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Whitehead Manifold
In mathematics, the Whitehead manifold is an open 3-manifold that is contractible, but not homeomorphic to \R^3. discovered this puzzling object while he was trying to prove the Poincaré conjecture, correcting an error in an earlier paper where he incorrectly claimed that no such manifold exists. A contractible manifold is one that can continuously be shrunk to a point inside the manifold itself. For example, an open ball is a contractible manifold. All manifolds homeomorphic to the ball are contractible, too. One can ask whether ''all'' contractible manifolds are homeomorphic to a ball. For dimensions 1 and 2, the answer is classical and it is "yes". In dimension 2, it follows, for example, from the Riemann mapping theorem. Dimension 3 presents the first counterexample: the Whitehead manifold. Construction Take a copy of S^3, the three-dimensional sphere. Now find a compact unknotted solid torus T_1 inside the sphere. (A solid torus is an ordinary three-dimensional doughnut, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Whitehead Manifold
In mathematics, the Whitehead manifold is an open 3-manifold that is contractible, but not homeomorphic to \R^3. discovered this puzzling object while he was trying to prove the Poincaré conjecture, correcting an error in an earlier paper where he incorrectly claimed that no such manifold exists. A contractible manifold is one that can continuously be shrunk to a point inside the manifold itself. For example, an open ball is a contractible manifold. All manifolds homeomorphic to the ball are contractible, too. One can ask whether ''all'' contractible manifolds are homeomorphic to a ball. For dimensions 1 and 2, the answer is classical and it is "yes". In dimension 2, it follows, for example, from the Riemann mapping theorem. Dimension 3 presents the first counterexample: the Whitehead manifold. Construction Take a copy of S^3, the three-dimensional sphere. Now find a compact unknotted solid torus T_1 inside the sphere. (A solid torus is an ordinary three-dimensional doughnut, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Disk (mathematics)
In geometry, a disk (also spelled disc). is the region in a plane bounded by a circle. A disk is said to be ''closed'' if it contains the circle that constitutes its boundary, and ''open'' if it does not. For a radius, r, an open disk is usually denoted as D_r and a closed disk is \overline. However in the field of topology the closed disk is usually denoted as D^2 while the open disk is \operatorname D^2. Formulas In Cartesian coordinates, the ''open disk'' of center (a, b) and radius ''R'' is given by the formula :D=\ while the ''closed disk'' of the same center and radius is given by :\overline=\. The area of a closed or open disk of radius ''R'' is π''R''2 (see area of a disk). Properties The disk has circular symmetry. The open disk and the closed disk are not topologically equivalent (that is, they are not homeomorphic), as they have different topological properties from each other. For instance, every closed disk is compact whereas every open disk is not compact ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dogbone Space
In geometric topology, the dogbone space, constructed by , is a quotient space of three-dimensional Euclidean space \R^3 such that all inverse images of points are points or tame arcs, yet it is not homeomorphic to \R^3. The name "dogbone space" refers to a fanciful resemblance between some of the diagrams of genus 2 surfaces in R. H. Bing's paper and a dog bone. showed that the product of the dogbone space with \R^1 is homeomorphic to \R^4. Although the dogbone space is not a manifold, it is a generalized homological manifold and a homotopy manifold. See also * List of topologies * Whitehead manifold, a contractible 3-manifold not homeomorphic to \R^3. References * * *{{Citation , last1=Bing , first1=R. H. , authorlink = R. H. Bing , title=The cartesian product of a certain nonmanifold and a line is E4 , jstor=1970322 , mr=0107228 , year=1959 , journal=Annals of Mathematics The ''Annals of Mathematics'' is a mathematical journal published every two months by ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Casson Handle
In 4-dimensional topology, a branch of mathematics, a Casson handle is a 4-dimensional topological 2-handle constructed by an infinite procedure. They are named for Andrew Casson, who introduced them in about 1973. They were originally called "flexible handles" by Casson himself, and introduced the name "Casson handle" by which they are known today. In that work he showed that Casson handles are topological 2-handles, and used this to classify simply connected compact topological 4-manifolds. Motivation In the proof of the h-cobordism theorem, the following construction is used. Given a circle in the boundary of a manifold, we would often like to find a disk embedded in the manifold whose boundary is the given circle. If the manifold is simply connected then we can find a map from a disc to the manifold with boundary the given circle, and if the manifold is of dimension at least 5 then by putting this disc in "general position" it becomes an embedding. The number 5 appears for the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
David Gabai
David Gabai is an American mathematician and the Hughes-Rogers Professor of Mathematics at Princeton University. Focused on low-dimensional topology and hyperbolic geometry, he is a leading researcher in those subjects. Biography David Gabai received his B.S. in mathematics from MIT in 1976 and his Ph.D. in mathematics from Princeton University in 1980. Gabai completed his doctoral dissertation, titled "Foliations and genera of links", under the supervision of William Thurston. After positions at Harvard and University of Pennsylvania, Gabai spent most of the period of 1986–2001 at Caltech, and has been at Princeton since 2001. Gabai was the Chair of the Department of Mathematics at Princeton University from 2012 to 2019. Honours and awards In 2004, David Gabai was awarded the Oswald Veblen Prize in Geometry, given every three years by the American Mathematical Society. He was an invited speaker in the International Congress of Mathematicians 2010, Hyderabad on the top ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simply Connected At Infinity
