Multiplicative gender

A multiplicative gender, also called Hirzebruch gender, is an object of mathematics . It is examined in the sub-areas of differential topology and algebraic topology . As a topological invariant , it can help to distinguish manifolds that are not equivalent to one another ( homeomorphic ).

In the late 1950s, Friedrich Hirzebruch developed a method in which he defined multiplicative genders using multiplicative sequences (also known as multiplicative sequences ). These genders, which can be defined by multiplicative sequences, include the Todd gender, the Geschlecht gender , the L gender and the class of the elliptical genders . These objects are central to the definition of the topological index for the Atiyah-Singer index set . Hirzebruch proved in his signature theorem for the L-gender that it agrees with the signature of the manifold .

Multiplicative gender

A multiplicative gender is a mapping which assigns an element from an integrity ring to every closed, oriented smooth manifold of the dimension , so that for every two such manifolds and the three conditions ${\displaystyle \phi }$ ${\displaystyle n}$ ${\displaystyle R}$ ${\displaystyle X^{n}}$ ${\displaystyle Y^{n}}$

• ${\displaystyle \phi (X^{n}\sqcup Y^{n})=\phi (X^{n})+\phi (Y^{n})}$, where the disjoint union is, ${\displaystyle \sqcup }$
• ${\displaystyle \phi (X^{n}\times Y^{n})=\phi (X^{n})\phi (Y^{n})}$
• ${\displaystyle \phi (X^{n})=0}$if there is a compact oriented manifold of dimension with ${\displaystyle W^{n+1}}$ ${\displaystyle n+1}$ ${\displaystyle \partial W^{n+1}=X^{n}}$

are fulfilled. A multiplicative sex can therefore (equivalent) as a ring homomorphism (which also includes the one-element considered) from Kobordismusring to be understood. Often the set of rational numbers is used as the integrity ring . ${\displaystyle \phi }$ ${\displaystyle R}$

Multiplicative sequence

Let be a formal power series with rational coefficients and constant term and be a positive integer. The formal power series is then symmetric . Hence there exist polynomials such that ${\displaystyle f\in \mathbb {Q} [[x]]}$ ${\displaystyle 1}$ ${\displaystyle n\in \mathbb {Z} _{+}}$ ${\displaystyle f(x_{1})\cdots f(x_{n})}$ ${\displaystyle F_{k}}$

${\displaystyle f(x_{1})\cdots f(x_{n})=1+F_{1}(\sigma _{1})+F_{2}(\sigma _{1},\sigma _{2})+F_{3}(\sigma _{1},\sigma _{2},\sigma _{3})+\ldots }$

holds, where

${\displaystyle \sigma _{k}(x_{1},\ldots ,x_{k}):=\sum _{i_{1}<\cdots <i_{k}}x_{i_{1}}\cdots x_{i_{k}}}$

denotes the k-th elementary symmetric polynomial . The sequence of polynomials is called multiplicative sequence or multiplicative sequence with respect to the formal power series . ${\displaystyle (F_{k})}$ ${\displaystyle f}$

Gender of a multiplicative sequence

In this section the gender of a manifold is defined in terms of a multiplicative sequence. This gender is a multiplicative gender in the above sense. The definition occurs separately according to smooth or complex manifolds. However, both definitions are similar.

For smooth manifolds

Let be an oriented smooth -dimensional manifold, its tangent bundle , which is a real vector bundle, and a multiplicative sequence to the formal power series . Then the multiplicative gender of is defined by ${\displaystyle X}$ ${\displaystyle n}$ ${\displaystyle TX}$ ${\displaystyle (F_{k})}$ ${\displaystyle f}$ ${\displaystyle X}$

${\displaystyle F(X):=\langle F_{k}(p_{1},\ldots ,p_{k}),[X]\rangle }$,

if is and otherwise through . The -th Pontryagin class denotes by , the fundamental class of and the natural pairing between homology and cohomology . ${\displaystyle n=4k}$ ${\displaystyle F(X)=0}$ ${\displaystyle p_{i}}$ ${\displaystyle i}$ ${\displaystyle TM}$ ${\displaystyle [X]}$ ${\displaystyle X}$ ${\displaystyle \langle \cdot ,\cdot \rangle }$

For complex manifolds

Let be an oriented complex manifold with , let its tangent bundle, which is a complex vector bundle , and be a multiplicative sequence to the formal power series . Then the multiplicative gender of is defined by ${\displaystyle X}$ ${\displaystyle \dim _{\mathbb {C} }X=n}$ ${\displaystyle TX}$ ${\displaystyle (F_{k})}$ ${\displaystyle f}$ ${\displaystyle X}$

${\displaystyle F(E):=\langle F_{k}(c_{1},\ldots ,c_{k}),[X]\rangle }$,

if is and otherwise through . The -th Chern class denotes by , the fundamental class of and the natural pairing between homology and cohomology. ${\displaystyle n=2k}$ ${\displaystyle F(E)=0}$ ${\displaystyle c_{i}}$ ${\displaystyle i}$ ${\displaystyle TX}$ ${\displaystyle [X]}$ ${\displaystyle X}$ ${\displaystyle \langle \cdot ,\cdot \rangle }$

Special multiplicative genders

In this section special, central multiplicative genders are listed.

