scholarly journals Toric objects associated with the dodecahedron

Filomat ◽  
2020 ◽  
Vol 34 (7) ◽  
pp. 2329-2356
Author(s):  
Djordje Baralic ◽  
Jelena Grbic ◽  
Ivan Limonchenko ◽  
Aleksandar Vucic
Keyword(s):  
Homotopy Theory ◽  
Cohomology Ring ◽  
Massey Products ◽  
Small Cover ◽  
Toric Topology ◽  
Quasitoric Manifold

In this paper we illustrate a tight interplay between homotopy theory and combinatorics within toric topology by explicitly calculating homotopy and combinatorial invariants of toric objects associated with the dodecahedron. In particular, we calculate the cohomology ring of the (complex and real) moment-angle manifolds over the dodecahedron, and of a certain quasitoric manifold and of a related small cover. We finish by studying Massey products in the cohomology ring of moment-angle manifolds over the dodecahedron and how the existence of nontrivial Massey products influences the behaviour of the Poincar? series of the corresponding Pontryagin algebra.

scholarly journals Higher order Massey products and applications

2021 ◽  
pp. 209-240
Author(s):  
Ivan Limonchenko ◽  
Dmitry Millionshchikov
Keyword(s):  
Lie Algebras ◽  
Homotopy Theory ◽  
Homology Theory ◽  
Higher Order ◽  
Local Rings ◽  
Main Theme ◽  
Massey Products ◽  
Research Areas ◽  
Cohomology Of Lie Algebras ◽  
Toric Topology

In this survey, we discuss two research areas related to Massey’s higher operations. The first direction is connected with the cohomology of Lie algebras and the theory of representations. The second main theme is at the intersection of toric topology, homotopy theory of polyhedral products, and the homology theory of local rings, Stanley–Reisner rings of simplicial complexes.

scholarly journals Example of C-rigid polytopes which are not B-rigid

2019 ◽  
Vol 69 (2) ◽  
pp. 437-448
Author(s):  
Suyoung Choi ◽  
Kyoungsuk Park
Keyword(s):  
Cohomology Ring ◽  
Combinatorial Structure ◽  
Simple Polytope ◽  
Quasitoric Manifold

Abstract A simple polytope P is said to be B-rigid if its combinatorial structure is characterized by its Tor-algebra, and is said to be C-rigid if its combinatorial structure is characterized by the cohomology ring of a quasitoric manifold over P. It is known that a B-rigid simple polytope is C-rigid. In this paper, we show that the B-rigidity is not equivalent to the C-rigidity.

scholarly journals Massey products, toric topology and combinatorics of polytopes

2019 ◽  
Vol 83 (6) ◽  
pp. 1081-1136
Author(s):  
V. M. Buchstaber ◽  
I. Yu. Limonchenko
Keyword(s):  
Massey Products ◽  
Toric Topology

scholarly journals HIGHER CONJUGATION COHOMOLOGY IN COMMUTATIVE HOPF ALGEBRAS

2001 ◽  
Vol 44 (1) ◽  
pp. 19-26 ◽  
Author(s):  
M. D. Crossley ◽  
Sarah Whitehouse
Keyword(s):  
Hopf Algebra ◽  
Tensor Product ◽  
Symmetric Group ◽  
Group Action ◽  
Hopf Algebras ◽  
Homotopy Theory ◽  
Cohomology Ring ◽  
Stable Homotopy ◽  
Stable Homotopy Theory ◽  
Primary 16W30

AbstractLet $A$ be a graded, commutative Hopf algebra. We study an action of the symmetric group $\sSi_n$ on the tensor product of $n-1$ copies of $A$; this action was introduced by the second author in 1 and is relevant to the study of commutativity conditions on ring spectra in stable homotopy theory 2.We show that for a certain class of Hopf algebras the cohomology ring $H^*(\sSi_n;A^{\otimes n-1})$ is independent of the coproduct provided $n$ and $(n-2)!$ are invertible in the ground ring. With the simplest coproduct structure, the group action becomes particularly tractable and we discuss the implications this has for computations.AMS 2000 Mathematics subject classification: Primary 16W30; 57T05; 20C30; 20J06; 55S25

scholarly journals The Milnor-Moore theorem for $$L_\infty $$ algebras in rational homotopy theory

2021 ◽  
Author(s):  
José Manuel Moreno Fernández
Keyword(s):  
Homotopy Theory ◽  
Homotopy Groups ◽  
Rational Homotopy Theory ◽  
Rational Homotopy ◽  
Massey Products ◽  
Homotopy Types ◽  
Homology Algebra ◽  
Simply Connected ◽  
Space Homology ◽  
Loop Space Homology

AbstractWe give a construction of the universal enveloping $$A_\infty $$ A ∞ algebra of a given $$L_\infty $$ L ∞ algebra, alternative to the already existing versions. As applications, we derive a higher homotopy algebras version of the classical Milnor-Moore theorem. This proposes a new $$A_\infty $$ A ∞ model for simply connected rational homotopy types, and uncovers a relationship between the higher order rational Whitehead products in homotopy groups and the Pontryagin-Massey products in the rational loop space homology algebra.

scholarly journals Homotopy theory in toric topology

2016 ◽  
Vol 71 (2) ◽  
pp. 185-251 ◽  
Author(s):  
J Grbić ◽  
S Theriault
Keyword(s):  
Homotopy Theory ◽  
Toric Topology

A twisted tale of cochains and connections

2010 ◽  
Vol 17 (1) ◽  
pp. 203-215
Author(s):  
Jim Stasheff
Keyword(s):  
Homotopy Theory ◽  
Graded Algebra ◽  
Massey Products ◽  
Differential Graded Algebra ◽  
Personal Tribute ◽  
History Of ◽  
Homotopy Algebra ◽  
Homotopy Invariants

Abstract Early in the history of higher homotopy algebra [Stasheff, Trans. Am. Math. Soc. 108: 293–312, 1963], it was realized that Massey products are homotopy invariants in a special sense, but it was the work of Tornike Kadeishvili that showed they were but a shadow of an 𝐴∞-structure on the homology of a differential graded algebra. Here we relate his work to that of Victor Gugenheim [J. Pure Appl. Algebra 25: 197–205, 1982] and K. T. (Chester) Chen [Ann. of Math. (2) 97: 217–246, 1973]. This paper is a personal tribute to Tornike and the Georgian school of homotopy theory as well as to Gugenheim and Chen, who unfortunately are not with us to appreciate this convergence.

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