arXiv subjects
Publications and source records attributed to William Browder.
In this article we prove that each integral cycle $T$ in an oriented Riemannian manifold $\mathcal{M}$ can be approximated in flat norm by an integral cycle in the same homology class which is a smooth submanifold $\Sigma$ of nearly the same area, up to a singular set of codimension 5. Moreover, if the homology class $\tau$ is representable by a smooth submanifold, then $\Sigma$ can be chosen free of singularities.
A space is defined to be "$n$-spheroidal" if it has the homotopy type of an $n$-dimensional CW-complex $X$ with $H_{n}(X, \mathbb{Z})$ not zero and finitely generated. A group $G$ is called "$n$-spheroidal" if its classifying space $K(G,1)$ is $n$-spheroidal. Examples include fundamental groups of compact manifold $K(G,1)$'s. Moreover, the class of groups $G$ which are $n$-spheroidal for some $n$, is closed under products, free products, and group extensions. If $Y$ is a space with $π_{1}(Y)$ $n$-spheroidal, and if $H_{k}(Y;\mathbb{F}_{p})$ is non-zero and finitely generated, and if $H_{i}(Y;\mathbb{F}_{p}) = 0$ for $i>k$, then $H_{n+k}(\overline{Y};\mathbb{F}_{p}) \neq 0$ for $\overline{Y}$ a finite sheeted covering space of $Y$. Hence, dim$(Y) \geq n+k$. Thus, it follows that if dim$(Y) < n$, and if $H_{k}(Y;\mathbb{F}_{p}) \neq 0$ and $H_{i}(Y;\mathbb{F}_{p}) = 0$ for $i>k>0$, then $H_{k}(Y;\mathbb{F}_{p})$ is not finitely generated. Similar results follow for $Y\subset K(G,1)$.
Studies the cohomology of p-central, powerful, p-groups with a certain extension property. These groups are naturally associated to Lie algebras. The paper develops a machinery that calculates the first few terms of the Bockstein spectral sequence in terms of the associated Lie algebras. This is then used to obtain results on the integral cohomology of these groups.
Studies among other things, the question of whether a Lie algebra over Z/(p^k)Z lifts to one over Z/(p^(k+1))Z. An obstruction theory is developed and examples of Fp-Lie algebras which don't lift to Lie algebras over Z/p^2Z are discussed. An example of an application of the result: A Fp-Lie algebra L with H^3(L, ad)=0 will lift to a p-adic Lie algebra.