Todd gender

→ Main article : Todd class

The one through the (formal) power series

${\displaystyle {\frac {z}{1-\exp(-z)}}=1+{\frac {1}{2}}z+\sum _{i=1}^{\infty }(-1)^{i+1}{\frac {B_{2i}}{(2i)!}}z^{2i}}$,

where the Bernoulli numbers are a defined multiplicative sequence , called the Todd sequence. The first terms of the sequence with coefficients in the Chern classes are: ${\displaystyle B_{2i}}$ ${\displaystyle (\operatorname {Td} _{i})}$

${\displaystyle {\begin{aligned}\operatorname {Td} _{0}&=1\\\operatorname {Td} _{1}(c_{1})&=c_{1}/2\\\operatorname {Td} _{2}(c_{1},c_{2})&=(c_{2}+c_{1}^{2})/12\\\operatorname {Td} _{3}(c_{1},c_{2},c_{3})&=(c_{1}c_{2})/24\,.\end{aligned}}}$

The total Todd class is then given by ${\displaystyle \mathbf {Td} _{\mathbb {C} }}$

${\displaystyle \mathbf {Td} _{\mathbb {C} }(E)=\sum _{i}\operatorname {Td} _{i}(c_{1},\cdots ,c_{i})}$.

For a compact complex manifold of the (real) dimension , the Todd gender is defined by ${\displaystyle X}$ ${\displaystyle 2n}$

${\displaystyle \operatorname {Td} (X):=\langle \operatorname {Td} _{n}(TX),[X]\rangle }$.

Â-sex

The one through the (formal) power series

${\displaystyle {\hat {a}}(x)={\frac {\frac {\sqrt {x}}{2}}{\sinh \left({\frac {\sqrt {x}}{2}}\right)}}=1-{\frac {1}{24}}x+{\frac {7}{2^{7}\cdot 3^{2}\cdot 5}}x^{2}+\cdots }$

defined multiplicative sequence , is called Â-sequence (pronounced: A-roof sequence). The first terms of the sequence with coefficients in the Pontryagin classes are: ${\displaystyle ({\hat {A}}_{i})}$

${\displaystyle {\begin{aligned}{\hat {A}}_{0}&=1\\{\hat {A}}_{1}(p_{1})&=-{\frac {p_{1}}{24}}\\{\hat {A}}_{2}(p_{1},p_{2})&={\frac {1}{2^{7}\cdot 3^{2}\cdot 5}}(-4p_{2}+7p_{1}^{2})\\{\hat {A}}_{3}(p_{1},p_{2},p_{3})&={\frac {1}{2^{10}\cdot 3^{3}\cdot 5\cdot 7}}(16p_{3}-44p_{2}p_{1}+31p_{1}^{3})\,.\end{aligned}}}$

The Â-class is then defined by ${\displaystyle \mathbf {\hat {A}} }$

${\displaystyle \mathbf {\hat {A}} (E)=\sum _{i}{\hat {A}}_{i}(p_{1},\cdots ,p_{i})}$.

The class is the real analog of the Todd class. For every oriented real vector bundle the following applies . The gender is just like the Todd gender previously defined as the class paired with the fundamental class. ${\displaystyle E}$ ${\displaystyle \mathbf {Td} _{\mathbb {C} }(E)=\mathbf {\hat {A}} (E)^{2}}$ ${\displaystyle {\hat {A}}}$

L-gender

The one through the (formal) power series

${\displaystyle L(x)={\frac {\sqrt {x}}{\tanh \left({\sqrt {x}}\right)}}=\sum _{k\geq 0}{2^{2k}B_{2k}z^{k} \over (2k)!}=1+{z \over 3}-{z^{2} \over 45}+\cdots }$,

where the Bernoulli numbers are a defined multiplicative sequence , called the sequence of L-polynomials. The first terms of the sequence with coefficients in the Pontryagin classes are: ${\displaystyle B_{2i}}$ ${\displaystyle (L_{i})}$

