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Ă‚-genus
In mathematics, a genus of a multiplicative sequence is a ring homomorphism from the ring of smooth compact manifolds up to the equivalence of bounding a smooth manifold with boundary (i.e., up to suitable cobordism) to another ring, usually the rational numbers, having the property that they are constructed from a sequence of polynomials in characteristic classes that arise as coefficients in formal power series with good multiplicative properties. Definition A genus \varphi assigns a number \Phi(X) to each manifold ''X'' such that # \Phi(X \sqcup Y) = \Phi(X) + \Phi(Y) (where \sqcup is the disjoint union); # \Phi(X \times Y) = \Phi(X)\Phi(Y); # \Phi(X) = 0 if ''X'' is the boundary of a manifold with boundary. The manifolds and manifolds with boundary may be required to have additional structure; for example, they might be oriented, spin, stably complex, and so on (see list of cobordism theories for many more examples). The value \Phi(X) is in some ring, often the ring of rati ...
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Cobordism
In mathematics, cobordism is a fundamental equivalence relation on the class of compact space, compact manifolds of the same dimension, set up using the concept of the boundary (topology), boundary (French ''wikt:bord#French, bord'', giving ''cobordism'') of a manifold. Two manifolds of the same dimension are ''cobordant'' if their disjoint union is the ''boundary'' of a compact manifold one dimension higher. The boundary of an (n+1)-dimensional manifold W is an n-dimensional manifold \partial W that is closed, i.e., with empty boundary. In general, a closed manifold need not be a boundary: cobordism theory is the study of the difference between all closed manifolds and those that are boundaries. The theory was originally developed by René Thom for smooth manifolds (i.e., differentiable), but there are now also versions for Piecewise linear manifold, piecewise linear and topological manifolds. A ''cobordism'' between manifolds M and N is a compact manifold W whose boundary is th ...
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Orientability
In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "anticlockwise". A space is orientable if such a consistent definition exists. In this case, there are two possible definitions, and a choice between them is an orientation of the space. Real vector spaces, Euclidean spaces, and spheres are orientable. A space is non-orientable if "clockwise" is changed into "counterclockwise" after running through some loops in it, and coming back to the starting point. This means that a geometric shape, such as , that moves continuously along such a loop is changed into its own mirror image . A Möbius strip is an example of a non-orientable space. Various equivalent formulations of orientability can be given, depending on the desired application and level of generality. Formulations applicable to general topological manifolds oft ...
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Hirzebruch–Riemann–Roch Theorem
In mathematics, the Hirzebruch–Riemann–Roch theorem, named after Friedrich Hirzebruch, Bernhard Riemann, and Gustav Roch, is Hirzebruch's 1954 result generalizing the classical Riemann–Roch theorem on Riemann surfaces to all complex algebraic varieties of higher dimensions. The result paved the way for the Grothendieck–Hirzebruch–Riemann–Roch theorem proved about three years later. Statement of Hirzebruch–Riemann–Roch theorem The Hirzebruch–Riemann–Roch theorem applies to any holomorphic vector bundle ''E'' on a compact complex manifold ''X'', to calculate the holomorphic Euler characteristic of ''E'' in sheaf cohomology, namely the alternating sum : \chi(X,E) = \sum_^ (-1)^ \dim_ H^(X,E) of the dimensions as complex vector spaces, where ''n'' is the complex dimension of ''X''. Hirzebruch's theorem states that χ(''X'', ''E'') is computable in terms of the Chern classes ''ck''(''E'') of ''E'', and the Todd classes \operatorname_(X) of the holomorphi ...
