The Mapping Cylinder and Mapping Cone

2020 ◽  
pp. 83-88
Author(s):  
Robert Penner
Keyword(s):  
Mapping Cylinder ◽  
Mapping Cone

Simplicial Crossed Modules and Mapping Cones

2003 ◽  
Vol 10 (4) ◽  
pp. 623-636
Author(s):  
D. Conduché
Keyword(s):  
Homotopy Equivalent ◽  
Simplicial Group ◽  
Mapping Cone ◽  
Crossed Modules ◽  
Special Case

Abstract Given a bisimplicial group 𝐺∗∗ such that 𝑁(𝐺)∗𝑞 = {1} for 𝑞 ⩾ 2, a simplicial group is obtained whose Moore complex is a mapping cone of the chain morphism 𝑁(𝐺)∗1 → 𝑁(𝐺)∗0. This simplicial group is homotopy equivalent to the diagonal of 𝐺∗∗. In the last section a special case is considered.

THE CONE LENGTH OF A PRODUCT OF CO-H-SPACES AND A PROBLEM OF GANEA

2001 ◽  
Vol 33 (6) ◽  
pp. 735-742
Author(s):  
MARTIN ARKOWITZ ◽  
DONALD STANLEY
Keyword(s):  
Positive Solution ◽  
Mapping Cone

It is proved that the cone length or strong category of a product of two co-H-spaces is less than or equal to two. This yields the following positive solution to a problem of Ganea. Let α ∈ π2p(S3) be an element of order p, p a prime [ges ] 3, and let X(p) = S3∪αe2p+1. Then X(p) × X(p) is the mapping cone of some map φ : Y → Z where Z is a suspension.

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 From grids to pseudo-grids of lines: Resolution and seminormality

2020 ◽  
pp. 2150213
Author(s):  
Francesca Cioffi ◽  
Margherita Guida ◽  
Luciana Ramella
Keyword(s):  
Analogous Result ◽  
Betti Numbers ◽  
Free Resolution ◽  
Infinite Field ◽  
Minimal Free Resolution ◽  
Mapping Cone ◽  
Configurations Of Lines

Over an infinite field [Formula: see text], we investigate the minimal free resolution of some configurations of lines. We explicitly describe the minimal free resolution of complete grids of lines and obtain an analogous result about the so-called complete pseudo-grids. Moreover, we characterize the total Betti numbers of configurations that are obtained posing a multiplicity condition on the lines of either a complete grid or a complete pseudo-grid. Finally, we analyze when a complete pseudo-grid is seminormal, differently from a complete grid. The main tools that have been involved in our study are the mapping cone procedure and properties of liftings, of pseudo-liftings and of weighted ideals. Although complete grids and pseudo-grids are hypersurface configurations and many results about such type of configurations have already been stated in literature, we give new contributions, in particular about the maps of the resolution.

scholarly journals Complex of Characteristic Zero in the Skew-Shape (8, 6, 3) / (u,1) where u = 1 and 2

2020 ◽  
pp. 86-91
Author(s):  
Shaymaa N. Abd-Alridah ◽  
Haytham R. Hassan
Keyword(s):  
Characteristic Zero ◽  
Mapping Cone

In this work, we find the terms of the complex of characteristic zero in the case of the skew-shape (8,6, 3)/(u,1), where u = 1 and 2. We also study this complex as a diagram by using the mapping Cone and other concepts.

scholarly journals Mapping cone specific activity in primary visual cortex

2005 ◽  
Vol 5 (8) ◽  
pp. 1018-1018
Author(s):  
E. N. Johnson ◽  
T. R. Tucker ◽  
D. Fitzpatrick
Keyword(s):  
Visual Cortex ◽  
Primary Visual Cortex ◽  
Specific Activity ◽  
Mapping Cone

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.

Sign in / Sign up

Export Citation Format

Share Document