Chern numbers of Chern submanifolds

Abstract

We present a solution of the generalized Hirzebruch problem on the relations between the Chern numbers of a stably almost complex manifold and the Chern numbers of its virtual Chern submanifolds.

Miraculous cancellation and Pick’s theorem

“…where α(n) is the number of ones in the binary expansion of the number n. The general formula which expresses all relations between Chern characteristic numbers of virtual submanifolds was obtained in [13]. Various applications of miraculous cancellation type formulae to the questions of divisibility of topological invariants were found in [12,13,16,22]. In this paper we show that Cappell-Shaneson version [8] of Pick's theorem is, in fact, also a consequence of a cancellation formula for the Chern characteristic numbers of virtual submanifolds.…”

“…where α(n) is the number of ones in the binary expansion of the number n. The general formula which expresses all relations between Chern characteristic numbers of virtual submanifolds was obtained in [13].…”

Комплексные Кобордизмы И Формальные Группы

“…Важные результаты по общей проблеме Милнора-Хирцебруха получены К. Е. Фельдманом в [102]. В частности, доказано соотношение…”

“…где L и α(n) -такие же, как выше, и π k -операция, отвечающая характеристическому классу π k (M 4n ) = (−1) k (cf 2k (τ (M 4n ))) + cf 2k (τ (M 4n )), где cf 2k (·)характеристические классы Чженя-Коннера-Флойда. Отметим также следующий результат работы [102]: Пусть M 2n -почти комплексное многообразие, касательное расслоение которого допускает линейное подрасслоение. Тогда эйлерова характеристика многообразия M 2n является четной.…”

Functional Equations and the Generalised Elliptic Genus

“…In topology the German and Russian schools applied functional equations powerfully to formal group laws and genera [8,11,12,13,16,20,21,22,23]. They have arisen in the study of integrable systems in several different ways.…”