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Arithmetic Genus
In mathematics, the arithmetic genus of an algebraic variety is one of a few possible generalizations of the genus of an algebraic curve or Riemann surface. Projective varieties Let ''X'' be a projective scheme of dimension ''r'' over a field ''k'', the ''arithmetic genus'' p_a of ''X'' is defined asp_a(X)=(-1)^r (\chi(\mathcal_X)-1).Here \chi(\mathcal_X) is the Euler characteristic of the structure sheaf \mathcal_X. Complex projective manifolds The arithmetic genus of a complex projective manifold of dimension ''n'' can be defined as a combination of Hodge numbers, namely :p_a=\sum_^ (-1)^j h^. When ''n=1'', the formula becomes p_a=h^. According to the Hodge theorem, h^=h^. Consequently h^=h^1(X)/2=g, where ''g'' is the usual (topological) meaning of genus of a surface, so the definitions are compatible. When ''X'' is a compact Kähler manifold, applying ''h''''p'',''q'' = ''h''''q'',''p'' recovers the earlier definition for projective varieties. Kähler manifolds By u ...
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Smooth Structure
In mathematics, a smooth structure on a manifold allows for an unambiguous notion of smooth function. In particular, a smooth structure allows mathematical analysis to be performed on the manifold. Definition A smooth structure on a manifold M is a collection of smoothly equivalent smooth atlases. Here, a smooth atlas for a topological manifold M is an atlas for M such that each transition function is a smooth map, and two smooth atlases for M are smoothly equivalent provided their union is again a smooth atlas for M. This gives a natural equivalence relation on the set of smooth atlases. A smooth manifold is a topological manifold M together with a smooth structure on M. Maximal smooth atlases By taking the union of all atlases belonging to a smooth structure, we obtain a maximal smooth atlas. This atlas contains every chart that is compatible with the smooth structure. There is a natural one-to-one correspondence between smooth structures and maximal smooth atlases. ...
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PL Manifold
In mathematics, a piecewise linear manifold (PL manifold) is a topological manifold together with a piecewise linear structure on it. Such a structure can be defined by means of an atlas (topology), atlas, such that one can pass from chart (topology), chart to chart in it by piecewise linear functions. This is slightly stronger than the topological notion of a Triangulation (topology), triangulation. An isomorphism of PL manifolds is called a PL homeomorphism. Relation to other categories of manifolds PL, or more precisely PDIFF, sits between DIFF (the category of smooth manifolds) and TOP (the category of topological manifolds): it is categorically "better behaved" than DIFF — for example, the Generalized Poincaré conjecture is true in PL (with the possible exception of dimension 4, where it is equivalent to DIFF), but is false generally in DIFF — but is "worse behaved" than TOP, as elaborated in surgery theory. Smooth manifolds Smooth manifolds have canonical PL structu ...
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John Milnor
John Willard Milnor (born February 20, 1931) is an American mathematician known for his work in differential topology, algebraic K-theory and low-dimensional holomorphic dynamical systems. Milnor is a distinguished professor at Stony Brook University and the only mathematician to have won the Fields Medal, the Wolf Prize, the Abel Prize and all three Steele prizes. Early life and career Milnor was born on February 20, 1931, in Orange, New Jersey. His father was J. Willard Milnor, an engineer, and his mother was Emily Cox Milnor. As an undergraduate at Princeton University he was named a Putnam Fellow in 1949 and 1950 and also proved the Fáry–Milnor theorem when he was only 19 years old. Milnor graduated with an A.B. in mathematics in 1951 after completing a senior thesis, titled "Link groups", under the supervision of Ralph Fox. He remained at Princeton to pursue graduate studies and received his Ph.D. in mathematics in 1954 after completing a doctoral dissertation, t ...
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Hirzebruch Signature Theorem
In differential topology, an area of mathematics, the Hirzebruch signature theorem (sometimes called the Hirzebruch index theorem) is Friedrich Hirzebruch's 1954 result expressing the signature of a smooth closed oriented manifold by a linear combination of Pontryagin numbers called the L-genus. It was used in the proof of the Hirzebruch–Riemann–Roch theorem. Statement of the theorem The L-genus is the genus for the multiplicative sequence of polynomials associated to the characteristic power series : = \sum_ = 1 + - +\cdots . The first two of the resulting L-polynomials are: * L_1 = \tfrac13 p_1 * L_2 = \tfrac1(7p_2 - p_1^2) (for further ''L''-polynomials see McTague, Carl (2014"Computing Hirzebruch L-Polynomials" or ). By taking for the p_i the Pontryagin classes p_i(M) of the tangent bundle of a 4''n'' dimensional smooth closed oriented manifold M one obtains the L-classes of M. Hirzebruch showed that the n-th L-class of M evaluated on the fundamental class of M, /m ...
