scholarly journals Smooth approximations and their applications to homotopy types

2020 ◽  
Vol 13 (2) ◽  
pp. 68-108
Author(s):  
Олександра Олександрівна Хохлюк ◽  
Sergiy Ivanovych Maksymenko
Keyword(s):  
Compact Manifold ◽  
Homotopy Equivalence ◽  
Smooth Manifolds ◽  
Homotopy Types ◽  
Weak Homotopy ◽  
Smooth Approximations ◽  
Weak Homotopy Equivalence

Let $M, N$ the be smooth manifolds, $\mathcal{C}^{r}(M,N)$ the space of ${C}^{r}$ maps endowed with the corresponding weak Whitney topology, and $\mathcal{B} \subset \mathcal{C}^{r}(M,N)$ an open subset.It is proved that for $0<r<s\leq\infty$ the inclusion $\mathcal{B} \cap \mathcal{C}^{s}(M,N) \subset \mathcal{B}$ is a weak homotopy equivalence.It is also established a parametrized variant of such a result.In particular, it is shown that for a compact manifold $M$, the inclusion of the space of $\mathcal{C}^{s}$ isotopies $\eta:[0,1]\times M \to M$ fixed near $\{0,1\}\times M$ into the space of loops $\Omega(\mathcal{D}^{r}(M), \mathrm{id}_{M})$ of the group of $\mathcal{C}^{r}$ diffeomorphisms of $M$ at $\mathrm{id}_{M}$ is a weak homotopy equivalence.

The stable parametrized h-cobordism theorem

2013 ◽  
Author(s):  
Friedhelm Waldhausen ◽  
Bjørn Jahren ◽  
John Rognes
Keyword(s):  
Piecewise Linear ◽  
Homotopy Equivalence ◽  
Smooth Manifolds ◽  
Weak Homotopy ◽  
Topological Manifolds ◽  
Weak Homotopy Equivalence

This chapter deals with the stable parametrized h-cobordism theorem. It begins with a discussion of the manifold part; here DIFF is written for the category of Csuperscript infinity smooth manifolds, PL for the category of piecewise-linear manifolds, and TOP for the category of topological manifolds. CAT is generically written for any one of these geometric categories. Relevant terms such as stabilization map, simple map, pullback map, PL Serre fibrations, weak homotopy equivalence, PL Whitehead space, and cofibration are also defined. The chapter proceeds by describing the non-manifold part, the algebraic K-theory of spaces, and the relevance of simple maps to the study of PL homeomorphisms of manifolds.

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.

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 Orbit spaces of gradient vector fields

2012 ◽  
Vol 33 (6) ◽  
pp. 1732-1747 ◽  
Author(s):  
JACK S. CALCUT ◽  
ROBERT E. GOMPF
Keyword(s):  
Vector Fields ◽  
Orbit Space ◽  
Homotopy Equivalence ◽  
Gradient Vector ◽  
Generalized Gradient ◽  
Orbit Spaces ◽  
Morse Functions ◽  
Weak Homotopy ◽  
Weak Homotopy Equivalence

AbstractWe study orbit spaces of generalized gradient vector fields for Morse functions. Typically, these orbit spaces are non-Hausdorff. Nevertheless, they are quite structured topologically and are amenable to study. We show that these orbit spaces are locally contractible. We also show that the quotient map associated to each such orbit space is a weak homotopy equivalence and has the path lifting property.

scholarly journals ON A NOTION OF MAPS BETWEEN ORBIFOLDS II: HOMOTOPY AND CW-COMPLEX

2006 ◽  
Vol 08 (06) ◽  
pp. 763-821 ◽  
Author(s):  
WEIMIN CHEN
Keyword(s):  
Homotopy Groups ◽  
Homotopy Equivalence ◽  
Homotopy Classes ◽  
Complex Theory ◽  
Comprehensive Theory ◽  
Cw Complex ◽  
Algebraic Invariants ◽  
Weak Homotopy ◽  
Weak Homotopy Equivalence

This is the second of a series of papers which is devoted to a comprehensive theory of maps between orbifolds. In this paper, we develop a basic machinery for studying homotopy classes of such maps. It contains two parts: (1) the construction of a set of algebraic invariants — the homotopy groups, and (2) an analog of CW-complex theory. As a corollary of this machinery, the classical Whitehead theorem (which asserts that a weak homotopy equivalence is a homotopy equivalence) is extended to the orbifold category.

A Counterexample to a Conjecture of D. B. Fuks

1970 ◽  
Vol 13 (2) ◽  
pp. 261-266
Author(s):  
Luc Demers
Keyword(s):  
Homotopy Types ◽  
Weak Homotopy

In [3] D. B. Fuks defined a duality of functors in the category of weak homotopy types. In general this duality is more difficult to work with than the duality of functors of the category of pointed Kelley spaces [2]. It happens however that all so-called strong functors [2] F of induce functors of , and if we denote the duality operators of and by and D respectively, then there are many cases where .

Rational Homotopy Types with the Rational Cohomology Algebra of Stunted Complex Projective Space

1992 ◽  
Vol 44 (6) ◽  
pp. 1241-1261 ◽  
Author(s):  
Gregory Lupton ◽  
Ronald Umble
Keyword(s):  
Projective Space ◽  
Spectral Sequence ◽  
Complex Projective Space ◽  
Classification Theory ◽  
Cohomology Algebra ◽  
Homotopy Equivalence ◽  
Rational Homotopy ◽  
Homotopy Types ◽  
Rational Cohomology ◽  
Complex Projective

AbstractWe consider the number of spaces, up to rational homotopy equivalence, which have rational cohomology algebra isomorphic to that of stunted complex projective space . Using a classification theory due to Schlessinger and Stasheff, we determine the number of rational homotopy types with rational comology algebra isomorphic to , for any given n and k. The necessary computations make use of a spectral sequence introduced by the second named author.

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