Chas and Sullivan recently defined an intersection product on the homology $H_*(LM)$ of the space of smooth loops in a closed, oriented manifold $M$. In this paper we will use the homotopy theoretic realization of this product described by the first two authors to construct a second quadrant spectral sequence of algebras converging to the loop homology multiplicatively, when $M$ is simply connected. The $E_2$ term of this spectral sequence is $H^*(M;H_*(ΩM))$ where the product is given by the cup product on the cohomology of the manifold $H^* (M)$ with coefficients in the Pontryagin ring structure on the homology of its based loop space $H_*(ΩM)$. We then use this spectral sequence to compute the ring structures of $H_* (LS^n)$ and $H_* (L\bcp^n)$.
We study a spectral sequence that computes the (mod 2) S^1-equivariant homology of the free loop space LM of a manifold M (the "string homology" of M). Using it and knowledge of the string topology operations on the homology of LM, we compute the string homology of M when M is a sphere or a projective space.
We consider the rational vector space generated by all rational homology spheres up to orientation-preserving homeomorphism, and the filtration defined on this space by Lagrangian-preserving rational homology handlebody replacements. We identify the graded space associated with this filtration with a graded space of augmented Jacobi diagrams.
We show that the homology over a field of the space of free maps from the n-sphere to the n-fold suspension of X depends only on the cohomology algebra of X and compute it explicitly. We compute also the homology of the closely related labelled configuration space on the n-sphere with labels in X and of its completion, that depends only on the homology of X. In many but not all cases the homology of the configuration space coincides with the homology of the mapping space. In particular we obtain the homology of the unordered configuration spaces on a sphere.
As an application of Behrens and Rezk's spectral algebra model for unstable v_n-periodic homotopy theory, we give explicit presentations for the completed E-homology of the Bousfield-Kuhn functor on odd-dimensional spheres at chromatic level 2, and compare them to the level 1 case. The latter reflects earlier work in the literature on K-theory localizations.
Chas and Sullivan introduced string homology, which is the equivariant homology of the loop space with the $S^1$ action on loops by rotation. Craig Westerland computed the string homology for spheres with coefficients in $\mathbb{Z} /2\mathbb{Z}$ and Somnath Basu computed the string homology and string bracket for spheres over rational coefficients and found that the bracket is trivial in his dissertation. In this paper, we compute string homology and the string bracket for spheres with integer coefficients, treating the odd- and even-dimensional cases separately. We use the Gysin sequence and…
We show that counterexamples of Iyengar and Walker to the algebraic version of Gunnar Carlsson's conjecture on the rank of the homology of a free complex can be extended to examples over any finite group with many choices of the complex.
We show that if K is a non-trivial knot inside a homology sphere X, the rank of the knot Floer homology group associated with K is strictly bigger than the rank of the Heegaard Floer homology group associated with X.
For smooth embeddings of an integral homology 3-sphere in the 6-sphere, we define an integer invariant in terms of their Seifert surfaces. Our invariant gives a bijection between the set of smooth isotopy classes of such embeddings and the integers. It also gives rise to a complete invariant for homology bordism classes of all embeddings of homology 3-spheres in the 6-sphere. As a consequence, we show that two embeddings of an oriented integral homology 3-sphere in the 6-sphere are isotopic if and only if they are homology bordant. We also relate our invariant to the Rohlin invariant and accor…
Working in a semi-abelian context, we use Janelidze's theory of generalised satellites to study universal properties of the Everaert long exact homology sequence. This results in a new definition of homology which does not depend on the existence of projective objects. We explore the relations with other notions of homology, and thus prove a version of the higher Hopf formulae. We also work out some examples.
We use methods from the cohomology of groups to describe the finite groups which can act freely and homologically trivially on closed 3-manifolds which are rational homology spheres.
Given a genus-g Heegaard splitting of a 3-sphere, the genus-g Goeritz group is defined to be the group of the isotopy classes of orientation preserving homeomorphism of the 3-sphere that preserve the splitting. In this paper, we determine the twisted first (co)homology group of the genus-2 Goeritz group of 3-sphere.
In the present paper we study hyperbolic manifolds that are rational homology 3-spheres obtained by colouring of right--angled polytopes. We study the existence (or absence) of different kinds of symmetries of rational homology spheres that preserve the tessellation of the manifold into polytopes. We also describe how to create colourings with given symmetries.
We compute topological Hochschild homology mod $p$ and $v_1$ of the connective cover of the $K(1)$-local sphere spectrum for all primes $p\ge 3$. This is accomplished using a May-type spectral sequence in topological Hochschild homology constructed from a filtration of a commutative ring spectrum.