${\displaystyle {\begin{aligned}L_{0}&=1\\L_{1}(p_{1})&={\frac {p_{1}}{3}}\\L_{2}(p_{1},p_{2})&={\frac {1}{45}}(7p_{2}-p_{1}^{2})\\L_{3}(p_{1},p_{2},p_{3})&={\frac {1}{945}}(62p_{3}-13p_{1}p_{2}+2p_{1}^{3})\\L_{4}(p_{1},p_{2},p_{3},p_{4})&={\frac {1}{14175}}(381p_{4}-71p_{1}p_{3}-19p_{2}^{2}+22p_{1}^{2}p_{2}-3p_{1}^{4})\,.\end{aligned}}}$

For a compact complex manifold of dimension the L gender is also given by ${\displaystyle X}$ ${\displaystyle 4n}$

${\displaystyle L(X):=\langle L_{n}(TX),[X]\rangle }$.

Hirzebruch proved with the signature theorem that the L gender agrees with the signature of the manifold .

Elliptical gender

A multiplicative gender is called an elliptical gender if the formal power series is the differential equation ${\displaystyle Q(z)={\tfrac {z}{f(z)}}}$

${\displaystyle {f'}^{2}=1-2\delta f^{2}+\epsilon f^{4}}$

with constants and met. ${\displaystyle \delta }$ ${\displaystyle \epsilon }$

An explicit representation of is ${\displaystyle f}$

${\displaystyle f(z)={\frac {{\rm {sn}}\left(az,{\frac {\sqrt {\epsilon }}{{a}^{2}}}\right)}{a}}}$,

in which

${\displaystyle a={\sqrt {\delta +{\sqrt {{\delta }^{2}-\epsilon }}}}}$

and is the Jacobian elliptic function . So the logarithm of the multiplicative gender is the elliptic integral of the first kind ${\displaystyle \operatorname {sn} }$

${\displaystyle \log(\phi )(z)=\int _{0}^{z}{\frac {\mathrm {d} t}{\sqrt {1-2\delta f^{2}+\epsilon f^{4}}}}}$.

This was used in the first definition of the elliptical gender and therefore today also bears the attribute elliptical in the name. If or , then the corresponding elliptical gender is called degenerate. ${\displaystyle \delta ^{2}=\epsilon }$ ${\displaystyle \epsilon =0}$

For example , if you put and , you get the L-gender. The Geschlecht gender is obtained when one and . ${\displaystyle \delta =\epsilon =1}$ ${\displaystyle f(z)=\tanh(z)}$ ${\displaystyle \delta =-1/8,\ \epsilon =0}$ ${\displaystyle f(z)=2\sinh(z/2)}$

Web links

• The manifold atlas project: Formal group laws and genera
• Elliptic genera . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).

Individual evidence

1. ↑ Sergei Petrovich Novikov: Topics in Topology and Mathematical Physics . American Mathematical Soc., 1995, ISBN 978-0-8218-0455-1 , pp. 25 ( google.com ).
2. ↑ Ruedi Seiler, Volker Enss, Werner Müller : Geometry and Physics (Academy of Sciences in Berlin. Research reports). De Gruyter, 1997, ISBN 978-3110139440 , p. 170.
3. ↑ Matthias Kreck: An invariant for stable parallelized manifolds . Dissertation. ( Online )
4. ^ HB Lawson, M. Michelson: Spin Geometry . Princeton University Press, 1989, ISBN 978-0691085425 , pp. 228-229.
5. ^ Charles B. Thomas: Elliptic Cohomology (University Series in Mathematics) . Springer, 1999, ISBN 978-0-306-46097-5 , pp. 10 .
6. ^ HB Lawson, M. Michelson: Spin Geometry . Princeton University Press, 1989, ISBN 978-0691085425 , pp. 230-231.
7. ↑ Friedrich Hirzebruch : Topological methods in algebraic geometry ( Grundlehren der Mathematischen Wissenschaften 131). 2nd corrected printing of the 3rd edition. Springer, Berlin et al. 1978, ISBN 3-540-03525-7 , p. 77.
8. ↑ ^{a b} H. B. Lawson, M. Michelson: Spin Geometry . Princeton University Press, 1989, ISBN 978-0691085425 , p. 230.
9. ^ HB Lawson, M. Michelson: Spin Geometry . Princeton University Press, 1989, ISBN 978-0691085425 , pp. 231-232.
10. ↑ John W. Milnor, James D. Stasheff: Characteristic classes. Princeton, NJ, Princeton University Press, ISBN 0691081220 , 224.
11. ↑ S. Ochanine, "Sur les genres multiplicatifs définis par des intégrales elliptiques" Topology, 26 (1987) pp. 143–151 MR0895567 Zbl 0626.57014
12. ↑ Serge Ochanine, What is… an elliptic genus ?, Notices of the AMS, volume 56, number 6 (2009) ( online )