Lectures on quasi-isometric rigidity
... Note: For every finitely-generated group G there exists a compact Riemannian manifold M (of every dimension ≥ 2) with an epimorphism π1 (M ) → G. Thus, we get another correspondence Groups −→ Metric Spaces: Riemann : G → X = a covering space of some M as above. Thus, we have a problem on our hands, ...
... Note: For every finitely-generated group G there exists a compact Riemannian manifold M (of every dimension ≥ 2) with an epimorphism π1 (M ) → G. Thus, we get another correspondence Groups −→ Metric Spaces: Riemann : G → X = a covering space of some M as above. Thus, we have a problem on our hands, ...
Compact operators on Banach spaces
... This is the Fredholm alternative for operators T − λ with T compact and λ 6= 0: either T − λ is bijective, or has non-trivial kernel and non-trivial cokernel, of the same dimension. As above, the compactness of T implies the finite-dimensionality of ker(T − λ) for λ 6= 0. Dually, for y1 , . . . , yn ...
... This is the Fredholm alternative for operators T − λ with T compact and λ 6= 0: either T − λ is bijective, or has non-trivial kernel and non-trivial cokernel, of the same dimension. As above, the compactness of T implies the finite-dimensionality of ker(T − λ) for λ 6= 0. Dually, for y1 , . . . , yn ...
26 - HKU
... reduction of the structure group of the (real) tangent bundle from the general linear group GL(m, R) to the orthogonal group O(m). Riemannian geometry may be regarded as the geometry of smooth O(m)-structures. A Riemannian manifold (M, g) is locally isometric to the Euclidean space if and only if th ...
... reduction of the structure group of the (real) tangent bundle from the general linear group GL(m, R) to the orthogonal group O(m). Riemannian geometry may be regarded as the geometry of smooth O(m)-structures. A Riemannian manifold (M, g) is locally isometric to the Euclidean space if and only if th ...
Topology Proceedings - Topology Research Group
... α ∈ Λ and n ∈ ω, we write Pα,n = {Pβ : β ∈ Aα,n }, where Aα,n is countable. Put An = ∪{Aα,n : α ∈ Λ}, then Pn = ∪{Pα,n : α ∈ Λ} = {Pβ : β ∈ An }, and {Pn : n ∈ ω} is a sequence of point-finite covers of X. We can assume {Aα,n : α ∈ Λ} and {An : n ∈ ω} are all mutually disjoint. Let every An be endow ...
... α ∈ Λ and n ∈ ω, we write Pα,n = {Pβ : β ∈ Aα,n }, where Aα,n is countable. Put An = ∪{Aα,n : α ∈ Λ}, then Pn = ∪{Pα,n : α ∈ Λ} = {Pβ : β ∈ An }, and {Pn : n ∈ ω} is a sequence of point-finite covers of X. We can assume {Aα,n : α ∈ Λ} and {An : n ∈ ω} are all mutually disjoint. Let every An be endow ...
Connectedness and path-connectedness
... In the course of proving this, we proved that a closed interval was connected. We’d also like to assert that a half-open interval is connected - to do this, we could go through the whole proof again, but it’s simpler to prove the following more general result from the book: Lemma 5. Let A be a conne ...
... In the course of proving this, we proved that a closed interval was connected. We’d also like to assert that a half-open interval is connected - to do this, we could go through the whole proof again, but it’s simpler to prove the following more general result from the book: Lemma 5. Let A be a conne ...
Math 525 More notes about compactness (sec 26
... Heine-Borel Theorem A subset of (R, τu ) is compact iff it is closed and bounded. Proof : By Thm 27.1, any closed, bounded interval [a, b] in R is compact. Any closed, bounded set in R is a closed subset of some interval [a, b], and hence compact by Prop A52(a). Conversely, any compact set A in R is ...
... Heine-Borel Theorem A subset of (R, τu ) is compact iff it is closed and bounded. Proof : By Thm 27.1, any closed, bounded interval [a, b] in R is compact. Any closed, bounded set in R is a closed subset of some interval [a, b], and hence compact by Prop A52(a). Conversely, any compact set A in R is ...
Math 396. Paracompactness and local compactness 1. Motivation
... For example, the covering of R by open intervals (n − 1, n + 1) for n ∈ Z is locally finite, whereas the covering of (−1, 1) by intervals (−1/n, 1/n) (for n ≥ 1) barely fails to be locally finite: there is a problem at the origin (but nowhere else). Definition 2.5. A topological space X is paracompa ...
