EXISTENCE AND PROPERTIES OF GEOMETRIC QUOTIENTS
... written in a more general setting than needed for our generalization of the Keel–Mori theorem. For example, the results apply to non-flat equivalence relations [Kol11]. The impatient reader mainly interested in the Keel–Mori theorem is encouraged to go directly to §6. The first step in the proof of ...
... written in a more general setting than needed for our generalization of the Keel–Mori theorem. For example, the results apply to non-flat equivalence relations [Kol11]. The impatient reader mainly interested in the Keel–Mori theorem is encouraged to go directly to §6. The first step in the proof of ...
Algebraic characterization of finite (branched) coverings
... P r o o f. If g ∈ C ∗ (U ), it is clear that the function g 0 defined by g 0 = g · f on U = coz(f ) and g 0 = 0 on Z(f ) is continuous on X, and in the ring of fractions C ∗ (U )f we can write g/1 = g 0 /f . It is well known that the maximal spectrum of C(X), i.e., the subspace of Spec C(X) consisti ...
... P r o o f. If g ∈ C ∗ (U ), it is clear that the function g 0 defined by g 0 = g · f on U = coz(f ) and g 0 = 0 on Z(f ) is continuous on X, and in the ring of fractions C ∗ (U )f we can write g/1 = g 0 /f . It is well known that the maximal spectrum of C(X), i.e., the subspace of Spec C(X) consisti ...
spaces of finite length
... On completing the induction, set W = UneN Un\ ^ is open-and-closed because its complement is LlieN Vn, and of course C* c W c Z \ £ . (c) It follows that C* is connected. For if G and / / are open sets in X with C * c G U / / , C * n G D / / = 0 , then C* n G = C*\// and C*HH are disjoint closed set ...
... On completing the induction, set W = UneN Un\ ^ is open-and-closed because its complement is LlieN Vn, and of course C* c W c Z \ £ . (c) It follows that C* is connected. For if G and / / are open sets in X with C * c G U / / , C * n G D / / = 0 , then C* n G = C*\// and C*HH are disjoint closed set ...
A topological manifold is homotopy equivalent to some CW
... V of x in X1 such that f (V ) = f1 (V ) ⊂ U . Now V is open in X1 and so V = X1 ∩ W for some open subset W of X, and hence V ′ = V ∩ Int(X1 ) = W ∩ Int(X1 ) is an open neighborhood of x in X and f (V ′ ) ⊂ f (V ) ⊂ U . Hence f is continuous in x. Definition 1.4.3 (Neighborhood-finiteness (also call ...
... V of x in X1 such that f (V ) = f1 (V ) ⊂ U . Now V is open in X1 and so V = X1 ∩ W for some open subset W of X, and hence V ′ = V ∩ Int(X1 ) = W ∩ Int(X1 ) is an open neighborhood of x in X and f (V ′ ) ⊂ f (V ) ⊂ U . Hence f is continuous in x. Definition 1.4.3 (Neighborhood-finiteness (also call ...
![arXiv:1205.2342v1 [math.GR] 10 May 2012 Homogeneous compact](http://s1.studyres.com/store/data/014455921_1-3cd5e27d3d632291f2fbeb8ac68a29b6-300x300.png)
arXiv:1205.2342v1 [math.GR] 10 May 2012 Homogeneous compact
... We classify compact homogeneous geometries which look locally like compact spherical buildings. Geometries which look locally like buildings arise naturally in various recognition problems in group theory. Tits’ seminal paper A local approach to buildings [51] is devoted to them. Among other things, ...
... We classify compact homogeneous geometries which look locally like compact spherical buildings. Geometries which look locally like buildings arise naturally in various recognition problems in group theory. Tits’ seminal paper A local approach to buildings [51] is devoted to them. Among other things, ...
