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 p₁,... weighted so that p_i has weight i (with p₀ = 1) and all the terms in K_n have weight n (so that K_n is a polynomial in p₁, ..., p_n). The sequence K_n is multiplicative if an identity

${\displaystyle \sum _{i}p_{i}z^{i}=\sum _{i}p'_{i}z^{i}\cdot \sum _{i}p''_{i}z^{i}}$

implies

${\displaystyle \sum _{i}K_{i}(p_{1},\ldots ,p_{i})z^{i}=\sum _{j}K_{j}(p'_{1},\ldots ,p'_{j})z^{j}\cdot \sum _{k}K_{k}(p''_{1},\ldots ,p''_{k})z^{k}}$

In other words, ${\displaystyle p\mapsto K(p)}$ is required to be an endomomorphism of the multiplicative monoid ${\displaystyle (A[[X]],\cdot )}$.

The power series

${\displaystyle \sum K_{n}(1,0,\ldots ,0)z^{n}}$

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

${\displaystyle \prod _{i=1}^{m}Q(\beta _{i}z)\ }$

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(p₁, ..., 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

${\displaystyle Q(z)={\frac {\sqrt {z}}{\tanh {\sqrt {z}}}}=1-\sum _{k=1}^{\infty }(-1)^{k}{\frac {2^{2k}}{(2k)!}}B_{k}z^{k}\ }$

where B_k is the k-th Bernoulli number. The multiplicative sequence with Q as characteristic power series is denoted L_j(p₁, ..., p_j).

The multiplicative sequence with characteristic power series

${\displaystyle Q(z)={\frac {2{\sqrt {z}}}{\sinh 2{\sqrt {z}}}}\ }$

is denoted A_j(p₁,...,p_j).

The multiplicative sequence with characteristic power series

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

is denoted T_j(p₁,...,p_j): these are the Todd polynomials.

Genus

Main article: Genus of a multiplicative sequence

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 ${\displaystyle {\frac {z}{1-\exp(-z)}}}$.

References

• Hirzebruch, Friedrich (1995) [1978]. Topological methods in algebraic geometry. Classics in Mathematics. Translation from the German and appendix one by R. L. E. Schwarzenberger. Appendix two by A. Borel (Reprint of the 2nd, corr. print. of the 3rd ed.). Berlin: Springer-Verlag. ISBN 3-540-58663-6. Zbl 0843.14009.