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In mathematics, a genus of a multiplicative sequence is a
ring homomorphism In ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function such that ''f'' is: :addition preser ...
from the
ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...
of smooth
compact manifold In mathematics, a closed manifold is a manifold without boundary that is compact. In comparison, an open manifold is a manifold without boundary that has only ''non-compact'' components. Examples The only connected one-dimensional example is ...
s up to the equivalence of bounding a smooth manifold with boundary (i.e., up to suitable cobordism) to another ring, usually the
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rat ...
s, having the property that they are constructed from a
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
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 rational numbers, though it can be other rings such as \Z/2\Z or the ring of modular forms. The conditions on \Phi can be rephrased as saying that \varphi is a ring homomorphism from the cobordism ring of manifolds (with additional structure) to another ring. Example: If \Phi(X) is the
signature A signature (; from la, signare, "to sign") is a handwritten (and often stylized) depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and intent. The writer of a ...
of the oriented manifold ''X'', then \Phi is a genus from oriented manifolds to the ring of integers.


The genus associated to a formal power series

A sequence of polynomials K_1, K_2,\ldots in variables p_1, p_2,\ldots is called multiplicative if :1 + p_1z + p_2z^2 + \cdots = (1 + q_1z + q_2z^2 + \cdots) (1 + r_1z + r_2z^2 + \cdots) implies that :\sum_j K_j(p_1, p_2, \ldots)z^j = \sum_j K_j (q_1, q_2, \ldots) z^j\sum_k K_k (r_1, r_2, \ldots)z^k If Q(z) is a formal power series in ''z'' with constant term 1, we can define a multiplicative sequence :K = 1+ K_1 + K_2 + \cdots by :K(p_1, p_2, p_3, \ldots) = Q(z_1)Q(z_2)Q(z_3)\cdots, where p_k is the ''k''th elementary symmetric function of the indeterminates z_i. (The variables p_k will often in practice be
Pontryagin class In mathematics, the Pontryagin classes, named after Lev Pontryagin, are certain characteristic classes of real vector bundles. The Pontryagin classes lie in cohomology groups with degrees a multiple of four. Definition Given a real vector bundl ...
es.) The genus \Phi of
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...
,
connected Connected may refer to: Film and television * ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular'' * '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film * ''Connected'' (2015 TV ...
,
smooth Smooth may refer to: Mathematics * Smooth function, a function that is infinitely differentiable; used in calculus and topology * Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions * Smooth algebrai ...
, oriented manifolds corresponding to ''Q'' is given by :\Phi(X) = K(p_1, p_2, p_3, \ldots) where the p_k are the
Pontryagin class In mathematics, the Pontryagin classes, named after Lev Pontryagin, are certain characteristic classes of real vector bundles. The Pontryagin classes lie in cohomology groups with degrees a multiple of four. Definition Given a real vector bundl ...
es of ''X''. The power series ''Q'' is called the characteristic power series of the genus \Phi. A theorem of René Thom, which states that the rationals tensored with the cobordism ring is a polynomial algebra in generators of degree 4''k'' for positive integers ''k'', implies that this gives a bijection between formal power series ''Q'' with rational coefficients and leading coefficient 1, and genera from oriented manifolds to the rational numbers.


L genus

The L genus is the genus of the formal power series : = \sum_ \frac = 1 + - + \cdots where the numbers B_ are the
Bernoulli number In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, ...
s. The first few values are: :\begin L_0 &= 1 \\ L_1 &= \tfrac13 p_1 \\ L_2 &= \tfrac1\left(7p_2 - p_1^2\right) \\ L_3 &= \tfrac1\left(62 p_3 - 13 p_1 p_2 + 2 p_1^3\right) \\ L_4 &= \tfrac1\left(381 p_4 - 71 p_1 p_3 - 19 p_2^2 + 22 p_1^2 p_2 - 3 p_1^4\right) \end (for further ''L''-polynomials see or ). Now let ''M'' be a closed smooth oriented manifold of dimension 4''n'' with Pontrjagin classes p_i = p_i(M).
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 ...
showed that the ''L'' genus of ''M'' in dimension 4''n'' evaluated on the
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 fundam ...
of M, denoted /math>, is equal to \sigma(M), the
signature A signature (; from la, signare, "to sign") is a handwritten (and often stylized) depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and intent. The writer of a ...
of ''M'' (i.e., the signature of the intersection form on the 2''n''th cohomology group of ''M''): :\sigma(M) = \langle L_n(p_1(M), \ldots, p_n(M)), rangle. This is now known as the
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 combi ...
(or sometimes the Hirzebruch index theorem). The fact that L_2 is always integral for a smooth manifold was used by
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 Univ ...
to give an example of an 8-dimensional
PL manifold PL, P.L., Pl, or .pl may refer to: Businesses and organizations Government and political * Partit Laburista, a Maltese political party * Liberal Party (Brazil, 2006), a Brazilian political party * Liberal Party (Moldova), a Moldovan political ...
with no
smooth structure In mathematics, a smooth structure on a manifold allows for an unambiguous notion of smooth function. In particular, a smooth structure allows one to perform mathematical analysis on the manifold. Definition A smooth structure on a manifold M is ...
. Pontryagin numbers can also be defined for PL manifolds, and Milnor showed that his PL manifold had a non-integral value of p_2, and so was not smoothable.


