LECTURE NOTES IN TOPOLOGICAL GROUPS 1. Lecture 1
... (1) the inversion map, left and right translations, all conjugations, are homeomorphisms G → G. (2) G is homogeneous as a topological space. (3) * For every pair (x, y) ∈ G × G there exists f ∈ Homeo(G, G) such that f (x) = y and f (y) = x. (4) Which of the following topological spaces are of the gr ...
... (1) the inversion map, left and right translations, all conjugations, are homeomorphisms G → G. (2) G is homogeneous as a topological space. (3) * For every pair (x, y) ∈ G × G there exists f ∈ Homeo(G, G) such that f (x) = y and f (y) = x. (4) Which of the following topological spaces are of the gr ...
Centre de Recerca Matem`atica
... Let F be a transversely parallelizable foliation of codimension n on a compact manifold M . Then: (1) the closures of the leaves are submanifolds which are fibres of a locally trivial fibration π : M −→ W where W is a compact manifold, (2) there exists a simply connected Lie group G0 such that the r ...
... Let F be a transversely parallelizable foliation of codimension n on a compact manifold M . Then: (1) the closures of the leaves are submanifolds which are fibres of a locally trivial fibration π : M −→ W where W is a compact manifold, (2) there exists a simply connected Lie group G0 such that the r ...
ON SEQUENTIALLY COHEN-MACAULAY
... As was mentioned, the notion of sequential Cohen-Macaulayness was first defined in terms of commutative algebra by Stanley. In [17] he also gave a homological characterization, see Appendix II, where the connection is outlined. Starting from Stanley’s homological characterization, two other homologi ...
... As was mentioned, the notion of sequential Cohen-Macaulayness was first defined in terms of commutative algebra by Stanley. In [17] he also gave a homological characterization, see Appendix II, where the connection is outlined. Starting from Stanley’s homological characterization, two other homologi ...
Notes on Real and Complex Analytic and Semianalytic Singularities
... manifolds with a prescribed atlas. When discussing submanifolds, the space in which the submanifold sits is frequently referred to as the ambient space. Note that the mx here is the dimension of the manifold M at x. If M is a pure-dimensional submanifold of the pure-dimensional manifold N , then the ...
... manifolds with a prescribed atlas. When discussing submanifolds, the space in which the submanifold sits is frequently referred to as the ambient space. Note that the mx here is the dimension of the manifold M at x. If M is a pure-dimensional submanifold of the pure-dimensional manifold N , then the ...
Dimension theory of arbitrary modules over finite von Neumann
... The p-th L2-Betti number measures the size of the space of smooth harmonic L2-integrable p-forms on M and vanishes precisely if there is no such non-trivial form. For a survey on L2-Betti numbers and related invariants like Novikov-Shubin invariants and L2-torsion and their applications and relatio ...
... The p-th L2-Betti number measures the size of the space of smooth harmonic L2-integrable p-forms on M and vanishes precisely if there is no such non-trivial form. For a survey on L2-Betti numbers and related invariants like Novikov-Shubin invariants and L2-torsion and their applications and relatio ...
On right-angled reflection groups in hyperbolic spaces
... that any k-dimensional face of an acute-angled polyhedron P ⊂ Hn belongs only to n − k hyperfaces. In particular, any (ordinary) vertex belongs only to n hyperfaces, so the local combinatorial structure of P at any vertex is the same as that of a simplicial cone. The local combinatorial structure of ...
... that any k-dimensional face of an acute-angled polyhedron P ⊂ Hn belongs only to n − k hyperfaces. In particular, any (ordinary) vertex belongs only to n hyperfaces, so the local combinatorial structure of P at any vertex is the same as that of a simplicial cone. The local combinatorial structure of ...
When does a manifold admit a metric with positive scalar curvature?
... by taking the disjoint union of M and the “handle” Dk × Dn−k and identifying the points in S k−1 × Dn−k with their image in ∂M . We say that M̃ is obtained by attaching a k-handle to M , or M̃ is obtained by surgery (i.e. by removing S k ×Dn−k and replacing it by Dk+1 ×S n−k−1 ). It is natural to as ...
