We define and study (derived) character maps of finite-dimensional representations of 1-groups.As models for 1-groups we take homotopy simplicial groups, ie the homotopy simplicial algebras over the algebraic theory of groups (in the sense of Badzioch ( 2002)).We introduce cyclic, symmetric and representation homology for "group algebras" kOE of such groups and construct canonical trace maps (natural transformations) relating these homology theories.We show that, in the case of one-dimensional representations, our trace maps are of topological origin: they are induced by natural maps of (iterated) loop spaces known in stable homotopy theory.Using this topological interpretation, we deduce some algebraic results on representation homology: in particular, we prove that the symmetric homology of group algebras and one-dimensional representation homology are naturally isomorphic, provided the base ring k is a field of characteristic zero.We also study the stable behavior of the derived character maps of n-dimensional representations as n ! 1, in which case we show that these maps "converge" to become isomorphisms.18A25, 18G15, 19D55, 55N35; 14A30, 55P42

We discuss the Mayer-Vietoris spectral sequence as an invariant in the context of persistent homology. In particular, we introduce the notion of εacyclic carriers and ε-acyclic equivalences between filtered regular CW-complexes and study stability conditions for the associated spectral sequences. We also look at the Mayer-Vietoris blowup complex and the geometric realization, finding stability properties under compatible noise; as a result we prove a version of an approximate nerve theorem. Adapting work by Serre we find conditions under which ε-interleavings exist between the spectral sequences associated to two different covers.

Given a finitely presented group Q and a compact special cube complex X with non-elementary hyperbolic fundamental group, we produce a non-elementary, torsion-free, cocompactly cubulated hyperbolic group Γ that surjects onto Q, with kernel isomorphic to a quotient of G=π1X and such that max{cd(G),2}≥cd(Γ)≥cd(G)−1.

Finite presentations of the mapping class groups of once-stabilized Heegaard splittings DAIKI IGUCHILet g 2 and assume that we are given a genus g Heegaard splitting of a closed orientable 3-manifold with distance greater than 2g C 2. We prove that the mapping class group of the once-stabilization of such a Heegaard splitting is finitely presented.57K30, 57M60 1 Introduction Let .M; / be a Heegaard splitting of a compact orientable 3-manifold M .The mapping class group MCG.M; / of the Heegaard splitting .M; / is defined to be the group 0 .Diff.M; // of path-connected components of the group Diff.M; /, where we denote by Diff.M; / the group of diffeomorphisms of M that preserve setwise.There is a natural homomorphism from MCG.M; / to the mapping class group MCG.M / of M .Following Johnson [2011], we call the kernel of this natural homomorphism the isotopy subgroup of MCG.M; /, and denote it by Isot.M; /.In this paper, we are interested in the isotopy subgroup of the mapping class group of a once-stabilized Heegaard splitting.Let .M; 0 / be a genus g. 0 / 2 Heegaard splitting of a closed orientable 3manifold M .We say that a Heegaard splitting .M; / is a (once-)stabilization of .M; 0 / if it is obtained from .M; 0 / by adding a 1-handle whose core is parallel into 0 .Corresponding to two handlebodies V 0 and V C 0 in M with @V 0 D @V C 0 D 0 , there are two obvious subgroups of Isot.M; /: one is Isot.V 0 ; / and the other is Isot.V C 0 ; C /, where (resp. C ) is the Heegaard surface obtained by pushing into V 0 (resp.V C 0 ) slightly.It is natural to ask when these subgroups generate Isot.M; /.Johnson [2011] proved that if the distance d. 0 / of the Heegaard splitting .M; 0 / is greater than 2g. 0 / C 2, then the two subgroups defined above generate Isot.M; /.As a consequence of this fact, together with a result of Scharlemann [2013] that says Isot.V 0 ; / are finitely generated, it follows that Isot.M; / and MCG.M; / are finitely generated.In that paper, Johnson conjectured that Isot.M; / is an amalgamation of the two groups Isot.V 0 ; / and Isot.V C 0 ; C /.This is the main result of the paper:Theorem 1.1 Suppose that .M; 0 / is Heegaard splitting of a closed orientable 3-manifold M with d. 0 / > 2g. 0 / C 2, and that .M; / is a once-stabilization of .M; 0 /.Suppose that .V 0 ; / (resp.

The group of compactly supported Hamiltonian diffeomorphisms of a symplectic manifold is endowed with a natural bi-invariant distance, due to Viterbo, Schwarz, Oh, Frauenfelder and Schlenk, coming from spectral invariants in Hamiltonian Floer homology.This distance has found numerous applications in symplectic topology.However, its diameter is still unknown in general.In fact, for closed symplectic manifolds there is no unifying criterion for the diameter to be infinite.We prove that for any Liouville domain this diameter is infinite if and only if its symplectic cohomology does not vanish.This generalizes a result of Monzner, Vichery and Zapolsky and has applications in the setting of closed symplectic manifolds.57R17, 51F99, 53D05, 57R58 1. Introduction and results 3759 2. Liouville domains and admissible Hamiltonians 3766 3. Filtered Floer and symplectic cohomology 3773 4. Spectral invariants and spectral norm 3779 5. Cohomological barricades on Liouville domains 3781

On products of beta and gamma elements in the homotopy of the first Smith-Toda spectrumWe determine the first cohomology of the monochromatic comodule M 1 2 at an odd prime, and apply the results to show nontrivialities of some products of beta and gamma elements in the homotopy groups of the Smith-Toda spectrum V .1/.The cohomology for a prime greater than three was previously determined by the first author.Here we verify them and determine the cohomology at the prime 3 by elementary calculation.The cohomology will be a stepping stone for computing the cohomology of the monochromatic comodule M 3 0 , which we hope to determine for a long time.55Q45; 55Q51, 55T15 Let .

We show that the homology of any Temperley-Lieb algebra T Ln(a) on an odd number of strands vanishes in positive degrees. This improves a result obtained by Boyd-Hepworth. In addition we present alternative arguments for the following two vanishing results of Boyd-Hepworth. (1) The stable homology of Temperley-Lieb algebras is trivial. (2) If the parameter a ∈ R is a unit, then the homology of any Temperley-Lieb algebra is concentrated in degree zero.

We consider in parallel pointed homotopy automorphisms of iterated wedge sums of finite CW–complexes and boundary-relative homotopy automorphisms of iterated connected sums of manifolds minus a disk. Under certain conditions on the spaces and manifolds, we prove that the rational homotopy groups of these homotopy automorphisms form finitely generated FI–modules, and thus satisfy representation stability for symmetric groups in the sense of Church and Farb. We also calculate explicit bounds on the weights and stability degrees of these FI–modules.

We compute the topological Hochschild homology of sufficiently structured forms of truncated Brown-Peterson spectra with coefficients. In particular, we compute THH .BhniI H Z .p/ / for all n, where Bhni is an E 3 form of BPhni for certain primes p, and THH .Bh2iI M / for M 2 fk.1/; k.2/g. For example, this gives a computation of THH.taf D I M / for M 2 fH Z .3/ ; k.1/; k.2/g where taf D is the E 1 form of BPh2i constructed by Hill and Lawson.