On Normal Stratified Pseudomanifolds
... For a detailed treatment of the results contained in this section, see [8]. Manifolds considered in this paper will always be topological manifolds. A topological space is stratified if it can be written as a disjoint union of manifolds which are related by an incidence condition. Definition 1.1. Le ...
... For a detailed treatment of the results contained in this section, see [8]. Manifolds considered in this paper will always be topological manifolds. A topological space is stratified if it can be written as a disjoint union of manifolds which are related by an incidence condition. Definition 1.1. Le ...
PDF version - University of Warwick
... n–complex K and permutation π ∈ Σn+1 . Permutation homology is a convenient device (implicit in Goresky and MacPherson [2]) for studying intersection homology. We prove that, for a PL manifold, all permutation homology groups are the same as ordinary homology groups. In section 3 we prove that the g ...
... n–complex K and permutation π ∈ Σn+1 . Permutation homology is a convenient device (implicit in Goresky and MacPherson [2]) for studying intersection homology. We prove that, for a PL manifold, all permutation homology groups are the same as ordinary homology groups. In section 3 we prove that the g ...
Surveys on Surgery Theory : Volume 1 Papers dedicated to C. T. C.
... The investigation of simply connected 4-dimensional manifolds suggested the more general problem of classifying (n − 1)-connected 2n-dimensional manifolds, for all n. The intersection form on middle homology again appears as a fundamental algebraic invariant of oriented homotopy type. In fact this i ...
... The investigation of simply connected 4-dimensional manifolds suggested the more general problem of classifying (n − 1)-connected 2n-dimensional manifolds, for all n. The intersection form on middle homology again appears as a fundamental algebraic invariant of oriented homotopy type. In fact this i ...
first four chapters - Jesse Johnson`s Website
... of another simplex. For shorthand, we will write P (σ) for the power set of σ, i.e. the set of all subsets of σ. By the second condition on a simplicial complex, if σ ∈ S then P (σ) ⊂ S. Therefore, we can write S as the union of the power sets of its maximal simplices. 2. Example. Figure 2 shows the ...
... of another simplex. For shorthand, we will write P (σ) for the power set of σ, i.e. the set of all subsets of σ. By the second condition on a simplicial complex, if σ ∈ S then P (σ) ⊂ S. Therefore, we can write S as the union of the power sets of its maximal simplices. 2. Example. Figure 2 shows the ...
Repovš D.: Topology and Chaos
... fundamental group of the knot complement, and the Alexander polynomial. Actually, there are two trefoil knots, called the right and left-handed trefoils, which are mirror images of each other. These are not equivalent to each other. This was shown by Max Dehn (Dehn 1914). ...
... fundamental group of the knot complement, and the Alexander polynomial. Actually, there are two trefoil knots, called the right and left-handed trefoils, which are mirror images of each other. These are not equivalent to each other. This was shown by Max Dehn (Dehn 1914). ...
Chapter 6 Manifolds, Tangent Spaces, Cotangent Spaces, Vector
... shows that if U is any open subset of a C k -manifold, M , then U is also a C k -manifold whose charts are the restrictions of charts on M to U . Example 1. The sphere S n. Using the stereographic projections (from the north pole and the south pole), we can define two charts on S n and show that S n ...
... shows that if U is any open subset of a C k -manifold, M , then U is also a C k -manifold whose charts are the restrictions of charts on M to U . Example 1. The sphere S n. Using the stereographic projections (from the north pole and the south pole), we can define two charts on S n and show that S n ...
topological group
... A topological group is a group G endowed with a topology such that the multiplication and inverse operations of G are continuous. That is, the map G × G → G defined by (x, y) 7→ xy is continuous, where the topology on G×G is the product topology, and the map G → G defined by x 7→ x−1 is also continu ...
... A topological group is a group G endowed with a topology such that the multiplication and inverse operations of G are continuous. That is, the map G × G → G defined by (x, y) 7→ xy is continuous, where the topology on G×G is the product topology, and the map G → G defined by x 7→ x−1 is also continu ...
Differential Algebraic Topology
... (co)homology see [B-R-S]. As indicated above, the key for passing from singular bordism to ordinary homology is to introduce generalized manifolds that are a certain kind of stratified space. These are topological spaces S together with a decomposition of S into manifolds of increasing dimension call ...
... (co)homology see [B-R-S]. As indicated above, the key for passing from singular bordism to ordinary homology is to introduce generalized manifolds that are a certain kind of stratified space. These are topological spaces S together with a decomposition of S into manifolds of increasing dimension call ...