... by taking the disjoint union of M and the “handle” Dk × Dn−k and identifying the points in S k−1 × Dn−k with their image in ∂M . We say that M̃ is obtained by attaching a k-handle to M , or M̃ is obtained by surgery (i.e. by removing S k ×Dn−k and replacing it by Dk+1 ×S n−k−1 ). It is natural to as ...
STABLE COHOMOLOGY OF FINITE AND PROFINITE GROUPS 1
... The cohomology groups constructed above depend only on the homotopy type of the space BG. In particular,other models of the space BG give the same groups. In general, let E be a contractible topological space with a free action of G, then E/G has a homotopy type of BG above. The space BG (modulo hom ...
... The cohomology groups constructed above depend only on the homotopy type of the space BG. In particular,other models of the space BG give the same groups. In general, let E be a contractible topological space with a free action of G, then E/G has a homotopy type of BG above. The space BG (modulo hom ...
Lie theory for non-Lie groups - Heldermann
... See [25] for a study of the properties of dim for separable metric spaces. Although it is quite intuitive, our dimension function does not work well for arbitrary spaces. Other dimension functions, notably covering dimension [38, 3.1.1], have turned out to be better suited for general spaces, while ...
... See [25] for a study of the properties of dim for separable metric spaces. Although it is quite intuitive, our dimension function does not work well for arbitrary spaces. Other dimension functions, notably covering dimension [38, 3.1.1], have turned out to be better suited for general spaces, while ...
An Introduction to K-theory
... 1.7 K0 of Quasi-projective Varieties . . . . 1.8 K1 of rings . . . . . . . . . . . . . . . 1.9 K2 of rings . . . . . . . . . . . . . . . ...
... 1.7 K0 of Quasi-projective Varieties . . . . 1.8 K1 of rings . . . . . . . . . . . . . . . 1.9 K2 of rings . . . . . . . . . . . . . . . ...
SUFFICIENTLY GENERIC ORTHOGONAL GRASSMANNIANS 1
... In the case where the quadratic form q is “sufficiently generic” (the precise condition is formulated in terms of the J-invariant of q introduced in [18], its definition and meaning are recalled in the beginnings of Sections 3, 4, and 5), we are going to show (see Theorems 3.1, 4.1, and 5.1) that the m ...
... In the case where the quadratic form q is “sufficiently generic” (the precise condition is formulated in terms of the J-invariant of q introduced in [18], its definition and meaning are recalled in the beginnings of Sections 3, 4, and 5), we are going to show (see Theorems 3.1, 4.1, and 5.1) that the m ...
Automorphisms of 2--dimensional right
... act naturally on CAT(0) cube complexes. Also known as graph groups, they occur in many different mathematical contexts; for some particularly interesting examples we refer to the work of Bestvina and Brady [3] on finiteness properties of groups, Croke and Kleiner [10] on boundaries of CAT(0) spaces, ...
... act naturally on CAT(0) cube complexes. Also known as graph groups, they occur in many different mathematical contexts; for some particularly interesting examples we refer to the work of Bestvina and Brady [3] on finiteness properties of groups, Croke and Kleiner [10] on boundaries of CAT(0) spaces, ...
Chapter 2: A few introductory remarks on topology, Abstract: November 12, 2014
... 2.2.1 Relation to the Hopf invariant The above innocent observation is actually related to a deep mathematical construction known as the Hopf fibration, which is a particularly beautiful map pH : S 3 → S 2 . We will describe it later when we discuss the representation theory of SU (2). The following ...
... 2.2.1 Relation to the Hopf invariant The above innocent observation is actually related to a deep mathematical construction known as the Hopf fibration, which is a particularly beautiful map pH : S 3 → S 2 . We will describe it later when we discuss the representation theory of SU (2). The following ...
REMARKS ON ALGEBRAIC GEOMETRY 1. Algebraic varieties
... be a little bit confusing to use the same notation for real and complex objects). Is Hλ always a hyperbola? Almost always. If λ 6= 0, then Hλ is a hyperbola. What is Hλ if λ = 0? It is just a union of two lines! Not just any lines. These lines, given by x = 0 and y = 0, are asymptotes of the hyperbo ...
... be a little bit confusing to use the same notation for real and complex objects). Is Hλ always a hyperbola? Almost always. If λ 6= 0, then Hλ is a hyperbola. What is Hλ if λ = 0? It is just a union of two lines! Not just any lines. These lines, given by x = 0 and y = 0, are asymptotes of the hyperbo ...