ROLLING OF COXETER POLYHEDRA ALONG MIRRORS 1
... Figure 3. Even and odd labels. Proof of Rolling Lemma. 1.12. A more general view. Nikolas Bourbaki5 proposed a way to build topological spaces from Coxeter groups. M. Davis used this approach in numerous papers (see e.g. [9], [10]) and the book [11]; in particular he constructed nice examples/counte ...
... Figure 3. Even and odd labels. Proof of Rolling Lemma. 1.12. A more general view. Nikolas Bourbaki5 proposed a way to build topological spaces from Coxeter groups. M. Davis used this approach in numerous papers (see e.g. [9], [10]) and the book [11]; in particular he constructed nice examples/counte ...
EXISTENCE AND PROPERTIES OF GEOMETRIC QUOTIENTS
... Keel-Mori theorem. The impatient reader mainly interested in the KeelMori theorem is encouraged to go directly to §6. The first step in the proof of the Keel-Mori theorem is to find a flat and locally quasi-finite presentation of a stack X with quasi-finite diagonal. The existence of such a presenta ...
... Keel-Mori theorem. The impatient reader mainly interested in the KeelMori theorem is encouraged to go directly to §6. The first step in the proof of the Keel-Mori theorem is to find a flat and locally quasi-finite presentation of a stack X with quasi-finite diagonal. The existence of such a presenta ...
subgroups of free topological groups and free
... Our objectives are topological versions of the Nielsen-Schreier Theorem on subgroups of free groups, and the Kurosh Theorem on subgroups of free products of groups. It is known that subgroups of free topological groups need not be free topological [2, 6, and 9]. However we might expect a subgroup th ...
... Our objectives are topological versions of the Nielsen-Schreier Theorem on subgroups of free groups, and the Kurosh Theorem on subgroups of free products of groups. It is known that subgroups of free topological groups need not be free topological [2, 6, and 9]. However we might expect a subgroup th ...
Sheaf Cohomology 1. Computing by acyclic resolutions
... For the particular functor ‘take global sections’, other conditions on a sheaf still guarantee acyclicity and at the same time are ‘intrinsic’ in that they do not refer to any ‘ambient category’. Threfore, in principle these other conditions are more readily verifiable. For the moment, we merely cat ...
... For the particular functor ‘take global sections’, other conditions on a sheaf still guarantee acyclicity and at the same time are ‘intrinsic’ in that they do not refer to any ‘ambient category’. Threfore, in principle these other conditions are more readily verifiable. For the moment, we merely cat ...
QUOTIENTS IN ALGEBRAIC AND SYMPLECTIC GEOMETRY 1
... Moreover, 0 is a regular value of µ if and only if X s = X ss . In this case, the GIT quotient is a projective variety which is an orbit space for the action of G on X s . There is a further generalisation of this result which gives a correspondence between an algebraic and symplectic stratification ...
... Moreover, 0 is a regular value of µ if and only if X s = X ss . In this case, the GIT quotient is a projective variety which is an orbit space for the action of G on X s . There is a further generalisation of this result which gives a correspondence between an algebraic and symplectic stratification ...
Ordered Quotients and the Semilattice of Ordered
... if A = d(A), and convex if A = i(A) ∩ d(A). The closed increasing hull of A, denoted by I(A), is the intersection of all closed increasing sets containing A. The closed decreasing hull of A is defined similarly and denoted by D(A). An ordered topological space is a triple (X, τ, ≤) consisting of a se ...
... if A = d(A), and convex if A = i(A) ∩ d(A). The closed increasing hull of A, denoted by I(A), is the intersection of all closed increasing sets containing A. The closed decreasing hull of A is defined similarly and denoted by D(A). An ordered topological space is a triple (X, τ, ≤) consisting of a se ...
Arithmetic fundamental groups and moduli of curves
... a homeomorphism of Fb is not unique, but its isotopy class is well de ned. Thus, 1 (B; b) acts on various homotopy invariants of Fb , like cohomologies and homotopy groups. Such a representation of 1 (B; b) is called monodromy representation associated to F ! B . Let us concentrate on the monodrom ...
