Degree Formulas

This chapter uses algebraic cobordism to establish some degree formulas. It presents δ‎ as a function from a class of smooth projective varieties over a field 𝑘 to some abelian group. Here, a degree formula for δ‎ is a formula relating δ‎(𝑋), δ‎(𝑌), and deg(𝑓) for any generically finite map 𝑓 : 𝑌 → 𝑋 in this class. The formula is usually δ‎(𝑌)=deg(𝑓)δ‎(𝑋). These degree formulas are used to prove that any norm variety over 𝑘 is a ν‎ n−1-variety. Using a standard result for the complex bordism ring 𝑀𝑈*, which uses a gluing argument of equivariant bordism theory, this chapter establishes Rost's DN (Degree and Norm Principle) Theorem for degrees, and defines the invariant η‎(𝑋/𝑆) of a pseudo-Galois cover.

Stiefel–Whitney numbers for singular varieties

This paper determines which Stiefel–Whitney numbers can be defined for singular varieties compatibly with small resolutions. First an upper bound is found by identifying theF2-vector space of Stiefel–Whitney numbers invariant under classical flops, equivalently by computing the quotient of the unoriented bordism ring by the total spaces ofRP3bundles. These Stiefel–Whitney numbers are then defined for any real projective normal Gorenstein variety and shown to be compatible with small resolutions whenever they exist. In light of Totaro's result [Tot00] equating the complex elliptic genus with complex bordism modulo flops, equivalently complex bordism modulo the total spaces of3bundles, these findings can be seen as hinting at a new elliptic genus, one for unoriented manifolds.

On a construction in bordism theory

In [2], R. Arthan and S. Bullett pose the problem of representing generators of the complex bordism ring MU* by manifolds which are totally normally split; i.e. whose stable normal bundles are split into a sum of complex line bundles. This has recently been solved by Ochanine and Schwartz [8] who use a mixture of J-theory and surgery theory to establish several results, including the following.

Realizing symplectic bordism classes

This is the second of two papers elaborating (4), and is concerned with the more geometrical aspects of the symplectic bordism ring in low dimensions. As such, it is a natural sequel to (2). However, the reader who does not care for the algebra therein need only consult (3), in which appears all the notation and prerequisites for this paper.

Some results in generalized homology, K-theory and bordism

This paper is designed to pave the way for the author's work on the symplectic bordism ring MSp*(12), (13). We here discuss all the notation used, and collect together all the theorems quoted therein. At the same time, I hope that some of the results presented here might be of interest in their own right. Our central theme is the study of the hurewicz map e: S*(X) → E*(X), both in general and certain specific cases.