Totalization of simplicial homotopy types

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.

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Strong Homotopy Types, Nerves and Collapses

Discrete & Computational Geometry ◽

10.1007/s00454-011-9357-5 ◽

2011 ◽

Vol 47 (2) ◽

pp. 301-328 ◽

Cited By ~ 36

Author(s):

Jonathan Ariel Barmak ◽

Elias Gabriel Minian

Keyword(s):

Homotopy Types

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COMBINATORIAL AND TOPOLOGICAL PHASE STRUCTURE OF NON-PERTURBATIVE n-DIMENSIONAL QUANTUM GRAVITY

International Journal of Modern Physics B ◽

10.1142/s0217979292001055 ◽

1992 ◽

Vol 06 (11n12) ◽

pp. 2109-2121

Author(s):

M. CARFORA ◽

M. MARTELLINI ◽

A. MARZUOLI

Keyword(s):

Quantum Gravity ◽

Phase Structure ◽

Gravity Models ◽

Topological Phase ◽

Geometrical Characterization ◽

Homotopy Types ◽

Geometrical Constraints ◽

Riemannian Geometries

We provide a non-perturbative geometrical characterization of the partition function of ndimensional quantum gravity based on a rough classification of Riemannian geometries. We show that, under natural geometrical constraints, the theory admits a continuum limit with a non-trivial phase structure parametrized by the homotopy types of the class of manifolds considered. The results obtained qualitatively coincide, when specialized to dimension two, with those of two-dimensional quantum gravity models based on random triangulations of surfaces.

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