Multiplicative Sequence

In mathematics, a multiplicative sequence or ''m''-sequence is a sequence of polynomials associated with a formal group structure. They have application in the cobordism ring in algebraic topology.

Definition

Let ''K''_''n'' be polynomials over a ring ''A'' in indeterminates ''p''₁, ... weighted so that ''p''_''i'' has weight ''i'' (with ''p''₀ = 1) and all the terms in ''K''_''n'' have weight ''n'' (in particular ''K''_''n'' is a polynomial in ''p''₁, ..., ''p''_''n''). The sequence ''K''_''n'' is ''multiplicative'' if the map

:$K: \sum_^\infty q_nz^n\mapsto \sum_^\infty K_n(q_1,\cdots, q_n)z^n$

is an endomorphism of the multiplicative monoid (A _1, x_2,\cdots ,\cdot), where q_n\in A _1, x_2,\cdots/math>.

The power series

:$K(1+z) = \sum K_n(1,0,\ldots,0) z^n$

is the ''characteristic power series'' of the ''K''_''n''. A multiplicative sequence is determined by its characteristic power series ''Q''(''z''), and every power series with constant term 1 gives rise to a multiplicative sequence.

To recover a multiplicative sequence from a characteristic power series ''Q''(''z'') we consider the coefficient of ''z''^''j'' in the product

:$\prod_^m Q(\beta_i z) \$

for any ''m'' > ''j''. This is symmetric in the ''β''_''i'' and homogeneous of weight ''j'': so can be expressed as a polynomial ''K''_''j''(''p''₁, ..., ''p''_''j'') in the elementary symmetric functions ''p'' of the ''β''. Then ''K''_''j'' defines a multiplicative sequence.

Examples

As an example, the sequence ''K''_''n'' = ''p''_''n'' is multiplicative and has characteristic power series 1 + ''z''.

Consider the power series

:$Q(z) = \frac = 1 - \sum_^\infty (-1)^k \frac B_k z^k \$

where ''B''_''k'' is the ''k''-th Bernoulli number. The multiplicative sequence with ''Q'' as characteristic power series is denoted ''L''_''j''(''p''₁, ..., ''p''_''j'').

The multiplicative sequence with characteristic power series

:$Q(z) = \frac \$

is denoted ''A''_''j''(''p''₁,...,''p''_''j'').

The multiplicative sequence with characteristic power series

:$Q(z) = \frac = 1 + \frac - \sum_^\infty (-1)^k \frac z^ \$

is denoted ''T''_''j''(''p''₁,...,''p''_''j''): these are the ''Todd polynomials''.

Genus

The ''genus'' of a multiplicative sequence is a ring homomorphism, from the cobordism ring of smooth oriented compact manifolds to another ring, usually the ring of rational numbers.

For example, the Todd genus is associated to the Todd polynomials with characteristic power series $\frac$.

References

* {{cite book , zbl=0843.14009 , last=Hirzebruch , first=Friedrich , authorlink=Friedrich Hirzebruch , title=Topological methods in algebraic geometry , others=Translation from the German and appendix one by R. L. E. Schwarzenberger. Appendix two by A. Borel , edition=Reprint of the 2nd, corr. print. of the 3rd , origyear=1978 , series=Classics in Mathematics , location=Berlin , publisher=Springer-Verlag , year=1995 , isbn=3-540-58663-6

Polynomials
Topological methods of algebraic geometry