Multiplicative sequence: Difference between revisions

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In [[mathematics]], a '''multiplicative sequence''' or '''''m''-sequence''' is a [[sequence]] of [[polynomial]]s associated with a formal [[group theory(mathematics)|group]] structure. They have application in the [[cobordism|cobordism ring]] in [[algebraic topology]].

==Definition==

Let ''K''_''n'' be polynomials over a [[ring (mathematics)|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'' (soin thatparticular ''K''_''n'' is a polynomial in ''p''₁, ..., ''p''_''n''). The sequence ''K''_''n'' is ''multiplicative'' if anthe identitymap

:<math>\sum_i p_iK: z^i\sum_{n = 0}^\suminfty p'_i zq_nz^i n\cdotmapsto \sum_isum_{n=0}^\infty p''_iK_n(q_1,\cdots, q_n)z^in </math>

is an [[endomorphism]] of the multiplicative [[monoid]] <math>(A[x_1, x_2,\cdots][[z]],\cdot)</math>, where <math>q_n\in A[x_1, x_2,\cdots]</math>.

implies

The [[power series]]

:<math>\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 </math>

:<math>K(1+z) = \sum K_n(1,0,\ldots,0) z^n </math>

The power series

:<math>\sum K_n(1,0,\ldots,0) z^n </math>

is the ''characteristic power series'' of the&nbsp;''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''^{&thinsp;''j''} in the product

:<math> \prod_{i=1}^m Q(\beta_i z) \ </math>

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==Examples==

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

Consider the power series

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{{main|Genus of a multiplicative sequence}}

The ''genus'' of a multiplicative sequence is a [[ring homomorphism]], from the [[cobordism|cobordism ring]] of smooth oriented [[compact manifold]]s to another [[ring (mathematics)|ring]], usually the ring of [[rational number]]s.

For example, the [[Todd genus]] is associated to the Todd polynomials with characteristic power series <math>\frac{z}{1-\exp(-z)}</math>.

==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 | origyearorig-date=1978 | series=Classics in Mathematics | ___location=Berlin | publisher=[[Springer-Verlag]] | year=1995 | isbn=3-540-58663-6 }}

[[Category:Polynomials]]

[[Category:Topological methods of algebraic geometry]]

{{algebra-stub}}