Application on K3 surfaces

Since projective K3 surfaces are smooth complex manifolds of dimension two, their only non-trivial
Pontryagin class In mathematics, the Pontryagin classes, named after Lev Pontryagin, are certain characteristic classes of real vector bundles. The Pontryagin classes lie in cohomology groups with degrees a multiple of four. Definition Given a real vector bundl ...
is p_1 in H^4(X). It can be computed as -48 using the tangent sequence and comparisons with complex chern classes. Since L_1 = -16, we have its signature. This can be used to compute its intersection form as a unimodular lattice since it has \operatorname\left(H^2(X)\right) = 22 , and using the classification of unimodular lattices.


Todd genus

The Todd genus is the genus of the formal power series :\frac = 1 + \fracz + \sum_^\infty (-1)^\fracz^ with B_ as before, Bernoulli numbers. The first few values are :\begin Td_0 &= 1 \\ Td_1 &= \frac1 c_1 \\ Td_2 &= \frac1 \left (c_2 + c_1^2 \right ) \\ Td_3 &= \frac1 c_1 c_2 \\ Td_4 &= \frac1 \left(-c_1^4 + 4 c_2 c_1^2 + 3c_2^2 + c_3 c_1 - c_4\right) \end The Todd genus has the particular property that it assigns the value 1 to all complex projective spaces (i.e. \mathrm_n(\mathbb^n) = 1), and this suffices to show that the Todd genus agrees with the arithmetic genus for algebraic varieties as the arithmetic genus is also 1 for complex projective spaces. This observation is a consequence of the
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 algebra ...
, and in fact is one of the key developments that led to the formulation of that theorem.


 genus

The  genus is the genus associated to the characteristic power series : Q(z) = \frac = 1 - \frac + \frac - \cdots (There is also an  genus which is less commonly used, associated to the characteristic series Q(16z).) The first few values are :\begin \hat_0 &= 1 \\ \hat_1 &= -\tfrac1p_1 \\ \hat_2 &= \tfrac1\left(-4p_2 + 7 p_1^2\right) \\ \hat_3 &= \tfrac1\left(-16p_3 + 44p_2 p_1 - 31 p_1^3\right) \\ \hat_4 &= \tfrac1\left(-192p_4 + 512 p_3 p_1 + 208p_2^2 - 904p_2 p_1^2 + 381p_1^4\right) \end The  genus of a spin manifold is an integer, and an even integer if the dimension is 4 mod 8 (which in dimension 4 implies Rochlin's theorem) – for general manifolds, the  genus is not always an integer. This was proven by
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 ...
and
Armand Borel Armand Borel (21 May 1923 – 11 August 2003) was a Swiss mathematician, born in La Chaux-de-Fonds, and was a permanent professor at the Institute for Advanced Study in Princeton, New Jersey, United States from 1957 to 1993. He worked in alg ...
; this result both motivated and was later explained by the Atiyah–Singer index theorem, which showed that the  genus of a spin manifold is equal to the index of its
Dirac operator In mathematics and quantum mechanics, a Dirac operator is a differential operator that is a formal square root, or half-iterate, of a second-order operator such as a Laplacian. The original case which concerned Paul Dirac was to factorise forma ...
. By combining this index result with a Weitzenbock formula for the Dirac Laplacian,
André Lichnerowicz André Lichnerowicz (January 21, 1915, Bourbon-l'Archambault – December 11, 1998, Paris) was a noted French differential geometer and mathematical physicist of Polish descent. He is considered the founder of modern Poisson geometry. Biograp ...
deduced that if a compact spin manifold admits a metric with positive scalar curvature, its  genus must vanish. This only gives an obstruction to positive scalar curvature when the dimension is a multiple of 4, but
Nigel Hitchin Nigel James Hitchin FRS (born 2 August 1946) is a British mathematician working in the fields of differential geometry, gauge theory, algebraic geometry, and mathematical physics. He is a Professor Emeritus of Mathematics at the University o ...
later discovered an analogous \Z_2-valued obstruction in dimensions 1 or 2 mod 8. These results are essentially sharp. Indeed, Mikhail Gromov, H. Blaine Lawson, and Stephan Stolz later proved that the  genus and Hitchin's \Z_2-valued analog are the only obstructions to the existence of positive-scalar-curvature metrics on simply-connected spin manifolds of dimension greater than or equal to 5.


