10,691 research outputs found
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.Comment: 22 pages. Minor correction
On strong homotopy for quasi-schemoids
A quasi-schemoid is a small category with a particular partition of the set
of morphisms. We define a homotopy relation on the category of quasi-schemoids
and study its fundamental properties. As a homotopy invariant, the homotopy set
of self-homotopy equivalences on a quasi-schemoid is introduced. The main
theorem enables us to deduce that the homotopy invariant for the quasi-schemoid
induced by a finite group is isomorphic to the automorphism group of the given
group.Comment: 12 page
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.Comment: 20 pages. Some references are added. The proof of Theorem 4.3 is
revised. Final version, to appear in Annales de l'Institut Fourie
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