Embedding Certain Complexes up to Homotopy Type in Euclidean Space

1969 ◽  
Vol 90 (1) ◽  
pp. 144 ◽  
Author(s):  
George Cooke
Keyword(s):  
Euclidean Space ◽  
Homotopy Type

An n-dimensional Klein bottle

2019 ◽  
Vol 149 (5) ◽  
pp. 1207-1221
Author(s):  
Donald M. Davis
Keyword(s):  
Euclidean Space ◽  
Homotopy Type ◽  
Klein Bottle ◽  
Stable Homotopy ◽  
Cohomology Algebra ◽  
Topological Complexity ◽  
Integral Cohomology ◽  
Planar Polygon

AbstractAn n-dimensional analogue of the Klein bottle arose in our study of topological complexity of planar polygon spaces. We determine its integral cohomology algebra and stable homotopy type, and give an explicit immersion and embedding in Euclidean space.

scholarly journals Embedding up to homotopy type in Euclidean space

1993 ◽  
Vol 47 (1) ◽  
pp. 145-148 ◽  
Author(s):  
A.N. Dranišnikov ◽  
D. Repovš
Keyword(s):  
Euclidean Space ◽  
Homotopy Type ◽  
Short Proof ◽  
Dimensional Euclidean Space ◽  
Cellular Complex

We give 8 short proof of the classical Stallings theorem that every finite n-dimensional cellular complex embeds up to homotopy in the 2n-dimensional Euclidean space. As an application we solve a problem of M. Kreck.

scholarly journals Some small aspherical spaces

1973 ◽  
Vol 16 (3) ◽  
pp. 332-352 ◽  
Author(s):  
Eldon Dyer ◽  
A. T. Vasquez
Keyword(s):  
Euclidean Space ◽  
Fundamental Group ◽  
Homotopy Type ◽  
Homotopy Group ◽  
Continuous Mapping ◽  
Cell Complex ◽  
A Cell

Let Sn denote the sphere of all points in Euclidean space Rn + 1 at a distance of 1 from the origin and Dn + 1 the ball of all points in Rn + 1 at a distance not exceeding 1 from the origin The space X is said to be aspherical if for every n ≧ 2 and every continuous mapping: f: Sn → X, there exists a continuous mapping g: Dn + 1 → X with restriction to the subspace Sn equal to f. Thus, the only homotopy group of X which might be non-zero is the fundamental group τ1(X, *) ≅ G. If X is also a cell-complex, it is called a K(G, 1). If X and Y are K(G, l)'s, then they have the same homotopy type, and consequently

The smooth case

2017 ◽  
Author(s):  
Ehud Hrushovski ◽  
François Loeser
Keyword(s):  
Homotopy Type ◽  
Unit Vector ◽  
Smooth Case ◽  
Open Set ◽  
Birational Invariant

This chapter examines the simplifications occurring in the proof of the main theorem in the smooth case. It begins by stating the theorem about the existence of an F-definable homotopy h : I × unit vector X → unit vector X and the properties for h. It then presents the proof, which depends on two lemmas. The first recaps the proof of Theorem 11.1.1, but on a Zariski dense open set V₀ only. The second uses smoothness to enable a stronger form of inflation, serving to move into V₀. The chapter also considers the birational character of the definable homotopy type in Remark 12.2.4 concerning a birational invariant.

Symplectic capacities

2017 ◽  
Author(s):  
Dusa McDuff ◽  
Dietmar Salamon
Keyword(s):  
Euclidean Space ◽  
Generating Functions ◽  
Weinstein Conjecture ◽  
Finite Dimensional ◽  
Hofer Metric

This chapter returns to the problems which were formulated in Chapter 1, namely the Weinstein conjecture, the nonsqueezing theorem, and symplectic rigidity. These questions are all related to the existence and properties of symplectic capacities. The chapter begins by discussing some of the consequences which follow from the existence of capacities. In particular, it establishes symplectic rigidity and discusses the relation between capacities and the Hofer metric on the group of Hamiltonian symplectomorphisms. The chapter then introduces the Hofer–Zehnder capacity, and shows that its existence gives rise to a proof of the Weinstein conjecture for hypersurfaces of Euclidean space. The last section contains a proof that the Hofer–Zehnder capacity satisfies the required axioms. This proof translates the Hofer–Zehnder variational argument into the setting of (finite-dimensional) generating functions.

ANTI-SELF-DUAL SOLUTIONS OF THE YANG-MILLS EQUATIONS IN 4n DIMENSIONS

1992 ◽  
Vol 07 (23) ◽  
pp. 2077-2085 ◽  
Author(s):  
A. D. POPOV
Keyword(s):  
Scalar Field ◽  
Euclidean Space ◽  
Lie Algebra ◽  
Linear Equations ◽  
Dual Solutions ◽  
Gauge Fields ◽  
System Of Equations ◽  
Semisimple Lie Algebra ◽  
Yang Mills ◽  
Self Duality

The anti-self-duality equations for gauge fields in d = 4 and a generalization of these equations to dimension d = 4n are considered. For gauge fields with values in an arbitrary semisimple Lie algebra [Formula: see text] we introduce the ansatz which reduces the anti-self-duality equations in the Euclidean space ℝ4n to a system of equations breaking up into the well known Nahm's equations and some linear equations for scalar field φ.

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