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Intersection Theory
In mathematics, intersection theory is one of the main branches of algebraic geometry, where it gives information about the intersection of two subvarieties of a given variety. The theory for varieties is older, with roots in Bézout's theorem on curves and elimination theory. On the other hand, the topological theory more quickly reached a definitive form. There is yet an ongoing development of intersection theory. Currently the main focus is on: virtual fundamental cycles, quantum intersection rings, Gromov–Witten theory and the extension of intersection theory from schemes to stacks. Topological intersection form For a connected oriented manifold M of dimension 2n the intersection form is defined on the n-th cohomology group (what is usually called the 'middle dimension') by the evaluation of the cup product on the fundamental class /math> in H_(M,\partial M). Stated precisely, there is a bilinear form :\lambda_M \colon H^n(M,\partial M) \times H^n(M,\partial M)\ ...
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Signature (topology)
In the field of topology, the signature is an integer invariant which is defined for an oriented manifold ''M'' of dimension divisible by four. This invariant of a manifold has been studied in detail, starting with Rokhlin's theorem for 4-manifolds, and Hirzebruch signature theorem. Definition Given a connected and oriented manifold ''M'' of dimension 4''k'', the cup product gives rise to a quadratic form ''Q'' on the 'middle' real cohomology group :H^(M,\mathbf). The basic identity for the cup product :\alpha^p \smile \beta^q = (-1)^(\beta^q \smile \alpha^p) shows that with ''p'' = ''q'' = 2''k'' the product is symmetric. It takes values in :H^(M,\mathbf). If we assume also that ''M'' is compact, Poincaré duality identifies this with :H_(M,\mathbf) which can be identified with \mathbf. Therefore the cup product, under these hypotheses, does give rise to a symmetric bilinear form on ''H''2''k''(''M'',''R''); and therefore to a quadratic form ''Q''. The form ''Q'' ...
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Fundamental Class
In mathematics, the fundamental class is a homology class 'M''associated to a connected orientable compact manifold of dimension ''n'', which corresponds to the generator of the homology group H_n(M,\partial M;\mathbf)\cong\mathbf . The fundamental class can be thought of as the orientation of the top-dimensional simplices of a suitable triangulation of the manifold. Definition Closed, orientable When ''M'' is a connected orientable closed manifold of dimension ''n'', the top homology group is infinite cyclic: H_n(M;\mathbf) \cong \mathbf, and an orientation is a choice of generator, a choice of isomorphism \mathbf \to H_n(M;\mathbf). The generator is called the fundamental class. If ''M'' is disconnected (but still orientable), a fundamental class is the direct sum of the fundamental classes for each connected component (corresponding to an orientation for each component). In relation with de Rham cohomology it represents ''integration over M''; namely for ''M'' a smooth ...
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Friedrich Hirzebruch
Friedrich Ernst Peter Hirzebruch ForMemRS (17 October 1927 – 27 May 2012) was a German mathematician, working in the fields of topology, complex manifolds and algebraic geometry, and a leading figure in his generation. He has been described as "the most important mathematician in Germany of the postwar period." Education Hirzebruch was born in Hamm, Westphalia in 1927. His father, of the same name, was a maths teacher. Hirzebruch studied at the University of Münster from 1945 to 1950, with one year at ETH Zürich. Career Hirzebruch then held a position at Erlangen, followed by the years 1952–54 at the Institute for Advanced Study in Princeton, New Jersey. After one year at Princeton University 1955–56, he was made a professor at the University of Bonn, where he remained, becoming director of the '' Max-Planck-Institut für Mathematik'' in 1981. More than 300 people gathered in celebration of his 80th birthday in Bonn in 2007. The Hirzebruch–Riemann–Roch theorem (195 ...
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