... For example, the covering of R by open intervals (n − 1, n + 1) for n ∈ Z is locally finite, whereas the covering of (−1, 1) by intervals (−1/n, 1/n) (for n ≥ 1) barely fails to be locally finite: there is a problem at the origin (but nowhere else). Definition 2.5. A topological space X is paracompa ...
Locally nite spaces and the join operator - mtc-m21b:80
... Topological properties of digital images play an important role in image processing. Much of the theoretical development has been motivated by the needs in applications, for example digital Jordan curve theorems and the theory of digitization. A classical survey is [11] by Kong and Rosenfeld. Inspir ...
... Topological properties of digital images play an important role in image processing. Much of the theoretical development has been motivated by the needs in applications, for example digital Jordan curve theorems and the theory of digitization. A classical survey is [11] by Kong and Rosenfeld. Inspir ...
Geometry. - SchoolNova
... Proof 2. Perhaps, the most famous proof is that by Euclid, although it is neither the simplest, nor the most elegant. It is illustrated in Fig. 3 below. ...
... Proof 2. Perhaps, the most famous proof is that by Euclid, although it is neither the simplest, nor the most elegant. It is illustrated in Fig. 3 below. ...
On Top Spaces
... (ii) For each x in T there exists a unique e(x) in T such that xe(x) = e(x)x = x; (iii) For each x in T there exists y in T such that xy = yx = e(x); (iv) T is a Hausdorff topological space; (v) The mapping m2 and the mapping m1 : T ...
... (ii) For each x in T there exists a unique e(x) in T such that xe(x) = e(x)x = x; (iii) For each x in T there exists y in T such that xy = yx = e(x); (iv) T is a Hausdorff topological space; (v) The mapping m2 and the mapping m1 : T ...
Slides
... (1) The class of valid ∀∃ sentences without function symbols is decidable. (2) The class of satisfiable ∃∀ sentences without function symbols is decidable. This results can extended to important cases where function symbols occur. ...
... (1) The class of valid ∀∃ sentences without function symbols is decidable. (2) The class of satisfiable ∃∀ sentences without function symbols is decidable. This results can extended to important cases where function symbols occur. ...
ON MINIMAL, STRONGLY PROXIMAL ACTIONS OF LOCALLY
... of all automorphisms of the Cayley graph XΓ,Σ . In addition fundamental groups Γ = π1 M of compact locally symmetric manifolds M have natural cocompact lattice embeddings in the semi-simple Lie group G = Isom + (M̃ ), and in direct products H = G × K where K is an arbitrary compact group. In [3] coc ...
... of all automorphisms of the Cayley graph XΓ,Σ . In addition fundamental groups Γ = π1 M of compact locally symmetric manifolds M have natural cocompact lattice embeddings in the semi-simple Lie group G = Isom + (M̃ ), and in direct products H = G × K where K is an arbitrary compact group. In [3] coc ...
Introduction to symmetric spectra I
... above. An important property of Boardman’s stable homotopy category is that it has a symmetric monoidal structure; however, it is not induced from any such structure on the category of spectra. All other categories of spectra defined in a similar manner share this disadvantage. This has caused algeb ...
... above. An important property of Boardman’s stable homotopy category is that it has a symmetric monoidal structure; however, it is not induced from any such structure on the category of spectra. All other categories of spectra defined in a similar manner share this disadvantage. This has caused algeb ...
Selected Response and Written Response Pre and
... 8. A triangle with no congruent sides is called ______________________. A. Equilateral B. Isosceles C. Scalene 9. A triangle with at least two congruent sides is called ______________________. A. Equilateral B. Isosceles C. Scalene 10. How many total degrees are in the interior angles of a triangle? ...
... 8. A triangle with no congruent sides is called ______________________. A. Equilateral B. Isosceles C. Scalene 9. A triangle with at least two congruent sides is called ______________________. A. Equilateral B. Isosceles C. Scalene 10. How many total degrees are in the interior angles of a triangle? ...
2.5 Proving Statements About Segments
... perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. ...
... perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. ...
229 ACTION OF GENERALIZED LIE GROUPS ON
... spaces) and the stabilizer of the top spaces. We show that stabilizer is a top space, moreover we find the tangent space of the stabilizer. Also, by using the generalized action, we find the dimension of some top spaces. Now, we recall the definition of a generalized group of [1]. A generalized grou ...