A YOUNG PERSON`S GUIDE TO THE HOPF FIBRATION The
... is not is historical. This is because the first ways of thinking of something are not always the easiest to understand. In this section you are often asked to show algebraic facts by drawing a picture. Not only is this rather unusual, it is rather subtle, but it can be done with all the rigor of a t ...
... is not is historical. This is because the first ways of thinking of something are not always the easiest to understand. In this section you are often asked to show algebraic facts by drawing a picture. Not only is this rather unusual, it is rather subtle, but it can be done with all the rigor of a t ...
Berkovich spaces embed in Euclidean spaces - IMJ-PRG
... Remark 3.2. Proposition 3.1 was proved in the 1930s. Namely, following a 1928 sketch by Menger, in 1931 it was proved independently by Lefschetz [Le], Nöbeling [Nö], and Pontryagin and Tolstowa [PT] that any compact metrizable space of dimension at most d embeds in R2d C1 . The proofs proceed by usi ...
... Remark 3.2. Proposition 3.1 was proved in the 1930s. Namely, following a 1928 sketch by Menger, in 1931 it was proved independently by Lefschetz [Le], Nöbeling [Nö], and Pontryagin and Tolstowa [PT] that any compact metrizable space of dimension at most d embeds in R2d C1 . The proofs proceed by usi ...
Intersection Theory course notes
... intersecting all 4 lines, namely, the line passing through the intersection points l1 ∩ l2 and l3 ∩ l4 and the intersection line of two planes containing l1 , l2 and l3 , l4 . Then by ”conservation of number principle” the number of solutions in general case is also two. To solve problems in enumera ...
... intersecting all 4 lines, namely, the line passing through the intersection points l1 ∩ l2 and l3 ∩ l4 and the intersection line of two planes containing l1 , l2 and l3 , l4 . Then by ”conservation of number principle” the number of solutions in general case is also two. To solve problems in enumera ...
COMPLEX VARIETIES AND THE ANALYTIC TOPOLOGY Classical
... Before we give the proof, we note that the analytic topology on a prevariety X is locally metrizable (and indeed metrizable if X is separated, from Corollary 2.6). Thus we can work more conceptually with sequential limits and compactness. Proof. We prove the statement in several steps. We first prov ...
... Before we give the proof, we note that the analytic topology on a prevariety X is locally metrizable (and indeed metrizable if X is separated, from Corollary 2.6). Thus we can work more conceptually with sequential limits and compactness. Proof. We prove the statement in several steps. We first prov ...
Lecture 1: Introduction to bordism Overview Bordism is a notion
... cyclic group (isomorphic to Z). One of the recent results which is a focal point of the course, the cobordism hypothesis [L1, F1], is a vast generalization of this easy classical theorem. We will also study bordism invariants. These are homomorphisms out of a bordism group or category into an abstra ...
... cyclic group (isomorphic to Z). One of the recent results which is a focal point of the course, the cobordism hypothesis [L1, F1], is a vast generalization of this easy classical theorem. We will also study bordism invariants. These are homomorphisms out of a bordism group or category into an abstra ...
Dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it – for example, the point at 5 on a number line. A surface such as a plane or the surface of a cylinder or sphere has a dimension of two because two coordinates are needed to specify a point on it – for example, both a latitude and longitude are required to locate a point on the surface of a sphere. The inside of a cube, a cylinder or a sphere is three-dimensional because three coordinates are needed to locate a point within these spaces.In classical mechanics, space and time are different categories and refer to absolute space and time. That conception of the world is a four-dimensional space but not the one that was found necessary to describe electromagnetism. The four dimensions of spacetime consist of events that are not absolutely defined spatially and temporally, but rather are known relative to the motion of an observer. Minkowski space first approximates the universe without gravity; the pseudo-Riemannian manifolds of general relativity describe spacetime with matter and gravity. Ten dimensions are used to describe string theory, and the state-space of quantum mechanics is an infinite-dimensional function space.The concept of dimension is not restricted to physical objects. High-dimensional spaces frequently occur in mathematics and the sciences. They may be parameter spaces or configuration spaces such as in Lagrangian or Hamiltonian mechanics; these are abstract spaces, independent of the physical space we live in.