Elliptic genus

A genus is called an elliptic genus if the power series Q(z) = z/f(z) satisfies the condition :^2 = 1 - 2\delta f^2 + \epsilon f^4 for constants \delta and \epsilon. (As usual, ''Q'' is the characteristic power series of the genus.) One explicit expression for ''f''(''z'') is :f(z) = \frac\operatorname\left( az, \frac \right) where :a = \sqrt and ''sn'' is the Jacobi elliptic function. Examples: *\delta = \epsilon = 1, f(z) = \tanh(z). This is the L-genus. *\delta = -\frac, \epsilon = 0, f(z) = 2\sinh\left(\fracz\right). This is the  genus. *\epsilon = \delta^2 , f(z) = \frac. This is a generalization of the L-genus. The first few values of such genera are: :\frac\delta p_1 :\frac \left \left (-4\delta^2 +18\epsilon \right )p_2+ \left (7\delta^2-9\epsilon \right )p_1^2\right /math> :\frac \left \left (16\delta^3 + 108\delta \epsilon \right )p_3 + \left (-44\delta^3 +18\delta \epsilon \right )p_2p_1 + \left (31\delta^3 -27\delta \epsilon \right )p_1^3\right /math> Example (elliptic genus for quaternionic projective plane) : :\begin \Phi_(HP^2) &= \int_\tfrac1\big -4\delta^2 +18\epsilon )p_2+(7\delta^2-9\epsilon )p_1^2\big\\ &= \int_\tfrac1\big -4\delta^2 +18\epsilon )(7u^2)+(7\delta^2-9\epsilon )(2u)^2\big\\ &= \int_ ^2 \epsilon \\ &= \epsilon \int_ ^2\\ &= \epsilon * 1 = \epsilon \end Example (elliptic genus for octonionic projective plane, or Cayley plane): :\begin \Phi_(OP^2) &= \int_\tfrac1 \left -192\delta^4 + 1728\delta^2\epsilon + 1512\epsilon^2)p_4 + (208\delta^4 - 1872\delta^2\epsilon + 1512\epsilon^2)p_2^2\right\\ &= \int_\tfrac1\big -192\delta^4 + 1728\delta^2\epsilon + 1512\epsilon^2)(39u^2) + (208\delta^4 - 1872\delta^2\epsilon + 1512\epsilon^2)(6u)^2\big\\ &= \int_\big \epsilon^2 u^2 \big\\ &= \epsilon^2\int_ \big u^2 \big\\ &= \epsilon^2* 1 = \epsilon^2 \\ &= \Phi_(HP^2) ^2 \end


Witten genus

The Witten genus is the genus associated to the characteristic power series :Q(z) = \frac = \exp\left(\sum_ \right) where σL is the
Weierstrass sigma function In mathematics, the Weierstrass functions are special functions of a complex variable that are auxiliary to the Weierstrass elliptic function. They are named for Karl Weierstrass. The relation between the sigma, zeta, and \wp functions is analog ...
for the lattice ''L'', and ''G'' is a multiple of an
Eisenstein series Eisenstein series, named after German mathematician Gotthold Eisenstein, are particular modular forms with infinite series expansions that may be written down directly. Originally defined for the modular group, Eisenstein series can be generaliz ...
. The Witten genus of a 4''k'' dimensional compact oriented smooth spin manifold with vanishing first Pontryagin class is a modular form of weight 2''k'', with integral Fourier coefficients.


See also

* Atiyah–Singer index theorem *
List of cohomology theories This is a list of some of the ordinary and generalized (or extraordinary) homology and cohomology theories in algebraic topology that are defined on the categories of CW complexes or spectra. For other sorts of homology theories see the links at ...


Notes


References

*Friedrich Hirzebruch'' Topological Methods in Algebraic Geometry'' Text of the original German version: http://hirzebruch.mpim-bonn.mpg.de/120/6/NeueTopologischeMethoden_2.Aufl.pdf * Friedrich Hirzebruch, Thomas Berger, Rainer Jung ''Manifolds and Modular Forms'' *Milnor, Stasheff, ''Characteristic classes'', * *{{springer, title=Elliptic genera, id=p/e110070 Topological methods of algebraic geometry Complex manifolds