scholarly journals The homotopy classification of four-dimensional toric orbifolds

2021 ◽  
pp. 1-23
Author(s):  
Xin Fu ◽  
Tseleung So ◽  
Jongbaek Song
Keyword(s):  
Homotopy Type ◽  
Homotopy Equivalent ◽  
Homotopy Classification ◽  
Cohomology Rings ◽  
Moore Space ◽  
Integral Cohomology ◽  
Cw Complex ◽  
Primary Torsion

Let X be a 4-dimensional toric orbifold. If $H^{3}(X)$ has a non-trivial odd primary torsion, then we show that X is homotopy equivalent to the wedge of a Moore space and a CW-complex. As a corollary, given two 4-dimensional toric orbifolds having no 2-torsion in the cohomology, we prove that they have the same homotopy type if and only their integral cohomology rings are isomorphic.

Classification of Bott Manifolds up to Dimension 8

2014 ◽  
Vol 58 (3) ◽  
pp. 653-659 ◽  
Author(s):  
Suyoung Choi
Keyword(s):  
Cohomology Ring ◽  
Cohomology Rings ◽  
Integral Cohomology ◽  
Ring Isomorphism

AbstractWe show that three- and four-stage Bott manifolds are classified up to diffeomorphism by their integral cohomology rings. In addition, any cohomology ring isomorphism between two three-stage Bott manifolds can be realized by a diffeomorphism between the Bott manifolds.

scholarly journals Spherical posets from commuting elements

2018 ◽  
Vol 21 (4) ◽  
pp. 593-628 ◽  
Author(s):  
Cihan Okay
Keyword(s):  
Homotopy Type ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Universal Cover ◽  
Homotopy Equivalent ◽  
Classifying Space ◽  
Partially Ordered ◽  
Abelian Subgroups

AbstractIn this paper, we study the homotopy type of the partially ordered set of left cosets of abelian subgroups in an extraspecial p-group. We prove that the universal cover of its nerve is homotopy equivalent to a wedge of r-spheres where {2r\geq 4} is the rank of its Frattini quotient. This determines the homotopy type of the universal cover of the classifying space of transitionally commutative bundles as introduced in [2].

scholarly journals Criteria for components of a function space to be homotopy equivalent

2008 ◽  
Vol 145 (1) ◽  
pp. 95-106 ◽  
Author(s):  
GREGORY LUPTON ◽  
SAMUEL BRUCE SMITH
Keyword(s):  
Function Space ◽  
Homotopy Type ◽  
Homotopy Equivalent ◽  
General Method

AbstractWe give a general method that may be effectively applied to the question of whether two components of a function space map(X, Y) have the same homotopy type. We describe certain group-like actions on map(X, Y). Our basic results assert that if maps f, g: X → Y are in the same orbit under such an action, then the components of map(X, Y) that contain f and g have the same homotopy type.

Homotopy invariants and continuous mappings

1950 ◽  
Vol 202 (1069) ◽  
pp. 253-263 ◽  
Keyword(s):  
Homotopy Type ◽  
Betti Numbers ◽  
Continuous Mappings ◽  
Numerical Invariants ◽  
Homotopy Invariants

This paper contributes new numerical invariants to the topology of a certain class of polyhedra. These invariants, together with the Betti numbers and coefficients of torsion, characterize the homotopy type of one of these polyhedra. They are also applied to the classification of continuous mappings of an ( n + 2)-dimensional polyhedron into an ( n + 1)-sphere ( n > 2).

scholarly journals Homotopy classification of elliptic operators on stratified manifolds

2006 ◽  
Vol 73 (3) ◽  
pp. 407-411 ◽  
Author(s):  
V. E. Nazaikinskii ◽  
A. Yu. Savin ◽  
B. Yu. Sternin
Keyword(s):  
Elliptic Operators ◽  
Homotopy Classification

Cohomology of Homotopy 2-types

2019 ◽  
pp. 379-422
Author(s):  
Graham Ellis
Keyword(s):  
Perturbation Theory ◽  
Projective Plane ◽  
Integral Homology ◽  
Homotopy Classes ◽  
Cw Complex ◽  
Crossed Modules ◽  
Homological Perturbation Theory ◽  
Homological Perturbation ◽  
Manual Classification

This chapter introduces some of the basic ingredients in the classification of homotopy 2-types and describes datatypes and algorithms for implementing them on a computer. These are illustrated using computer examples involving: the fundamental crossed modules of a CW-complex, cat-1-groups, simplicial groups, Moore complexes, the Dold-Kan correspondence, integral homology of simplicial groups, homological perturbation theory. A manual classification of homotopy classes of maps from a surface to the projective plane is also included.

On Graded Categorical Groups and Equivariant Group Extensions

2002 ◽  
Vol 54 (5) ◽  
pp. 970-997 ◽  
Author(s):  
A. M. Cegarra ◽  
J. M. Garćia-Calcines ◽  
J. A. Ortega
Keyword(s):  
Group Cohomology ◽  
Group Extension ◽  
Extension Problem ◽  
Homotopy Classification ◽  
Group Extensions ◽  
Categorical Groups

AbstractIn this article we state and prove precise theorems on the homotopy classification of graded categorical groups and their homomorphisms. The results use equivariant group cohomology, and they are applied to show a treatment of the general equivariant group extension problem.

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