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Normal Invariants
In mathematics, a normal map is a concept in geometric topology due to William Browder which is of fundamental importance in surgery theory. Given a Poincaré complex ''X'' (more geometrically a Poincaré space), a normal map on ''X'' endows the space, roughly speaking, with some of the homotopy-theoretic global structure of a closed manifold. In particular, ''X'' has a good candidate for a stable normal bundle and a Thom collapse map, which is equivalent to there being a map from a manifold ''M'' to ''X'' matching the fundamental classes and preserving normal bundle information. If the dimension of ''X'' is \ge 5 there is then only the algebraic topology surgery obstruction due to C. T. C. Wall to ''X'' actually being homotopy equivalent to a closed manifold. Normal maps also apply to the study of the uniqueness of manifold structures within a homotopy type, which was pioneered by Sergei Novikov. The cobordism classes of normal maps on ''X'' are called normal invariants. Dependi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometric Topology
In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another. History Geometric topology as an area distinct from algebraic topology may be said to have originated in the 1935 classification of lens spaces by Reidemeister torsion, which required distinguishing spaces that are homotopy equivalent but not homeomorphic. This was the origin of ''simple'' homotopy theory. The use of the term geometric topology to describe these seems to have originated rather recently. Differences between low-dimensional and high-dimensional topology Manifolds differ radically in behavior in high and low dimension. High-dimensional topology refers to manifolds of dimension 5 and above, or in relative terms, embeddings in codimension 3 and above. Low-dimensional topology is concerned with questions in dimensions up to 4, or embeddings in codimension up to 2. Dimension 4 is special, in that in some respects (topologica ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poincaré Duality
In mathematics, the Poincaré duality theorem, named after Henri Poincaré, is a basic result on the structure of the homology and cohomology groups of manifolds. It states that if ''M'' is an ''n''-dimensional oriented closed manifold (compact and without boundary), then the ''k''th cohomology group of ''M'' is isomorphic to the (n-k)th homology group of ''M'', for all integers ''k'' :H^k(M) \cong H_(M). Poincaré duality holds for any coefficient ring, so long as one has taken an orientation with respect to that coefficient ring; in particular, since every manifold has a unique orientation mod 2, Poincaré duality holds mod 2 without any assumption of orientation. History A form of Poincaré duality was first stated, without proof, by Henri Poincaré in 1893. It was stated in terms of Betti numbers: The ''k''th and (n-k)th Betti numbers of a closed (i.e., compact and without boundary) orientable ''n''-manifold are equal. The ''cohomology'' concept was at that time about 40 y ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jim Stasheff
James Dillon Stasheff (born January 15, 1936, New York City) is an American mathematician, a professor emeritus of mathematics at the University of North Carolina at Chapel Hill. He works in algebraic topology and algebra as well as their applications to physics. Biography Stasheff did his undergraduate studies in mathematics at the University of Michigan, graduating in 1956. Stasheff then began his graduate studies at Princeton University; his notes for a 1957 course by John Milnor on characteristic classes first appeared in mimeographed form and later in 1974 in revised form book with Stasheff as a co-author. After his second year at Princeton, he moved to Oxford University on a Marshall Scholarship. Two years later in 1961, with a pregnant wife, needing an Oxford degree to get reimbursed for his return trip to the US, and yet still feeling attached to Princeton, he split his thesis into two parts (one topological, the other algebraic) and earned two doctorates, a D.Phil. from Ox ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Samuel Gitler Hammer
Samuel Carlos Gitler Hammer (July 14, 1933 – September 9, 2014) , retrieved 2012-05-19.Member biography , Colegio Nacional, retrieved 2012-05-19. was a Mexican mathematician. He was an expert in and is known for the [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pontrjagin-Thom Construction
In mathematics, the Thom space, Thom complex, or Pontryagin–Thom construction (named after René Thom and Lev Pontryagin) of algebraic topology and differential topology is a topological space associated to a vector bundle, over any paracompact space. Construction of the Thom space One way to construct this space is as follows. Let :p: E \to B be a rank ''n'' real number, real vector bundle over the paracompact space ''B''. Then for each point ''b'' in ''B'', the Fiber (mathematics)#Fiber in naive set theory, fiber E_b is an n-dimensional real vector space. Choose an orthogonal structure on E, a smoothly varying inner product on the fibers; we can do this using partitions of unity. Let D(E) be the unit ball bundle with respect to our orthogonal structure, and let S(E) be the unit sphere bundle, then the Thom space T(E) is the quotient T(E) := D(E)/S(E) of topological spaces. T(E) is a pointed space with the image of S(E) in the quotient as basepoint. If ''B'' is compact, then T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hassler Whitney
