Katsuhiko Kuribayashi

The cohomology of a pull-back on K-formal spaces

Abstract We prove a collapse theorem for the Eilenberg–Moore spectral sequence with coefficients in a field K converging to the cohomology of the pull-back of a fibration q :E→B by a continuous map f :X→B when E , X and B are K -formal. We also show that the cohomology algebra of the pull-back can be expressed via the torsion functor with the shc -minimal model for B in the sense of Ndombol and Thomas [Topology 41 (2002) 85] and its free extensions for E and X without the assumption of K -formality. Moreover not only does the shc -minimal models for E , B and X enable us to construct a model for the Eilenberg–Moore spectral sequence also they help in computing the spectral sequence.

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 groupoids 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.

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 U α 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 a: S k → X is greater than the Whitehead length WL(Y) of Y, then F*(X ∪ α e k+1 , Y) has the rational homotopy type of the product space F* (X, Y) x Ω 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.

On the vanishing problem of string classes

The ordinary string class is an obstruction to lift the structure group LSpin ( n ) of a loop group bundle LQ → LM to the universal central extension of LSpin ( n ) by the circle. The vanishing problem of the ordinary string class and generalized string classes are considered from the viewpoint of the ring structure of the cohomology H *( M ; R).

Mod p equivariant cohomology of homogeneous spaces

Let (G;K) be a compact Lie pair. Then K acts on the homogeneous space G=K by left translation. In [3], Shiga has studied the condition on the pair (G;K) that the bration G=K !EK K G=K !BK is totally non-cohomologous to zero with respect to the real eld from the viewpoint of the equivariant de Rham cohomology and its model. We generalize the main theorem in [3] on the ordinary cohomology theory with coecients in any eld. c 2000 Elsevier Science B.V. All rights reserved. MSC: 55N91; 55T15

THE VANISHING PROBLEM OF THE STRING CLASS WITH DEGREE 3

Let be an SO .n/-bundle over a simply connected manifold M with a spin structure Q! M. The string class is an obstruction to lift the structure group LSpin.n/ of the loop group bundle LQ ! LM to the universal central extension of LSpin.n/ by the circle. We prove that the string class vanishes if and only if 1=2 the first Pontrjagin class of vanishes when M is a compact simply connected homogeneous space of rank one, a simply connected 4-dimensional manifold or a finite product space of those manifolds. This result is deduced by using the Eilenberg-Moore spectral sequence converging to the mod p cohomology of LM whose E2-term is the Hochschild homology of the mod p cohomology algebra of M. The key to the consideration is existence of a morphism of algebras, which is injective below degree 3, from an important graded commutative algebra into the Hochschild homology of a certain graded commutative algebra.

The extension problem and the mod 2 cohomology of the space of loops on spin (N)

Let {E r , d r } be a spectral sequence converging to a Hopf algebra H*. We give a method of reconstructing H* from E∞**. By using our method, we determine the mod 2 cohomology of the space of loops on a simply-connected space whose mod 2 cohomology is isomorphic to that of Spin(N) as an algebra over the Steenrod algebra.

A topos associated with a colored category

We show that a functor category whose domain is a colored category is a topos.The topos structure enables us to introduce cohomology of colored categories including quasi-schemoids. If the given colored category arises from an association scheme, then the cohomology coincides with the group cohomology of the factor scheme by the thin residue. Moreover, it is shown that the cohomology of a colored category relates to the standard representation of an association scheme via the Leray spectral sequence.