A hypersphere is parallelizable if there are vector fields that are linearly independent at each point. There exist only three parallelizable spheres: , , and (Adams 1958, 1960, Le Lionnais 1983).More generally, an -dimensional manifold is parallelizable if its tangent bundle is a trivial bundle (i.e., if is globally of the form ).
The real projective plane with elliptic metric where the distance between two points and is defined as the radian angle between the projection of the points on the surface of a sphere (which is tangent to the plane at a point ) from the antipode of the tangent point.
The Möbius strip, also called the twisted cylinder (Henle 1994, p. 110), is a one-sided nonorientable surface obtained by cutting a closed band into a single strip, giving one of the two ends thus produced a half twist, and then reattaching the two ends (right figure; Gray 1997, pp. 322-323). The strip bearing his name was invented by Möbius in 1858, although it was independently discovered by Listing, who published it, while Möbius did not (Derbyshire 2004, p. 381). Like the cylinder, it is not a true surface, but rather a surface with boundary (Henle 1994, p. 110).The Möbius strip has Euler characteristic (Dodson and Parker 1997, p. 125).According to Madachy (1979), the B. F. Goodrich Company patented a conveyor belt in the form of a Möbius strip which lasts twice as long as conventional belts. M. C. Escher was fond of portraying Möbius strips, and..
The generalization of the Schönflies theorem to dimensions. A smoothly embedded -hypersphere in an -hypersphere separates the -hypersphere into two components, each homeomorphic to -balls. It can be proved using Morse theory.
Thurston's conjecture proposed a complete characterization of geometric structureson three-dimensional manifolds.Before stating Thurston's geometrization conjecture in detail, some background information is useful. Three-dimensional manifolds possess what is known as a standard two-level decomposition. First, there is the connected sum decomposition, which says that every compact three-manifold is the connected sum of a unique collection of prime three-manifolds.The second decomposition is the Jaco-Shalen-Johannson torus decomposition, which states that irreducible orientable compact 3-manifolds have a canonical (up to isotopy) minimal collection of disjointly embedded incompressible tori such that each component of the 3-manifold removed by the tori is either "atoroidal" or "Seifert-fibered."Thurston's conjecture is that, after you split a three-manifold into its connected sum and..
Every closed three-manifold with finite fundamental group has a metric of constant positive scalar curvature, and hence is homeomorphic to a quotient , where is a finite group of rotations that acts freely on .Since the trivial group is in particular a finite group, the elliptization conjecture implies the Poincaré conjecture.
The Klein bottle is a closed nonorientable surface of Euler characteristic 0 (Dodson and Parker 1997, p. 125) that has no inside or outside, originally described by Felix Klein (Hilbert and Cohn-Vossen 1999, p. 308). It can be constructed by gluing both pairs of opposite edges of a rectangle together giving one pair a half-twist, but can be physically realized only in four dimensions, since it must pass through itself without the presence of a hole. Its topology is equivalent to a pair of cross-caps with coinciding boundaries (Francis and Weeks 1999). It can be represented by connecting the side of a square in the orientations illustrated in the right figure above (Gardner 1984, pp. 15-17; Gray 1997, pp. 323-324).It can be cut in half along its length to make two Möbius strips (Dodson and Parker 1997, p. 88), but can also be cut into a single Möbius strip (Gardner 1984, pp. 14 and 17).The above picture..
A 4-hyperboloid has negative curvature, with(1)(2)Since(3)it follows that(4)To stay on the surface of the hyperboloid, the line element is given by(5)(6)(7)
A hole in a mathematical object is a topological structure which prevents the object from being continuously shrunk to a point. When dealing with topological spaces, a disconnectivity is interpreted as a hole in the space. Examples of holes are things like the "donut hole" in the center of the torus, a domain removed from a plane, and the portion missing from Euclidean space after cutting a knot out from it.Singular homology groups form a measure of the hole structure of a space, but they are one particular measure and they don't always detect all holes. homotopy groups of a space are another measure of holes in a space, as well as bordism groups, K-theory, cohomotopy groups, and so on.There are many ways to measure holes in a space. Some holes are picked up by homotopy groups that are not detected by homology groups, and some holes are detected by homology groups that are not picked up by homotopy groups. (For example, in the torus, homotopy..
The real projective plane is the closed topological manifold, denoted , that is obtained by projecting the points of a plane from a fixed point (not on the plane), with the addition of the line at infinity. It can be described by connecting the sides of a square in the orientations illustrated above (Gardner 1971, pp. 15-17; Gray 1997, pp. 323-324).There is then a one-to-one correspondence between points in and lines through not parallel to . Lines through that are parallel to have a one-to-one correspondence with points on the line at infinity. Since each line through intersects the sphere centered at and tangent to in two antipodal points, can be described as a quotient space of by identifying any two such points. The real projective plane is a nonorientable surface. The equator of (which, in the quotient space, is itself a projective line) corresponds to the line at infinity.The complete graph on 6 vertices can be drawn in the projective..
