Quasi-isometries and proper homotopy: The quasi-isometry invariance of proper $3$-realizability of groups

2018 ◽  
Vol 147 (4) ◽  
pp. 1797-1804 ◽  
Author(s):  
M. Cárdenas ◽  
F. F. Lasheras ◽  
A. Quintero ◽  
R. Roy
Keyword(s):  
Proper Homotopy

scholarly journals Extreme nonuniqueness of end-sum

2020 ◽  
pp. 1-43
Author(s):  
Jack S. Calcut ◽  
Craig R. Guilbault ◽  
Patrick V. Haggerty
Keyword(s):  
Abelian Groups ◽  
Cohomology Algebra ◽  
Homotopy Types ◽  
Proper Homotopy ◽  
Open Formula

We give explicit examples of pairs of one-ended, open [Formula: see text]-manifolds whose end-sums yield uncountably many manifolds with distinct proper homotopy types. This answers strongly in the affirmative a conjecture of Siebenmann regarding nonuniqueness of end-sums. In addition to the construction of these examples, we provide a detailed discussion of the tools used to distinguish them; most importantly, the end-cohomology algebra. Key to our Main Theorem is an understanding of this algebra for an end-sum in terms of the algebras of summands together with ray-fundamental classes determined by the rays used to perform the end-sum. Differing ray-fundamental classes allow us to distinguish the various examples, but only through the subtle theory of infinitely generated abelian groups. An appendix is included which contains the necessary background from that area.

Detecting cohomology classes for the proper LS category. The case of semistable 3-manifolds

2011 ◽  
Vol 152 (2) ◽  
pp. 223-249 ◽  
Author(s):  
MANUEL CÁRDENAS ◽  
FRANCISCO F. LASHERAS ◽  
ANTONIO QUINTERO
Keyword(s):  
Sufficient Conditions ◽  
Proper Homotopy ◽  
Lusternik Schnirelmann

AbstractWe give sufficient conditions for the existence of detecting elements for the Lusternik–Schnirelmann category in proper homotopy. As an application we determine the proper LS category of some semistable one-ended open 3-manifolds.

scholarly journals Calculations of cylindrical p-homotopy groups

1989 ◽  
Vol 32 (3) ◽  
pp. 401-413
Author(s):  
R. Ayala ◽  
E. Domínguez ◽  
A. Quintero
Keyword(s):  
Complex Structure ◽  
Homotopy Groups ◽  
Proper Homotopy

The definitions of the various proper homotopy groups correspond to three main geometrical ideas: sequences of spheres converging to a Freudenthal end (Brown groups); infinite cylinders giving the mobility of spheres towards a proper end (Čerin-Steenrod groups); sequences of spheres, each one movable to the next one following a proper end (Čech groups). The Brown and Čech groups have a rather complex structure and the calculations of these groups are very difficult (see [4]). The Čerin-Steenrod groups have a much simpler structure and this fact eases the computations.

scholarly journals On various relative proper homotopy groups

1980 ◽  
Vol 4 (2) ◽  
pp. 177-202 ◽  
Author(s):  
Zvonko \v{C}erin
Keyword(s):  
Homotopy Groups ◽  
Proper Homotopy

A theoretical framework for proper homotopy theory

1990 ◽  
Vol 107 (3) ◽  
pp. 475-482 ◽  
Author(s):  
R. Ayala ◽  
A. Quintero ◽  
E. Dominguez
Keyword(s):  
Homotopy Theory ◽  
Theoretical Framework ◽  
Homotopy Groups ◽  
Theoretical Treatment ◽  
Proper Homotopy

AbstractFollowing the techniques of ordinary homotopy theory, a theoretical treatment of proper homotopy theory, including the known proper homotopy groups, is provided within Baues's theory of cofibration categories.

scholarly journals Direct products and properly 3-realisable groups

2004 ◽  
Vol 70 (2) ◽  
pp. 199-205 ◽  
Author(s):  
Manuel Cárdenas ◽  
Francisco F. Lasheras ◽  
Ranja Roy
Keyword(s):  
Homotopy Type ◽  
Universal Cover ◽  
Hyperbolic Groups ◽  
Direct Products ◽  
Finitely Presented Groups ◽  
Finitely Presented Group ◽  
Manifold With Boundary ◽  
Proper Homotopy ◽  
Finitely Presented

In this paper, we show that the direct of infinite finitely presented groups is always properly 3-realisable. We also show that classical hyperbolic groups are properly 3-realisable. We recall that a finitely presented group G is said to be properly 3-realisable if there exists a compact 2-polyhedron K with π1 (K) ≅ G and whose universal cover K̃ has the proper homotopy type of a (p.1.) 3-manifold with boundary. The question whether or not every finitely presented is properly 3-realisable remains open.

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