The mapping cylinder lemma

1963 ◽  
Vol 15 (1) ◽  
pp. 90-93
Keyword(s):  
Mapping Cylinder

A proof of the comparison theorem for spectral sequences

1957 ◽  
Vol 53 (1) ◽  
pp. 57-62 ◽  
Author(s):  
E. C. Zeeman
Keyword(s):  
Comparison Theorem ◽  
Analogous Result ◽  
Spectral Sequences ◽  
Mapping Cylinder ◽  
Fibre Space ◽  
The Third

The comparison theorem is an algebraic theorem, which corresponds to the following topological situation. A map between two fibre spaces induces a homomorphism between the two corresponding spectral sequences relating the homologies of the base, fibre and fibre space; if the map induces an isomorphism on any two of these quantities, then it also does on the third. J. C. Moore (l) proved the theorem using a mapping cylinder, which requires that the spectral sequences arise from filtered differential groups. The present proof assumes only the existence of the spectral sequences from the E2 terms onwards. Moreover, we generalize the theorem, assuming that the homomorphisms on the two given quantities are isomorphic only up to given dimensions, and deducing the same for the third quantity. For algebraic translucence we use a simpler hypothesis (property (iii)) than Moore's concerning the E2 terms. However, if his hypothesis is assumed, together with the existence of E1 terms, then his theorem can be deduced as a corollary. The analogous result for cohomology is similar, and is stated at the end.

scholarly journals Dupin hypersurfaces, group actions and the double mapping cylinder

1987 ◽  
Vol 26 (3) ◽  
pp. 429-459 ◽  
Author(s):  
Karsten Grove ◽  
Stephen Halperin
Keyword(s):  
Group Actions ◽  
Mapping Cylinder ◽  
Dupin Hypersurfaces

On simple maps

2013 ◽  
Author(s):  
Friedhelm Waldhausen ◽  
Bjørn Jahren ◽  
John Rognes
Keyword(s):  
Homotopy Type ◽  
Geometric Realization ◽  
Homotopy Equivalence ◽  
Mapping Cylinder ◽  
Simplicial Sets ◽  
Weak Homotopy ◽  
Formal Properties ◽  
Weak Homotopy Equivalence

This chapter deals with simple maps of finite simplicial sets, along with some of their formal properties. It begins with a discussion of simple maps of simplicial sets, presenting a proposition for the conditions that qualify a map of finite simplicial sets as a simple map. In particular, it considers a simple map as a weak homotopy equivalence. Weak homotopy equivalences have the 2-out-of-3 property, which combines the composition, right cancellation and left cancellation properties. The chapter proceeds by defining some relevant terms, such as Euclidean neighborhood retract, absolute neighborhood retract, Čech homotopy type, and degeneracy operator. It also describes normal subdivision of simplicial sets, geometric realization and subdivision, the reduced mapping cylinder, how to make simplicial sets non-singular, and the approximate lifting property.

scholarly journals Mapping Cylinder Neighborhoods of Some ANR's

1976 ◽  
Vol 103 (3) ◽  
pp. 417 ◽  
Author(s):  
R. T. Miller
Keyword(s):  
Mapping Cylinder

Linear configurations of complete graphs K4 and K5 in ℝ3, and higher dimensional analogs

2017 ◽  
Vol 26 (05) ◽  
pp. 1750028
Author(s):  
Andrew Marshall
Keyword(s):  
Fundamental Group ◽  
Symmetric Group ◽  
Configuration Space ◽  
Braid Group ◽  
Simplicial Complexes ◽  
Alternating Group ◽  
Complete Graphs ◽  
Mapping Cylinder ◽  
Higher Dimensional ◽  
Special Cases

We investigate the space [Formula: see text] of images of linearly embedded finite simplicial complexes in [Formula: see text] isomorphic to a given complex [Formula: see text], focusing on two special cases: [Formula: see text] is the [Formula: see text]-skeleton [Formula: see text] of an [Formula: see text]-simplex, and [Formula: see text] is the [Formula: see text]-skeleton [Formula: see text] of an [Formula: see text]-simplex, so [Formula: see text] has codimension 2 in [Formula: see text], in both cases. The main result is that for [Formula: see text], [Formula: see text] (for either [Formula: see text]) deformation retracts to a subspace homeomorphic to the double mapping cylinder [Formula: see text] where [Formula: see text] is the alternating group and [Formula: see text] the symmetric group. The resulting fundamental group provides an example of a generalization of the braid group, which is the fundamental group of the configuration space of points in the plane.

Regular neighbourhoods and mapping cylinders

1971 ◽  
Vol 70 (1) ◽  
pp. 15-18
Author(s):  
C. Kearton
Keyword(s):  
Piecewise Linear ◽  
Linear Manifold ◽  
Mapping Cylinder ◽  
Simple Characterization ◽  
Regular Neighbourhood ◽  
Simplicial Map ◽  
Combinatorial Manifold ◽  
Piecewise Linear Manifold

It is well known that a regular neighbourhood of a polyhedron in a piecewise linear manifold may be regarded as a simplicial mapping cylinder. The aim of this paper is to show that if the polyhedron is a locally unknotted submanifold of the interior then the class of maps giving rise to such regular neighbourhoods has a simple characterization. At the same time, it is possible to answer the question: Given a simplicial map f defined on a combinatorial manifold, when is the image of f also a combinatorial manifold? Marshall Cohen has answered this question when the image is required to be isomorphic to the domain; the methods used here are those developed in (1), to which the reader is referred for definitions and notation.

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