... spaces) and the stabilizer of the top spaces. We show that stabilizer is a top space, moreover we find the tangent space of the stabilizer. Also, by using the generalized action, we find the dimension of some top spaces. Now, we recall the definition of a generalized group of [1]. A generalized grou ...
The Space of Metric Spaces
... Geodesic of L(γ) = d(γ(a), γ(b)). A Lengths space X is called Strictly intrinsic or Geodesic if for all pairs of points can be connected with a geodesic, x, y ∈ X, there is a length minimizing curve γ with γ(a) = x, ...
... Geodesic of L(γ) = d(γ(a), γ(b)). A Lengths space X is called Strictly intrinsic or Geodesic if for all pairs of points can be connected with a geodesic, x, y ∈ X, there is a length minimizing curve γ with γ(a) = x, ...
Section 29. Local Compactness - Faculty
... not locally compact. Recall that basis elements for the product topology are of the form B = (a1 , b1 ) × (a1, a2 ) × · · · × (an , bn ) × R × R × · · · (by Theorem 19.1). If C is a compact subspace of Rω that contains x ∈ Rω and there is a neighborhood of x in C, then the neighborhood contains a ba ...
... not locally compact. Recall that basis elements for the product topology are of the form B = (a1 , b1 ) × (a1, a2 ) × · · · × (an , bn ) × R × R × · · · (by Theorem 19.1). If C is a compact subspace of Rω that contains x ∈ Rω and there is a neighborhood of x in C, then the neighborhood contains a ba ...
SOME UNSOLVED PROBLEMS CONCERNING
... PROOF. Suppose that ( X , T ) is a a#-space and let *? = U ° | = 1 { ^ } be a a-closure preserving separating closed cover. For each t ^ l , let < V = l 4 J = 1
... PROOF. Suppose that ( X , T ) is a a#-space and let *? = U ° | = 1 { ^ } be a a-closure preserving separating closed cover. For each t ^ l , let < V = l 4 J = 1
Group actions in symplectic geometry
... This motivates the question whether there are interesting Hamiltonian actions of in nite discrete groups like, for example, lattices in semisimple Lie groups. In turns out that, under certain geometric conditions, there are restrictions. f → M be the universal cover. A symplectic form ω on Let p : M ...
... This motivates the question whether there are interesting Hamiltonian actions of in nite discrete groups like, for example, lattices in semisimple Lie groups. In turns out that, under certain geometric conditions, there are restrictions. f → M be the universal cover. A symplectic form ω on Let p : M ...
(pdf)
... The Dold-Thom theorem regarding quasifibrations and the infinite symmetric product will prove useful in demonstrating several important results. Before stating the theorem, we will establish some necessary definitions and show the long exact sequence generated by a quasifibration, which is what will ...
... The Dold-Thom theorem regarding quasifibrations and the infinite symmetric product will prove useful in demonstrating several important results. Before stating the theorem, we will establish some necessary definitions and show the long exact sequence generated by a quasifibration, which is what will ...
PROPERTIES OF FINITE-DIMENSIONAL GROUPS Topological
... In connection with 2 and 2', Zippin and the author have shown [10; 8; 16] that any compact connected group acting effectively on a three-dimensional manifold M must be a Lie group. If M = Ez, we showed further that G must be equivalent either to the group of all rigid motions about an axis or to the ...
... In connection with 2 and 2', Zippin and the author have shown [10; 8; 16] that any compact connected group acting effectively on a three-dimensional manifold M must be a Lie group. If M = Ez, we showed further that G must be equivalent either to the group of all rigid motions about an axis or to the ...
On Hausdorff compactifications - Mathematical Sciences Publishers
... Given a pair of spaces X and Y, a necessary and sufficient condition is found for Y to be homeomorphic to daχ(aX— X) for some compactification aX of X. From this follows a necessary and sufficient condition for Y to be homeomorphic to aX — X for some aX. As an application, a sufficient condition is ...
... Given a pair of spaces X and Y, a necessary and sufficient condition is found for Y to be homeomorphic to daχ(aX— X) for some compactification aX of X. From this follows a necessary and sufficient condition for Y to be homeomorphic to aX — X for some aX. As an application, a sufficient condition is ...