arXiv subjects
Publications and source records attributed to William Richter.
We construct periodic families of Poincare complexes, partially solving a question of Hodgson that was posed in the proceedings of the 1982 Northwestern homotopy theory conference. We also construct infinite families of Poincare complexes whose top cell falls off after one suspension but which fail to embed in a sphere of codimension one. We give a homotopy theoretic description of the four-fold periodicity in knot cobordism.
Let K be a connected finite complex. This paper studies the problem of whether one can attach a cell to some iterated suspension S^j K so that the resulting space satisfies Poincare duality. When this is possible, we say that S^j K is a spine. We introduce the notion of quadratic self duality and show that if K is quadratically self dual, then S^j K is a spine whenever j is a suitable power of two. The powers of two come from the James periodicity theorem. We briefly explain how our main results, considered up to bordism, give a new interpretation of the four-fold periodicity of the surgery obstruction groups. We therefore obtain a relationship between James periodicity and the four-fold periodicity in L-theory.
We study the purely topological restrictions on allowed spin and statistics of topological solitons in nonlinear sigma models. Taking as space the connected $d$-manifold $X$, and considering nonlinear sigma models with the connected manifold $M$ as target space, topological solitons are given by elements of $pi_d(M)$. Any topological soliton $α\in π_d(M)$ determines a quotient $\Stat_n(X,α)$ of the group of framed braids on $X$, such that choices of allowed statistics for solitons of type $α$ are given by unitary representations of $\Stat_n(X,α)$ when $n$ solitons are present. In particular, when $M = S^2$, as in the $O(3)$ nonlinear sigma model with Hopf term, and $α\in π_2(S^2)$ is a generator, we compute that $\Stat_n(\R^2,α) = \Z$, while $\Stat_n(S^2,α) = \Z_{2n}$. It follows that phase $\exp(iθ)$ for interchanging two solitons of type $α$ on $S^2$ must satisfy the constraint $θ= kπ/n$, $k \in \Z$, when $n$ such solitons are present.