scholarly journals On the failure of the first Čech homotopy group to register geometrically relevant fundamental group elements

2020 ◽  
Vol 52 (6) ◽  
pp. 1072-1092
Author(s):  
Jeremy Brazas ◽  
Hanspeter Fischer
Keyword(s):  
Fundamental Group ◽  
Homotopy Group

On the Asphericity of Knot Complements

1993 ◽  
Vol 45 (2) ◽  
pp. 340-356
Author(s):  
Vo Thanh Liem ◽  
Gerard A. Venema
Keyword(s):  
Fundamental Group ◽  
Homotopy Group ◽  
Homotopy Groups ◽  
Unusual Property ◽  
Knot Complements ◽  
Flat Embedding

AbstractTwo examples of topological embeddings of S2 in S4 are constructed. The first has the unusual property that the fundamental group of the complement is isomorphic to the integers while the second homotopy group of the complement is nontrivial. The second example is a non-locally flat embedding whose complement exhibits this property locally.Two theorems are proved. The first answers the question of just when good π1 implies the vanishing of the higher homotopy groups for knot complements in S4. The second theorem characterizes local flatness for 2-spheres in S4 in terms of a local π1 condition.

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

scholarly journals A Group of Automorphisms of the Homotopy Groups

1951 ◽  
Vol 2 ◽  
pp. 73-82
Author(s):  
Hiroshi Uehara
Keyword(s):  
Normal Subgroup ◽  
Topological Space ◽  
Fundamental Group ◽  
Homotopy Group ◽  
Factor Group ◽  
Complete Analysis ◽  
Homotopy Groups ◽  
Group Of Automorphisms ◽  
Arcwise Connected

It is well known that the fundamental group π1(X) of an arcwise connected topological space X operates on the n-th homotopy group πn(X) of X as a group of automorphisms. In this paper I intend to construct geometrically a group 𝒰(X) of automorphisms of πn(X), for every integer n ≥ 1, which includes a normal subgroup isomorphic to π1(X) so that the factor group of 𝒰(X) by π1(X) is completely determined by some invariant Σ(X) of the space X. The complete analysis of the operation of the group on πn(X) is given in §3, §4, and §5,

scholarly journals Embedding spheres in knot traces

2021 ◽  
Vol 157 (10) ◽  
pp. 2242-2279
Author(s):  
Peter Feller ◽  
Allison N. Miller ◽  
Matthias Nagel ◽  
Patrick Orson ◽  
Mark Powell ◽  
...  
Keyword(s):  
Fundamental Group ◽  
Homotopy Group ◽  
Alexander Polynomial ◽  
Homotopy Equivalent ◽  
Knot Invariants

Abstract The trace of the $n$ -framed surgery on a knot in $S^{3}$ is a 4-manifold homotopy equivalent to the 2-sphere. We characterise when a generator of the second homotopy group of such a manifold can be realised by a locally flat embedded $2$ -sphere whose complement has abelian fundamental group. Our characterisation is in terms of classical and computable $3$ -dimensional knot invariants. For each $n$ , this provides conditions that imply a knot is topologically $n$ -shake slice, directly analogous to the result of Freedman and Quinn that a knot with trivial Alexander polynomial is topologically slice.

Filling words in fundamental groups of Riemann surfaces

2013 ◽  
Vol 50 (1) ◽  
pp. 31-50
Author(s):  
C. Zhang
Keyword(s):  
Riemann Surface ◽  
Fundamental Group ◽  
Riemann Surfaces ◽  
Fundamental Groups ◽  
Closed Geodesics ◽  
New Words

The purpose of this article is to utilize some exiting words in the fundamental group of a Riemann surface to acquire new words that are represented by filling closed geodesics.

Representations of the fundamental group and the torsor of deformations. Global aspects

2017 ◽  
Author(s):  
Ahmed Abbes ◽  
Michel Gros
Keyword(s):  
Fundamental Group ◽  
Koszul Complex ◽  
Finiteness Conditions ◽  
Inverse Systems ◽  
Logarithmic Geometry

This chapter continues the construction and study of the p-adic Simpson correspondence and presents the global aspects of the theory of representations of the fundamental group and the torsor of deformations. After fixing the notation and general conventions, the chapter develops preliminaries and then introduces the results and complements on the notion of locally irreducible schemes. It also fixes the logarithmic geometry setting of the constructions and considers a number of results on the Koszul complex. Finally, it develops the formalism of additive categories up to isogeny and describes the inverse systems of a Faltings ringed topos, with a particular focus on the notion of adic modules and the finiteness conditions adapted to this setting. The chapter rounds up the discussion with sections on Higgs–Tate algebras and Dolbeault modules.

scholarly journals TOPOLOGICAL 4-MANIFOLDS WITH 4-DIMENSIONAL FUNDAMENTAL GROUP

2021 ◽  
pp. 1-8
Author(s):  
DANIEL KASPROWSKI ◽  
MARKUS LAND
Keyword(s):  
Fundamental Group ◽  
Homotopy Equivalent ◽  
Poincaré Duality ◽  
Poincare Duality ◽  
Canonical Map ◽  
Duality Space ◽  
Simply Connected Manifolds ◽  
Simply Connected ◽  
Poincaré Duality Space

Abstract Let $\pi$ be a group satisfying the Farrell–Jones conjecture and assume that $B\pi$ is a 4-dimensional Poincaré duality space. We consider topological, closed, connected manifolds with fundamental group $\pi$ whose canonical map to $B\pi$ has degree 1, and show that two such manifolds are s-cobordant if and only if their equivariant intersection forms are isometric and they have the same Kirby–Siebenmann invariant. If $\pi$ is good in the sense of Freedman, it follows that two such manifolds are homeomorphic if and only if they are homotopy equivalent and have the same Kirby–Siebenmann invariant. This shows rigidity in many cases that lie between aspherical 4-manifolds, where rigidity is expected by Borel’s conjecture, and simply connected manifolds where rigidity is a consequence of Freedman’s classification results.

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