closed oriented manifold, when the field of coefficients is of characteristic zero. © 2004 Elsevier B.V. All rights reserved