In topology, a branch of mathematics, a topological space ''X'' is said to be simply connected at infinity if for any compact subset ''C'' of ''X'', there is a compact set ''D'' in ''X'' containing ''C'' so that the induced map : \pi_1(X-D) \to \pi_1(X-C) is the zero map. Intuitively, this is the property that loops far away from a small subspace of ''X'' can be collapsed, no matter how bad the small subspace is. The Whitehead manifold is an example of a 3-manifold that is contractible but not simply connected at infinity. Since this property is invariant under homeomorphism, this proves that the Whitehead manifold is not homeomorphic to R3. However, it is a theorem of John R. Stallings John Robert Stallings Jr. (July 22, 1935 – November 24, 2008) was a mathematician known for his seminal contributions to geometric group theory and Low-dimensional topology, 3-manifold topology. Stallings was a Professor Emeritus in the Departme ... that for n \geq 5, a contractible ''n''- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Morton Brown
Morton Brown (born August 12, 1931, in New York City, New York) is an American mathematician, who specializes in geometric topology. In 1958 Brown earned his Ph.D. from the University of Wisconsin-Madison under R. H. Bing. From 1960 to 1962 he was at the Institute for Advanced Study. Afterwards he became a professor at the University of Michigan at Ann Arbor. With Barry Mazur in 1965 he won the Oswald Veblen prize for their independent and nearly simultaneous proofs of the generalized Schoenflies hypothesis in geometric topology. Brown's short proof was elementary and fully general. Mazur's proof was also elementary, but it used a special assumption which was removed via later work of Morse. In 2012 he became a fellow of the American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through it ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Whitehead Theorem
In homotopy theory (a branch of mathematics), the Whitehead theorem states that if a continuous mapping ''f'' between CW complexes ''X'' and ''Y'' induces isomorphisms on all homotopy groups, then ''f'' is a homotopy equivalence. This result was proved by J. H. C. Whitehead in two landmark papers from 1949, and provides a justification for working with the concept of a CW complex that he introduced there. It is a model result of algebraic topology, in which the behavior of certain algebraic invariants (in this case, homotopy groups) determines a topological property of a mapping. Statement In more detail, let ''X'' and ''Y'' be topological spaces. Given a continuous mapping :f\colon X \to Y and a point ''x'' in ''X'', consider for any ''n'' ≥ 1 the induced homomorphism :f_*\colon \pi_n(X,x) \to \pi_n(Y,f(x)), where π''n''(''X'',''x'') denotes the ''n''-th homotopy group of ''X'' with base point ''x''. (For ''n'' = 0, π0(''X'') just means the set of path components o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hurewicz Theorem
In mathematics, the Hurewicz theorem is a basic result of algebraic topology, connecting homotopy theory with homology theory via a map known as the Hurewicz homomorphism. The theorem is named after Witold Hurewicz, and generalizes earlier results of Henri Poincaré. Statement of the theorems The Hurewicz theorems are a key link between homotopy groups and homology groups. Absolute version For any path-connected space ''X'' and positive integer ''n'' there exists a group homomorphism :h_* \colon \pi_n(X) \to H_n(X), called the Hurewicz homomorphism, from the ''n''-th homotopy group to the ''n''-th homology group (with integer coefficients). It is given in the following way: choose a canonical generator u_n \in H_n(S^n), then a homotopy class of maps f \in \pi_n(X) is taken to f_*(u_n) \in H_n(X). The Hurewicz theorem states cases in which the Hurewitz homomorphism is an isomorphism. * For n\ge 2, if ''X'' is (n-1)-connected (that is: \pi_i(X)= 0 for all ''i''2 there exists ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Winding Number
In mathematics, the winding number or winding index of a closed curve in the plane around a given point is an integer representing the total number of times that curve travels counterclockwise around the point, i.e., the curve's number of turns. The winding number depends on the orientation of the curve, and it is negative if the curve travels around the point clockwise. Winding numbers are fundamental objects of study in algebraic topology, and they play an important role in vector calculus, complex analysis, geometric topology, differential geometry, and physics (such as in string theory). Intuitive description Suppose we are given a closed, oriented curve in the ''xy'' plane. We can imagine the curve as the path of motion of some object, with the orientation indicating the direction in which the object moves. Then the winding number of the curve is equal to the total number of counterclockwise turns that the object makes around the origin. When counting the total nu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Null-homotopic
In topology, a branch of mathematics, two continuous function (topology), continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deformation being called a homotopy (, ; , ) between the two functions. A notable use of homotopy is the definition of homotopy groups and cohomotopy groups, important invariant (mathematics), invariants in algebraic topology. In practice, there are technical difficulties in using homotopies with certain spaces. Algebraic topologists work with compactly generated spaces, CW complexes, or Spectrum (homotopy theory), spectra. Formal definition Formally, a homotopy between two continuous function (topology), continuous functions ''f'' and ''g'' from a topological space ''X'' to a topological space ''Y'' is defined to be a continuous function H: X \times [0,1] \to Y from the product topology, product of the space ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Whitehead Link
In knot theory, the Whitehead link, named for J. H. C. Whitehead, is one of the most basic links. It can be drawn as an alternating link with five crossings, from the overlay of a circle and a figure-eight shaped loop. Structure A common way of describing this knot is formed by overlaying a figure-eight shaped loop with another circular loop surrounding the crossing of the figure-eight. The above-below relation between these two unknots is then set as an alternating link, with the consecutive crossings on each loop alternating between under and over. This drawing has five crossings, one of which is the self-crossing of the figure-eight curve, which does not count towards the linking number. Because the remaining crossings have equal numbers of under and over crossings on each loop, its linking number is 0. It is not isotopic to the unlink, but it is link homotopic to the unlink. Although this construction of the knot treats its two loops differently from each other, the two ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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