... a homeomorphism of Fb is not unique, but its isotopy class is well de ned. Thus, 1 (B; b) acts on various homotopy invariants of Fb , like cohomologies and homotopy groups. Such a representation of 1 (B; b) is called monodromy representation associated to F ! B . Let us concentrate on the monodrom ...
COMBINATORIAL HOMOTOPY. I 1. Introduction. This is the first of a
... Let K be the universal covering complex of a CW-complex K. Since TI(K) = 1, irn(K) ^Tn(K) if n> 1 and since L is a Jm-complex if 7TV(L) = 0 for r = 1, • • • , m — 1 it follows from Theorem 10 that K is a J m -complex if 7r n (i£)=0 for n = 2, • • • , m — 1 . In particular if is a J^-complex if its u ...
... Let K be the universal covering complex of a CW-complex K. Since TI(K) = 1, irn(K) ^Tn(K) if n> 1 and since L is a Jm-complex if 7TV(L) = 0 for r = 1, • • • , m — 1 it follows from Theorem 10 that K is a J m -complex if 7r n (i£)=0 for n = 2, • • • , m — 1 . In particular if is a J^-complex if its u ...
Semi-quotient mappings and spaces
... element, and for a given x 2 G, `x W G ! G, y 7! x ı y, and rx W G ! G, y 7! y ı x, denote the left and the right translation by x, respectively. The operation we call the multiplication mapping m W G G ! G, and the inverse operation x 7! x 1 is denoted by i . In 1963, N. Levine [1] defined semi ...
... element, and for a given x 2 G, `x W G ! G, y 7! x ı y, and rx W G ! G, y 7! y ı x, denote the left and the right translation by x, respectively. The operation we call the multiplication mapping m W G G ! G, and the inverse operation x 7! x 1 is denoted by i . In 1963, N. Levine [1] defined semi ...
TOPOLOGICAL GROUPS 1. Introduction Topological groups are
... surprising result due to Keller. Thus, homogeneity can improve when taking products, but can never be lost. Recall that an action of a group G on a space X is called transitive if the action has only one orbit, i.e. X = G·x for some x ∈ X; and it is called free if the mapping x 7→ g · · · x has no f ...
... surprising result due to Keller. Thus, homogeneity can improve when taking products, but can never be lost. Recall that an action of a group G on a space X is called transitive if the action has only one orbit, i.e. X = G·x for some x ∈ X; and it is called free if the mapping x 7→ g · · · x has no f ...
Analogues of Cayley graphs for topological groups
... π : G → Aut(X) given by the action of G on X is continuous, the kernel of this homomorphism is compact and the image of π is closed in Aut(X). Conversely, if G acts as a group of automorphisms on a locally finite connected graph X such that G is transitive on the vertex set of X and the stabilizers ...
... π : G → Aut(X) given by the action of G on X is continuous, the kernel of this homomorphism is compact and the image of π is closed in Aut(X). Conversely, if G acts as a group of automorphisms on a locally finite connected graph X such that G is transitive on the vertex set of X and the stabilizers ...
Topology I - School of Mathematics
... out: the global calculus of variations, global geometry, the topology of Lie groups and homogeneous spaces, the topology of complex manifolds and algebraic varieties, the qualitative (topological) theory of dynamical systems and foliations, the topology of elliptic and hyperbolic partial differentia ...
... out: the global calculus of variations, global geometry, the topology of Lie groups and homogeneous spaces, the topology of complex manifolds and algebraic varieties, the qualitative (topological) theory of dynamical systems and foliations, the topology of elliptic and hyperbolic partial differentia ...
The Simplicial Lusternik
... K & L. The definition of the simplicial category given in [1] is based on the definition of geometric category given by Fox [7] and represents the minimum, among the simplicial complexes L such that K & L, of the smallest number of collapsible subcomplexes that can cover L. However, the concept of c ...
