3,081 research outputs found
The ghost length and duality on the chain and cochain type levels
We establish equalities between cochain and chain type levels of maps by
making use of exact functors which connect appropriate derived and coderived
categories. Relevant conditions for levels of maps to be finite are extracted
from the equalities which we call duality on the levels. Moreover, we give a
lower bound of the cochain type level of the diagonal map on the classifying
space of a Lie group by considering the ghostness of a shriek map which appears
in derived string topology. A variant of Koszul duality for a differential
graded algebra is also discussed.Comment: 23 pages. This is a new verision of the preprint "Duality on the
(co)chain type levels". The title is change
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
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.Comment: 28 page
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