Self-Conjugate Cobordism and the Rectified Adams-Novikov Spectral Sequence

Abstract

This thesis considers the problem of computing the cobordism groups associated to manifolds with self-conjugate and double-real structures. In the first two chapters, we discuss the historical and mathematical background relevant to the problem, and highlight the parallels with our own arguments. In Chapter 3, we introduce a new spectral sequence, called the rectified Adams-Novikov spectral sequence, which we show converges to the relevant cobordism groups. This is a further generalization of both the classical Adams spectral sequence and the generalized Adams-Novikov spectral sequence. In particular, our spectral sequence relies on the resolution of the classical complex cobordism group as a comodule over two specific Hopf algebroids, one for each of self-conjugate and double-real cobordism. We give a complete computation of the algebraic structure of these Hopf algebroids, showing each is polynomial and giving a determination of the respective coproduct structures. Additional useful properties of these Hopf algebroids are also shown. In the case of self-conjugate cobordism, we show that our spectral sequence collapses, and we discuss the potential for collapse of the spectral sequence associated to double-real cobordism.

In Chapter 4, we discuss Sage computations which allow us to compute the self-conjugate and double-real cobordism groups to degree 16, which doubles the height of previous computations. We produce code which symbolically solves for the image of each polynomial generator in our given Hopf algebroids under their coproduct maps. We construct the reduced cobar complex and associated differentials coming from our spectral sequence, and compute the homology to recover the homotopy groups. Additional intermediate computations are also included. We conclude by including a list of tables containing the result of the computations given in Chapter 4.