In this paper we compute the singular homology of the space of immersions of the circle into the $n$-sphere. Equipped with Chas-Sullivan's loop product these homology groups are graded commutative algebras, we also compute these algebras. We enrich Morse spectral sequences for fibrations of free loop spaces together with loop products, this offers some new computational tools for string topology.
In this paper we prove that the Rohlin invariant is the unique invariant inducing a homomorphism on the Torelli group. Using this result we generalize the construction of invariants of homology $3$-spheres from families of trivial 2-cocycles on the Torelli group given by Pitsch to include invariants with values on an abelian group with $2$-torsion.
We give a generalization of Fukaya's Morse homotopy theoretic approach for 2-loop Chern--Simons perturbation theory to 3-valent graphs with arbitrary number of loops at least 2. We construct a sequence of invariants of integral homology 3-spheres with values in a space of 3-valent graphs (Jacobi diagrams or Feynman diagrams) by counting graphs in an integral homology 3-sphere satisfying certain condition described by a set of ordinary differential equations.
In this article, we give a classification of Alexander modules of null-homologous knots in rational homology spheres. We characterize these modules A equipped with their Blanchfield forms $ϕ$, and the modules A such that there is a unique isomorphism class of $(A,ϕ)$, and we prove that for the other modules A, there are infinitely many such classes. We realise all these $(A,ϕ)$ by explicit knots in rational homology spheres.
The main target of this thesis is to solve the Perron's conjecture. This conjecture affirms that some function on the mod p Torelli group, with values in Z/p, is an invariant of mod p homology 3-spheres. In order to solve this conjecture, in this thesis we first study the mod p homology 3-spheres, the rational homology 3-spheres and those that can be realized as a Heegaard splitting with gluing map an element of the mod p Torelli group. In particular we give a criterion to determine whenever a rational homology 3-sphere has a Heegaard splitting with gluing map an element of the Torelli group m…
We consider the following problem: for which classes of finite groups, and in particular finite simple groups, does the minimal dimension of a faithful, smooth action on a homology sphere coincide with the minimal dimension of a faithful, linear action on a sphere? We prove that the two minimal dimensions coincide for the linear fractional groups PSL(2,p) as well as for various classes of alternating and symmetric groups. We prove analogous results also for actions on Euclidean spaces.
We prove that for every integer k>1 there is a simply connected rational homology 5-sphere $\scriptstyle{M^5_k}$ with spin such that $\scriptstyle{H_2(M^5_k,\bbz)}$ has order $\scriptstyle{k^2},$ and $\scriptstyle{M^5_k}$ admits a Riemannian metric of positive Ricci curvature. Moreover, if the prime number decomposition of $\scriptstyle{k}$ has the form $\scriptstyle{k=p_1... p_r}$ for distinct primes $\scriptstyle{p_i}$ then $\scriptstyle{M^5_k}$ is uniquely determined.
Let $Σ$ be a simply connected rational homology sphere. A pair of disjoint closed submanifolds $M_+, M_-$ in $Σ$ are called dual to each other if the complement $Σ- M_+$ strongly homotopy retracts onto $M_-$ or vice-versa. In this paper we will give a complete answer of which integral triples $(n; m_+, m_-)$ can appear, where $n=dim Σ-1$, $m_+={codim}M_+ -1$ and $m_-={codim}M_- -1$.
Special generic maps are smooth maps between smooth manifolds with only definite fold points as their singularities. The problem of whether a closed $n$-manifold admits a special generic map into Euclidean $p$-space for $1 \leq p \leq n$ was studied by several authors including Burlet, de Rham, Porto, Furuya, Èliašberg, Saeki, and Sakuma. In this paper, we study rational homology $n$-spheres that admit special generic maps into $\mathbb{R}^{p}$ for $p<n$. We use the technique of Stein factorization to derive a necessary homological condition for the existence of such maps for odd $n$. We exami…
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 algebra…
In this note we compute several invariants (e.g. algebraic K-theory, cyclic homology and topological Hochschild homology) of the noncommutative projective schemes associated to Koszul algebras of finite global dimension.
We show that various classes of products of manifolds do not support transitive Anosov diffeomorphisms. Exploiting the Ruelle-Sullivan cohomology class, we prove that the product of a negatively curved manifold with a rational homology sphere does not support transitive Anosov diffeomorphisms. We extend this result to products of finitely many negatively curved manifolds of dimensions at least three with a rational homology sphere that has vanishing simplicial volume. As an application of this study, we obtain new examples of manifolds that do not support transitive Anosov diffeomorphisms, inc…
