Author
Gries, Daniel Joseph

Year
2000

Advisor
Mislin, Guido

Abstract

Let $\Gamma_{g,k}^n$ be the mapping class group of an oriented surface $S_{g,k}^n$ of genus $g$, with $k$ boundary components, and $n$ punctures. We set $\Gamma_g=\Gamma^0_{g,0}$ and $\Gamma^n=\Gamma^n_{0,0}$ . The focus of this dissertation is the \emph{hyperelliptic} mapping class group, denoted by $\Delta_g$, which is defined as the normalizer of a special order two element in $\Gamma_g$. It is a particularly nice subgroup of $\Gamma_g$ to study, as it is a quotient of Artin's braid group $B_{2g+2}$, which leads to a graphical interpretation of $\Delta_g$ in terms of braids, which we can manipulate to obtain information about $\Delta_g$.

The dissertation is divided into four chapters; the first presents the necessary background information, the second discusses the relationship between braid groups and $\Delta_g$, the third concerns the Yagita invariant of $\Delta_g$ at the prime 2, and the last presents some cohomology computations.

The introduction is divided into five sections. The first section provides some background on Farrell cohomology. The next two sections define the mapping class groups and the Yagita invariant. The fourth section provides information on Riemann surfaces, which is needed to study $\Delta_g$ from a topological viewpoint. For an algebraic description, the fifth section discusses known presentations of mapping class groups.

Viewing the presentations of $\Delta_g$ and Artin's braid group $B_n$, one can readily see a classical map $B_{2g+2} \twoheadrightarrow \Delta_g$. In Chapter 2 we describe the kernel of this map, and then get information about torsion and dihedral subgroups in $\Delta_g$ by studying elements of $B_{2g+2}$. This graphical tool is also used in Chapter 4 when we need some information about $\Gamma^4$.

The cohomology of $\Delta_g$ is $p$-periodic for all odd primes $p$, however it is never 2-periodic. This motivates a study in Chapter 3 of the Yagita invariant of $\Delta_g$ at the prime 2, which is a sort of generalized $2$-period. We obtain complete results for every even genus $g$, and partial results for odd $g$. We also determine the 2-rank of $\Delta_g$.

Finally, in the last chapter we gather information about $\Delta_g$ from the topological viewpoint, toward the calculation of the Farrell cohomology of $\Delta_g$. We then combine this information along with the information and techniques of Chapter 2 to determine the $p$-part of the Farrell cohomology of $\Delta_g$ for $g=(p-1)/2$ and $g=p-1$, which are the first two cases of $\Delta_g$ containing $p$-torsion.