Simple homotopy types of even dimensional manifolds
Csaba Nagy, John Nicholson, Mark Powell
TL;DR
The paper constructs infinite families of even-dimensional closed manifolds that are homotopy equivalent yet not simple homotopy equivalent, using a framework that ties simple homotopy manifold sets to Whitehead torsion, the surgery obstruction map, and homotopy automorphisms. Focusing on manifolds of the form $M^n_m=S^1\times L$ (with lens-space $L$ and fundamental group $G=C_{\infty}\times C_m$), the authors reduce the classification problem to the action of $\mathrm{hAut}(M)$ on $\mathrm{Wh}(G,w)$ and the corresponding Tate cohomology groups, together with surjectivity results for the Ranicki–Wall exact sequences. A central technical achievement is the detailed analysis of the involution on $\widetilde{K}_0(\mathbb{Z}C_m)$ via the Bass–Heller–Swan decomposition and cyclotomic class groups, yielding sharp finiteness, infinitude, and asymptotic growth results for the size of simple-homotopy manifold sets and their $h$-cobordism refinements. The paper integrates algebraic K- and L-theory, representation theory, and cyclotomic number theory to obtain explicit arithmetic criteria (involving class numbers $h_m$, $h_m^{-}$, and related invariants) that govern the simple homotopy classification, providing concrete, computable invariants for the even-dimensional case that parallels classical odd-dimensional lens-space phenomena. The results extend the scope of simple-homotopy distinctions beyond odd dimensions and furnish a comprehensive program connecting manifold topology with deep algebraic number-theoretic invariants.
Abstract
Given a closed $n$-manifold, we consider the set of simple homotopy types of $n$-manifolds within its homotopy type, called its simple homotopy manifold set. We characterise it in terms of algebraic K-theory, the surgery obstruction map, and the homotopy automorphisms of the manifold. We use this to construct the first examples, for all $n \ge 4$ even, of closed $n$-manifolds that are homotopy equivalent but not simple homotopy equivalent. In fact, we construct infinite families of manifolds that are all homotopy equivalent but pairwise not simple homotopy equivalent, and our examples can be taken to be smooth for $n \geq 6$. Our examples are homotopy equivalent to the product of a circle and a lens space. We analyse the simple homotopy manifold sets of these manifolds, determining exactly when they are trivial, finite, or infinite, and investigating their asymptotic behaviour. The proofs involve integral representation theory and class numbers of cyclotomic fields. We also compare with the relation of $h$-cobordism, and produce similar detailed quantitative descriptions of the manifold sets that arise.