Hassler Whitney (March 23, 1907 – May 10, 1989) was an American mathematician. He was one of the founders of singularity theory, and did foundational work in manifolds, embeddings, immersions, characteristic classes, and geometric integration theory. Biography Life Hassler Whitney was born on March 23, 1907, in New York City, where his father Edward Baldwin Whitney was the First District New York Supreme Court judge. His mother, A. Josepha Newcomb Whitney, was an artist and active in politics. He was the paternal nephew of Connecticut Governor and Chief Justice Simeon Eben Baldwin, his paternal grandfather was William Dwight Whitney, professor of Ancient Languages at Yale University, linguist and Sanskrit scholar. Whitney was the great-grandson of Connecticut Governor and US Senator Roger Sherman Baldwin, and the great-great-grandson of American founding father Roger Sherman. His maternal grandparents were astronomer and mathematician Simon Newcomb (1835-1909), a Steeves desce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Immersion (mathematics)
In mathematics, an immersion is a differentiable function between differentiable manifolds whose differential (or pushforward) is everywhere injective. Explicitly, is an immersion if :D_pf : T_p M \to T_N\, is an injective function at every point ''p'' of ''M'' (where ''TpX'' denotes the tangent space of a manifold ''X'' at a point ''p'' in ''X''). Equivalently, ''f'' is an immersion if its derivative has constant rank equal to the dimension of ''M'': :\operatorname\,D_p f = \dim M. The function ''f'' itself need not be injective, only its derivative must be. A related concept is that of an embedding. A smooth embedding is an injective immersion that is also a topological embedding, so that ''M'' is diffeomorphic to its image in ''N''. An immersion is precisely a local embedding – that is, for any point there is a neighbourhood, , of ''x'' such that is an embedding, and conversely a local embedding is an immersion. For infinite dimensional manifolds, this is sometimes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Embedding
In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup. When some object X is said to be embedded in another object Y, the embedding is given by some injective and structure-preserving map f:X\rightarrow Y. The precise meaning of "structure-preserving" depends on the kind of mathematical structure of which X and Y are instances. In the terminology of category theory, a structure-preserving map is called a morphism. The fact that a map f:X\rightarrow Y is an embedding is often indicated by the use of a "hooked arrow" (); thus: f : X \hookrightarrow Y. (On the other hand, this notation is sometimes reserved for inclusion maps.) Given X and Y, several different embeddings of X in Y may be possible. In many cases of interest there is a standard (or "canonical") embedding, like those of the natural numbers in the integers, the integers in the rational numbers, the rational n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Whitehead's 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 of '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thom
The surname Thom is of Scottish origin, from the city of Aberdeen, Aberdeenshire and Angus, and is a sept of the Clan MacThomas. Thom is also a first name variant of the abbreviation "Tom" of "Thomas" that holds the "h". People with the surname * Alexander Thom (other), multiple people * Andreas Thom (b. 1965), former German football player * Bing Thom (b. 1940), Canadian architect * Cameron E. Thom (1825–1915), early settler in California, Confederate officer and lawyer * Charles Thom (1872–1956), US microbiologist and mycologist with the standard author abbreviation "Thom" * Cristy Thom (b. 1971), American model, actress and artist * Graeme Thom (born 1967), Zimbabwean cricketer * H. B. Thom (c. 1905–1983), South African rector and Chancellor of the Stellenbosch University * James Thom (other), multiple people * Jess Thom (b. 1980), English comedian * John Thom (soldier) (1891–1941), British lieutenant-colonel, judge and politician * John Hamilton Thom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tangent Bundle
In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of manifold the tangent spaces and have no common vector. This is graphically illustrated in the accompanying picture for tangent bundle of circle , see tangent bundle#Examples, Examples section: all tangents to a circle lie in the plane of the circle. In order to make them disjoint it is necessary to align them in a plane perpendicular to the plane of the circle. of the tangent spaces of M . That is, : \begin TM &= \bigsqcup_ T_xM \\ &= \bigcup_ \left\ \times T_xM \\ &= \bigcup_ \left\ \\ &= \left\ \end where T_x M denotes the tangent space to M at the point x . So, an element of TM can be thought of as a ordered pair, pair (x,v), where x is a point in M and v is a tangent vector to M at x . There i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
