scholarly journals Homotopy equivalence in unbounded KK-theory

2020 ◽  
Vol 5 (3) ◽  
pp. 501-537
Author(s):  
Koen van den Dungen ◽  
Bram Mesland
Keyword(s):  
Homotopy Equivalence

Spaces of PL Manifolds and Categories of Simple Maps (AM-186)

2013 ◽  
Author(s):  
Friedhelm Waldhausen ◽  
Bjørn Jahren ◽  
John Rognes
Keyword(s):  
Main Tool ◽  
Homotopy Equivalence ◽  
Classifying Space ◽  
Deformation Retract ◽  
Simplicial Sets ◽  
Full Proof

Since its introduction by the author in the 1970s, the algebraic K-theory of spaces has been recognized as the main tool for studying parametrized phenomena in the theory of manifolds. However, a full proof of the equivalence relating the two areas has not appeared until now. This book presents such a proof, essentially completing the author's program from more than thirty years ago. The main result is a stable parametrized h-cobordism theorem, derived from a homotopy equivalence between a space of PL h-cobordisms on a space X and the classifying space of a category of simple maps of spaces having X as deformation retract. The smooth and topological results then follow by smoothing and triangulation theory. The proof has two main parts. The essence of the first part is a “desingularization,” improving arbitrary finite simplicial sets to polyhedra. The second part compares polyhedra with PL manifolds by a thickening procedure. Many of the techniques and results developed should be useful in other connections.

Constructing symplectic manifolds

2017 ◽  
Author(s):  
Dusa McDuff ◽  
Dietmar Salamon
Keyword(s):  
Final Section ◽  
Symplectic Manifolds ◽  
Homotopy Equivalence ◽  
Codimension Two ◽  
Open Manifolds ◽  
Connected Sums ◽  
Symplectic Forms ◽  
Ambient Manifold ◽  
Blowing Up ◽  
Four Manifolds

This chapter examines various ways to construct symplectic manifolds and submanifolds. It begins by studying blowing up and down in both the complex and the symplectic contexts. The next section is devoted to a discussion of fibre connected sums and describes Gompf’s construction of symplectic four-manifolds with arbitrary fundamental group. The chapter also contains an exposition of Gromov’s telescope construction, which shows that for open manifolds the h-principle rules and the inclusion of the space of symplectic forms into the space of nondegenerate 2-forms is a homotopy equivalence. The final section outlines Donaldson’s construction of codimension two symplectic submanifolds and explains the associated decompositions of the ambient manifold.

scholarly journals THE SYMMETRIC SIGNATURE OF A WITT SPACE

2013 ◽  
Vol 05 (02) ◽  
pp. 121-159 ◽  
Author(s):  
GREG FRIEDMAN ◽  
JAMES McCLURE
Keyword(s):  
Intersection Homology ◽  
Homotopy Equivalence ◽  
New Construction

Witt spaces are pseudomanifolds for which the middle-perversity intersection homology with rational coefficients is self-dual. We give a new construction of the symmetric signature for Witt spaces which is similar in spirit to the construction given by Miščenko for manifolds. Our construction has all of the expected properties, including invariance under stratified homotopy equivalence.

scholarly journals Spaces which Invert Weak Homotopy Equivalences

2018 ◽  
Vol 62 (2) ◽  
pp. 553-558
Author(s):  
Jonathan Ariel Barmak
Keyword(s):  
Empty Space ◽  
Weak Equivalence ◽  
Homotopy Equivalence ◽  
Homotopy Classes ◽  
Cw Complex ◽  
Cw Complexes ◽  
Weak Homotopy ◽  
Weak Homotopy Equivalence

AbstractIt is well known that if X is a CW-complex, then for every weak homotopy equivalence f : A → B, the map f* : [X, A] → [X, B] induced in homotopy classes is a bijection. In fact, up to homotopy equivalence, only CW-complexes have that property. Now, for which spaces X is f* : [B, X] → [A, X] a bijection for every weak equivalence f? This question was considered by J. Strom and T. Goodwillie. In this note we prove that a non-empty space inverts weak equivalences if and only if it is contractible.

Homotopy Equivalence of a Cofibre Fibre Composite

1977 ◽  
Vol 29 (6) ◽  
pp. 1152-1156 ◽  
Author(s):  
Philip R. Heath
Keyword(s):  
Commutative Diagram ◽  
Fibre Composite ◽  
Topological Spaces ◽  
Homotopy Equivalence ◽  
Image Position

Consider the following commutative diagram in Top, the category of topological spacesin which j and j' are cofibrations, p and p' are (Hurewicz) fibrations and ƒ0, ƒi and ƒ2 are homotopy equivalences.

On the Stable Homotopy Type of Thom Complexes

1973 ◽  
Vol 25 (6) ◽  
pp. 1285-1294 ◽  
Author(s):  
R. P. Held ◽  
D. Sjerve
Keyword(s):  
Vector Bundle ◽  
Homotopy Type ◽  
Stable Homotopy ◽  
Homotopy Equivalence ◽  
Real Vector ◽  
Fiber Homotopy

Let α be a real vector bundle over a finite CW complex X and let T(α;X) be its associated Thorn complex. We propose to study the S-type (stable homotopy type) of Thorn complexes in the framework of the Atiyah-Adams J-Theory. Therefore we focus our attention on the group JR(X) which is defined to be the group of orthogonal sphere bundles over X modulo stable fiber homotopy equivalence.

Decomposable Free Loop Spaces

1987 ◽  
Vol 39 (4) ◽  
pp. 938-955 ◽  
Author(s):  
J. Aguadé
Keyword(s):  
Topological Group ◽  
Loop Space ◽  
Base Point ◽  
Point Of View ◽  
Homotopy Equivalence ◽  
Compact Open Topology ◽  
Loop Spaces ◽  
Continuous Maps

In this paper we study the spaces X having the property that the space of free loops on X is equivalent in some sense to the product of X by the space of based loops on X. We denote by ΛX the space of all continuous maps from S1 to X, with the compact-open topology. ΩX denotes, as usual, the loop space of X, i.e., the subspace of ΛX formed by the maps from S1 to X which map 1 to the base point of X.If G is a topological group then every loop on G can be translated to the base point of G and the space of free loops ΛG is homeomorphic to G × ΩG. More generally, any H-space has this property up to homotopy. Our purpose is to study from a homotopy point of view the spaces X for which there is a homotopy equivalence between ΛX and X × ΩX which is compatible with the inclusion ΩX ⊂ ΛX and the evaluation map ΛX → X.

Homotopy Equivalence and Groups of Measure-Preserving Homeomorphisms

2006 ◽  
Vol 49 (3) ◽  
pp. 337-346
Author(s):  
R. Berlanga
Keyword(s):  
Direct Limit ◽  
Homotopy Equivalence ◽  
General Group ◽  
Compactly Supported

AbstractIt is shown that the group of compactly supported, measure-preserving homeomorphisms of a connected, second countable manifold is locally contractible in the direct limit topology. Furthermore, this group is weakly homotopically equivalent to the more general group of compactly supported homeomorphisms.

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