... In contrast, the 2-dimensional sphere has a closed triangulation, and is borderless. Edwin Moise established in [5] that any 3-dimensional manifold has a finite triangulation. Thus, a 3-manifold is borderless iff it is ¾ Connected, and ¾ Its finite triangulation is such that each face of a tetrahedr ...
E∞-Comodules and Topological Manifolds A Dissertation presented
... The goal of this dissertation is to relate the theory of algebraic surgery developed predominantly by Andrew Ranicki, with that of E∞ -structures on chain complexes. Steenrod’s construction of higher chain approximations to the diagonal inclusion has been encoded, by several authors, as a functor fr ...
... The goal of this dissertation is to relate the theory of algebraic surgery developed predominantly by Andrew Ranicki, with that of E∞ -structures on chain complexes. Steenrod’s construction of higher chain approximations to the diagonal inclusion has been encoded, by several authors, as a functor fr ...
ON MINIMAL, STRONGLY PROXIMAL ACTIONS OF LOCALLY
... measures. A standard construction of induction allows to induce linear or affine representations from a closed subgroup (here N0 ) to the larger group H (see Zimmer [11] 4.2 for details). In our case, consider the space L∞ (H/N0 , C(M )∗ ) = L1 (H/N0 , C(M ))∗ with the weak-* topology, and its conve ...
... measures. A standard construction of induction allows to induce linear or affine representations from a closed subgroup (here N0 ) to the larger group H (see Zimmer [11] 4.2 for details). In our case, consider the space L∞ (H/N0 , C(M )∗ ) = L1 (H/N0 , C(M ))∗ with the weak-* topology, and its conve ...
STABLE HOMOTOPY THEORY 1. Spectra and the stable homotopy
... A morphism f : X → Y which admits a homotopy inverse (as above) is called a homotopy equivalence. There is a weaker notion of equivalence, defined using homotopy groups. Definition 1.8. For n a natural number and X a pointed topological space, let πn (X) denote [S n , X]. The set π0 (X) is the set o ...
... A morphism f : X → Y which admits a homotopy inverse (as above) is called a homotopy equivalence. There is a weaker notion of equivalence, defined using homotopy groups. Definition 1.8. For n a natural number and X a pointed topological space, let πn (X) denote [S n , X]. The set π0 (X) is the set o ...
Topology
... so-called “pure mathematics” with remarkable successes and results in this subject. In the course of the twentieth century it has also 1. provided notions and concepts that are of core importance for all of mathematics, such as the notion of “compactness”, 2. contributed a great variety of important ...
... so-called “pure mathematics” with remarkable successes and results in this subject. In the course of the twentieth century it has also 1. provided notions and concepts that are of core importance for all of mathematics, such as the notion of “compactness”, 2. contributed a great variety of important ...
HYPERBOLIZATION OF POLYHEDRA
... The fact that polyhedral homology manifolds which are not PL manifolds have something to do with exotic universal covers was first recognized in [11], through the use of reflection groups. In the recent Ph.D. thesis of G. Moussong [24], it is shown that some of the results of [11] on reflection grou ...
... The fact that polyhedral homology manifolds which are not PL manifolds have something to do with exotic universal covers was first recognized in [11], through the use of reflection groups. In the recent Ph.D. thesis of G. Moussong [24], it is shown that some of the results of [11] on reflection grou ...
On acyclic and simply connected open manifolds - ICMC
... Proof of the Main Theorem B. Suppose the Generalized Poincaré Conjecture is true in dimension n. We will prove that the Main Theorem A is true in dimension n. Let X be an acyclic and simply connected open n-manifold and let X̄ = X ∪ {∞} be its onepoint compactification. If X is homeomorphic to Rn , ...
... Proof of the Main Theorem B. Suppose the Generalized Poincaré Conjecture is true in dimension n. We will prove that the Main Theorem A is true in dimension n. Let X be an acyclic and simply connected open n-manifold and let X̄ = X ∪ {∞} be its onepoint compactification. If X is homeomorphic to Rn , ...
Chapter 5 Manifolds, Tangent Spaces, Cotangent Spaces
... that is, the inversion of center O = (0, . . . , 0) and power 1. Clearly, this map is smooth on Rn {O}, so we conclude that (UN , N ) and (US , S ) form a smooth atlas for S n. Example 2. Smooth manifolds in RN . Any m-dimensional manifold M in RN is a smooth manifold, because by Lemma 2.22, the inv ...
... that is, the inversion of center O = (0, . . . , 0) and power 1. Clearly, this map is smooth on Rn {O}, so we conclude that (UN , N ) and (US , S ) form a smooth atlas for S n. Example 2. Smooth manifolds in RN . Any m-dimensional manifold M in RN is a smooth manifold, because by Lemma 2.22, the inv ...