A handlebody of type is an -dimensional manifold that is attained from the standard -disk by attaching only -D handles.
A topology on a set whose open sets are the unions of open ballswhere is a pseudometric on , is any point of , and .There is a remarkable difference between a metric and a pseudometric topology. The former is always , whereas the latter is, in general, not even . In fact, a pseudometric allows for some distinct points and , and then every open ball containing contains and conversely, so that no open set can separate the two points.
A handle is a topological structure which can be thought of as the object produced by puncturing a surface twice, attaching a zip around each puncture travelling in opposite directions, pulling the edges of the zips together, and then zipping up.Handles are to manifolds as cells are to CW-complexes. If is a manifold together with a -sphere embedded in its boundary with a trivial tubular neighborhood, we attach a -handle to by gluing the tubular neighborhood of the -sphere to the tubular neighborhood of the standard -sphere in the dim()-dimensional disk. In this way, attaching a -handle is essentially just the process of attaching a fattened-up -disk to along the -sphere . The embedded disk in this new manifold is called the -handle in the union of and the handle.Dyck's theorem states that handles and cross-handlesare equivalent in the presence of a cross-cap...
A surface constructed by placing a family of figure-eight curves into such that the first and last curves reduce to points. The surface has parametric equations(1)(2)(3)
The topology on the Cartesian product of two topological spaces whose open sets are the unions of subsets , where and are open subsets of and , respectively.This definition extends in a natural way to the Cartesian product of any finite number of topological spaces. The product topology ofwhere is the real line with the Euclidean topology, coincides with the Euclidean topology of the Euclidean space .In the definition of product topology of , where is any set, the open sets are the unions of subsets , where is an open subset of with the additional condition that for all but finitely many indices (this is automatically fulfilled if is a finite set). The reason for this choice of open sets is that these are the least needed to make the projection onto the th factor continuous for all indices . Admitting all products of open sets would give rise to a larger topology (strictly larger if is infinite), called the box topology.The product topology is also called..
Construct a chain of components in a solid torus . Now thicken each component of slightly to form a chain of solid tori in , wherevia inclusion. In each component of , construct a smaller chain of solid tori embedded in that component. Denote the union of these smaller solid tori . Continue this process a countable number of times, then the intersectionwhich is a nonempty compact subset of is called Antoine's necklace. Antoine's necklace is homeomorphic with the Cantor set.
A topological two-sphere in three-space whose exterior is not simply connected. The outer complement of Antoine's horned sphere is not simply connected. Furthermore, the group of the outer complement is not even finitely generated. Antoine's horned sphere is inequivalent to Alexander's horned sphere since the complement in of the bad points for Alexander's horned sphere is simply connected.
Milnor (1956) found more than one smooth structure on the seven-dimensional hypersphere. Generalizations have subsequently been found in other dimensions. Using surgery theory, it is possible to relate the number of diffeomorphism classes of exotic spheres to higher homotopy groups of spheres (Kosinski 1992).Kervaire and Milnor (1963) computed a list of the number of distinct (up to diffeomorphism) differential structures on spheres indexed by the dimension of the sphere. For , 2, ..., assuming the Poincaré conjecture, they are 1, 1, 1, , 1, 1, 28, 2, 8, 6, 992, 1, 3, 2, 16256, 2, 16, 16, ... (OEIS A001676). The status of is still unresolved, and it is not known whether there is 1, more than 1, or infinitely many smooth structures on the 4-sphere (Scorpan 2005). The claim that there is exactly one is known as the smooth Poincaré conjecture for .The only exotic Euclidean spaces are a continuum ofexotic R4 structures...
The above topological structure, composed of a countable union of compact sets, is called Alexander's horned sphere. It is homeomorphic with the ball , and its boundary is therefore a sphere. It is therefore an example of a wild embedding in . The outer complement of the solid is not simply connected, and its fundamental group is not finitely generated. Furthermore, the set of nonlocally flat ("bad") points of Alexander's horned sphere is a Cantor set.The horned sphere as originally drawn by Alexander (1924) is illustrated above.The complement in of the bad points for Alexander's horned sphere is simply connected, making it inequivalent to Antoine's horned sphere. Alexander's horned sphere has an uncountable infinity of wild points, which are the limits of the sequences of the horned sphere's branch points (roughly, the "ends" of the horns), since any neighborhood of a limit contains a horned complex.A humorous drawing..
A Tychonoff plank is a topological space that is an example of a normal space which has a non-normal subset, thus showing that normality is not a hereditary property. Let be the set of all ordinals which are less than or equal to , and the set of all ordinals which are less than or equal to . Consider the set with the product topology induced by the order topologies of and . Then is normal, but the subset is not. It can be shown that the set of all elements of whose first coordinate is equal to and the set of all elements of whose second coordinate is equal to are disjoint closed subsets , but there are no disjoint open subsets and of such that and .