The center of a rational cohomology algebra for some loop spaces with respect to the Pontryagin product

AbstractWe consider the number of spaces, up to rational homotopy equivalence, which have rational cohomology algebra isomorphic to that of stunted complex projective space . Using a classification theory due to Schlessinger and Stasheff, we determine the number of rational homotopy types with rational comology algebra isomorphic to , for any given n and k. The necessary computations make use of a spectral sequence introduced by the second named author.

Download Full-text

Rational Cohomology Algebra of Mapping Spaces between Complex Grassmannians

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/2020/9385153 ◽

2020 ◽

Vol 2020 ◽

pp. 1-4

Author(s):

Paul Antony Otieno ◽

Jean Baptiste Gatsinzi ◽

Vitalis Onyango-Otieno

Keyword(s):

Cohomology Algebra ◽

Mapping Spaces ◽

Rational Cohomology ◽

Natural Inclusion ◽

Sullivan Models

We consider the complex Grassmannian Grk,n of k-dimensional subspaces of ℂn. There is a natural inclusion in,r:Grk,n↪Grk,n+r. Here, we use Sullivan models to compute the rational cohomology algebra of the component of the inclusion in,r in the space of mappings from Grk,n to Grk,n+r for r≥1 and in particular to show that the cohomology of mapGrn,k,Grn,k+r;in,r contains a truncated algebra ℚx/xr+n+k2−nk, where x=2, for k≥2 and n≥4.

Download Full-text

The cohomology algebra of certain loop spaces

Proceedings of the American Mathematical Society ◽

10.1090/s0002-9939-1958-0099661-4 ◽

1958 ◽

Vol 9 (5) ◽

pp. 808-808 ◽

Cited By ~ 2

Author(s):

Edward Halpern

Keyword(s):

Cohomology Algebra ◽

Loop Spaces

Download Full-text

Homological epimorphisms, homotopy epimorphisms and acyclic maps

Forum Mathematicum ◽

10.1515/forum-2019-0249 ◽

2020 ◽

Vol 32 (6) ◽

pp. 1395-1406

Author(s):

Joseph Chuang ◽

Andrey Lazarev

Keyword(s):

Wide Class ◽

Topological Spaces ◽

Loop Spaces ◽

Graded Algebras ◽

Differential Graded Algebras ◽

Algebraic Description ◽

Acyclic Map ◽

Induced Maps

AbstractWe show that the notions of homotopy epimorphism and homological epimorphism in the category of differential graded algebras are equivalent. As an application we obtain a characterization of acyclic maps of topological spaces in terms of induced maps of their chain algebras of based loop spaces. In the case of a universal acyclic map we obtain, for a wide class of spaces, an explicit algebraic description for these induced maps in terms of derived localization.

Download Full-text

RATIONAL LOCAL SYSTEMS AND CONNECTED FINITE LOOP SPACES

Glasgow Mathematical Journal ◽

10.1017/s0017089520000658 ◽

2021 ◽

pp. 1-29

Author(s):

DREW HEARD

Keyword(s):

Lie Group ◽

Loop Space ◽

Algebraic Model ◽

Compact Lie Group ◽

Loop Spaces ◽

Local Systems ◽

Special Case ◽

Finite Loop ◽

Finite Loop Space

Abstract Greenlees has conjectured that the rational stable equivariant homotopy category of a compact Lie group always has an algebraic model. Based on this idea, we show that the category of rational local systems on a connected finite loop space always has a simple algebraic model. When the loop space arises from a connected compact Lie group, this recovers a special case of a result of Pol and Williamson about rational cofree G-spectra. More generally, we show that if K is a closed subgroup of a compact Lie group G such that the Weyl group W G K is connected, then a certain category of rational G-spectra “at K” has an algebraic model. For example, when K is the trivial group, this is just the category of rational cofree G-spectra, and this recovers the aforementioned result. Throughout, we pay careful attention to the role of torsion and complete categories.

Download Full-text