arXiv subjects

Katsuhiko Kuribayashi

Publications and source records attributed to Katsuhiko Kuribayashi.

27 records · Page 2Linked to original sources

arXiv2013

Association schemoids and their categories

We propose the notion of association schemoids generalizing that of association schemes from small categorical points of view. In particular, a generalization of the Bose-Mesner algebra of an association scheme appears as a subalgebra in the category algebra of the underlying category of a schemoid. In this paper, the equivalence between the categories of grouopids and that of thin association schemoids is established. Moreover linear extensions of schemoids are considered. A general theory of the Baues-Wirsching cohomology deduces a classification theorem for such extensions of a schemoid. We also introduce two relevant categories of schemoids into which the categories of schemes due to Hanaki and due to French are embedded, respectively.

math.CT↗

arXiv2013

Derived string topology and the Eilenberg-Moore spectral sequence

Let $M$ be any simply-connected Gorenstein space over any field. Félix and Thomas have extended to simply-connected Gorenstein spaces, the loop (co)products of Chas and Sullivan on the homology of the free loop space $H_*(LM)$. We describe these loop (co)products in terms of the torsion and extension functors by developing string topology in appropriate derived categories. As a consequence, we show that the Eilenberg-Moore spectral sequence converging to the loop homology of a Gorenstein space admits a multiplication and a comultiplication with shifted degree which are compatible with the loop product and the loop coproduct of its target, respectively. We also define a generalized cup product on the Hochschild cohomology $HH^*(A,A^\vee)$ of a commutative Gorenstein algebra $A$ and show that over $\mathbb{Q}$, $HH^*(A_{PL}(M),A_{PL}(M)^\vee)$ is isomorphic as algebras to $H_*(LM)$. Thus, when $M$ is a Poincaré duality space, we recover the isomorphism of algebras $\mathbb{H}_*(LM;\mathbb{Q})^\cong HH^*(A_{PL}(M),A_{PL}(M))$ of Félix and Thomas.

math.AT↗

arXiv2013

Behavior of the Eilenberg-Moore spectral sequence in derived string topology

The purpose of this paper is to give applications of the Eilenberg-Moore type spectral sequence converging to the relative loop homology algebra of a Gorenstein space, which is introduced in the previous paper due to the authors. Moreover, it is proved that the spectral sequence is functorial on the category of simply-connected Poincaré duality spaces over a space.

math.AT↗

arXiv2011

On extensions of a symplectic class

Let F be a fibration on a simply-connected base with symplectic fibre (M, ω). Assume that the fibre is nilpotent and T^{2k}-separable for some integer k or a nilmanifold. Then our main theorem, Theorem 1.8, gives a necessary and sufficient condition for the cohomology class [ω] to extend to a cohomology class of the total space of F. This allows us to describe Thurston's criterion for a symplectic fibration to admit a compatible symplectic form in terms of the classifying map for the underlying fibration. The obstruction due to Lalond and McDuff for a symplectic bundle to be Hamiltonian is also rephrased in the same vein. Furthermore, with the aid of the main theorem, we discuss a global nature of the set of the homotopy equivalence classes of fibrations with symplectic fibre in which the class [ω] is extendable.

math.AT↗

arXiv2011

On the levels of maps and topological realization of objects in a triangulated category

The level of a module over a differential graded algebra measures the number of steps required to build the module in an appropriate triangulated category. Based on this notion, we introduce a new homotopy invariant of spaces over a fixed space, called the level of a map. Moreover we provide a method to compute the invariant for spaces over a $\K$-formal space. This enables us to determine the level of the total space of a bundle over the 4-dimensional sphere with the aid of Auslander-Reiten theory for spaces due to Jørgensen. We also discuss the problem of realizing an indecomposable object in the derived category of the sphere by the singular cochain complex of a space. The Hopf invariant provides a criterion for the realization.

math.AT↗

arXiv2011

Upper and lower bounds of the (co)chain type level of a space

We establish an upper bound for the cochain type level of the total space of a pull-back fibration. It explains to us why the numerical invariant for a principal bundle over the sphere are less than or equal to two. Moreover computational examples of the levels of path spaces and Borel constructions, including biquotient spaces and Davis-Januszkiewicz spaces, are presented. We also show that the chain type level of the homotopy fibre of a map is greater than the E-category in the sense of Kahl, which is an algebraic approximation of the Lusternik-Schnirelmann category of the map. The inequality fits between the grade and the projective dimension of the cohomology of the homotopy fibre.

math.AT↗

arXiv2011

The Hochschild cohomology ring of the singular cochain algebra of a space

We determine the algebra structure of the Hochschild cohomology of the singular cochain algebra with coefficients in a field on a space whose cohomology is a polynomial algebra. A spectral sequence calculation of the Hochschild cohomology is also described. In particular, when the underlying field is of characteristic two, we determine the associated bigraded Batalin-Vilkovisky algebra structure on the Hochschild cohomology of the singular cochain on a space whose cohomology is an exterior algebra.

math.AT↗

arXiv2010

Rational visibility of a Lie group in the monoid of self-homotopy equivalences of a homogeneous space

Let M be a homogeneous space admitting a left translation by a connected Lie group G. The adjoint to the action gives rise to a map from G to the monoid of self-homotopy equivalences of M.The purpose of this paper is to investigate the injectivity of the homomorphism which is induced by the adjoint map on the rational homotopy. In particular, the visible degrees are determined explicitly for all the cases of simple Lie groups and their associated homogeneous spaces of rank one which are classified by Oniscik.

math.AT↗

arXiv2009

A rational splitting of a based mapping space

Let F_*(X, Y) be the space of base-point-preserving maps from a connected finite CW complex X to a connected space Y. Consider a CW complex of the form X cup_{alpha}e^{k+1} and a space Y whose connectivity exceeds the dimension of the adjunction space. Using a Quillen-Sullivan mixed type model for a based mapping space, we prove that, if the bracket length of the attaching map alpha: S^k --> X is greater than the Whitehead length WL(Y) of Y, then F_*(X cup_{alpha}e^{k+1}, Y) has the rational homotopy type of the product space F_*(X, Y) times Omega^{k+1}Y. This result yields that if the bracket lengths of all the attaching maps constructing a finite CW complex X are greater than WL(Y) and the connectivity of Y is greater than or equal to dim X, then the mapping space F_*(X, Y) can be decomposed rationally as the product of iterated loop spaces.

math.AT↗