... K & L. The definition of the simplicial category given in [1] is based on the definition of geometric category given by Fox [7] and represents the minimum, among the simplicial complexes L such that K & L, of the smallest number of collapsible subcomplexes that can cover L. However, the concept of c ...
Connected topological generalized groups
... group, and G is disjoint union of such topological groups. Example 1.1. The set G = R × (R − {0}) with the topology induced by a Euclidean metric, and with operation (a, b)(c, d) = (bc, bd) is a topological generalized group. Theorem 1.1. Let G be a topological group, and let a2 = e for all a ∈ G. T ...
... group, and G is disjoint union of such topological groups. Example 1.1. The set G = R × (R − {0}) with the topology induced by a Euclidean metric, and with operation (a, b)(c, d) = (bc, bd) is a topological generalized group. Theorem 1.1. Let G be a topological group, and let a2 = e for all a ∈ G. T ...
The Coarse Geometry of Groups
... The Milnor-Svarc Lemma One of the fundamental tools in studying the coarse geometry of groups is the following fact. ...
... The Milnor-Svarc Lemma One of the fundamental tools in studying the coarse geometry of groups is the following fact. ...
Connes–Karoubi long exact sequence for Fréchet sheaves
... Lemma 2.3. Let (X, OX ) denote the formal scheme obtained by completing an integral noetherian scheme X ′ of finite type over C along a closed primary integral subscheme X, as in example (a) above. Then, for each open subset U of X, the ring OX (U ) is an ultrametric Banach algebra, i.e., (X, OX ) d ...
... Lemma 2.3. Let (X, OX ) denote the formal scheme obtained by completing an integral noetherian scheme X ′ of finite type over C along a closed primary integral subscheme X, as in example (a) above. Then, for each open subset U of X, the ring OX (U ) is an ultrametric Banach algebra, i.e., (X, OX ) d ...
Lecture notes
... space theory, that the closure is the set of limit points of a set. Unfortunately limits are not always defined in topological spaces, so this definition does not generalise directly. 5In fact, one can prove that any metric on Rn of the form d(x, y) = kx − yk where k·k is a norm (see http: //en.wiki ...
... space theory, that the closure is the set of limit points of a set. Unfortunately limits are not always defined in topological spaces, so this definition does not generalise directly. 5In fact, one can prove that any metric on Rn of the form d(x, y) = kx − yk where k·k is a norm (see http: //en.wiki ...
Lecture notes for topology
... for a topology. And indeed, if B1 = S1 ∩ S2 ∩ · · · ∩ Sm and B2 = S1′ ∩ S2′ ∩ · · · ∩ Sn′ , then B1 ∩ B2 = S1 ∩ S2 ∩ · · · ∩ S1′ ∩ S2′ ∩ · ∩ Sn′ which is again in B. Done. Here is a criterion that allows us to recognize at first glance bases for topologies. Proposition 1.3.4. Let X be a topological ...
... for a topology. And indeed, if B1 = S1 ∩ S2 ∩ · · · ∩ Sm and B2 = S1′ ∩ S2′ ∩ · · · ∩ Sn′ , then B1 ∩ B2 = S1 ∩ S2 ∩ · · · ∩ S1′ ∩ S2′ ∩ · ∩ Sn′ which is again in B. Done. Here is a criterion that allows us to recognize at first glance bases for topologies. Proposition 1.3.4. Let X be a topological ...
Relative Stanley–Reisner theory and Upper Bound Theorems for
... From discrete geometry to combinatorial topology to commutative algebra. — An intriguing feature of the UBT is that its validity extends beyond the realm of convex polytopes and into combinatorial topology. Let be a triangulation of the (d − 1)-sphere and, as before, let us write fk () for the nu ...
... From discrete geometry to combinatorial topology to commutative algebra. — An intriguing feature of the UBT is that its validity extends beyond the realm of convex polytopes and into combinatorial topology. Let be a triangulation of the (d − 1)-sphere and, as before, let us write fk () for the nu ...