arXiv subjects

Oliver H. Wang

Publications and source records attributed to Oliver H. Wang.

5 recordsLinked to original sources

arXiv2026

The Rational Homotopy of Stable $C_p$-Smoothings

Smooth structures on high dimensional manifolds are classified by maps to the infinite loop space $TOP/O$. The homotopy groups of this space are known to be finite. Given a compact Lie group $G$, this space can be regarded as an equivariant infinite loop space and equivariant maps from a locally linear, high dimensional $G$-manifold to $TOP/O$ classify stable $G$-smoothings. We compute the equivariant homotopy groups $π_V^{C_p}TOP/O\otimes\mathbb{Q}$ where $C_p$ denotes the cyclic group of order $p$. By applying our methods to the group $C_4$, we prove a Chern class analogue of Novikov's theorem on rational Pontryagin classes.

math.AT↗

arXiv2025

The Whitehead group and stably trivial $G$-smoothings

A closed manifold $M$ of dimension at least $5$ has only finitely many smooth structures. Moreover, smooth structures of $M$ are in bijection with smooth structures of $M\times\mathbb{R}$. Both of these statements are false equivariantly. In this paper, we use controlled $h$-cobordisms to construct infinitely many $G$-smoothings of a $G$-manifold $X$. Moreover, these $G$-smoothings are isotopic after taking a product with $\mathbb{R}$.

math.GT↗

arXiv2024

Torus bundles over lens spaces

Let $p$ be an odd prime and let $ρ:\mathbb{Z}/p\rightarrow\operatorname{GL}_n(\mathbb{Z})$ be an action of $\mathbb{Z}/p$ on a lattice and let $Γ:=\mathbb{Z}^n\rtimes_ρ\mathbb{Z}/p$ be the corresponding semidirect product. The torus bundle $M:=T^n_ρ\times_{\mathbb{Z}/p}S^{\ell}$ over the lens space $S^{\ell}/\mathbb{Z}/p$ has fundamental group $Γ$. When $\mathbb{Z}/p$ fixes only the origin of $\mathbb{Z}^n$, Davis and Lück \cite{DavisLuckTorusBundles} compute the $L$-groups $L^{\langle j\rangle}_m(\mathbb{Z}[Γ])$ and the structure set $\mathcal{S}^{geo,s}(M)$. In this paper, we extend these computations to all actions of $\mathbb{Z}/p$ on $\mathbb{Z}^n$. In particular, we compute $L^{\langle j\rangle}_m(\mathbb{Z}[Γ])$ and $\mathcal{S}^{geo,s}(M)$ in a case where $\underline{E}Γ$ has a non-discrete singular set.

math.GT↗

arXiv2023

Chern class obstructions to smooth equivariant rigidity

By work of Kirby-Siebenmann \cite{KirbySiebenmann} and Kervaire-Milnor \cite{KervaireMilnor}, there are only finitely many smooth manifolds homeomorphic to a given closed topological manifold. A construction involving Whitehead torsion shows this is not the case equivariantly for smooth finite group actions on a product $M\times I$ (see \cite[p. 262-266]{BrowderHsiangProblem}). When $2$ has odd order in $\left(\mathbb{Z}/p\mathbb{Z}\right)^\times$, Schultz \cite{SchultzSpherelike} uses a different method involving the Atiyah-Singer index theorem and computations of Ewing \cite{EwingSpheresAsFPSets} to show that there are infinitely many equivariant smooth structures for certain actions of $G=\mathbb{Z}/p\mathbb{Z}$ on even dimensional spheres with fixed point set $S^2$. These examples are constructed by finding infinitely many $G$-vector bundles over $S^2$ with vanishing Atiyah-Singer class and using these vector bundles to replace the normal bundle of $S^2\subseteq S^{2n}$. We analyze when a manifold supports infinitely many $G$-vector bundles with vanishing Atiyah-Singer class and show that Schultz's examples of exotic equivariant manifolds can be extended to much greater generality. As a consequence, we see that, for infinitely many primes $p$, there are infinitely many stable $G$-smoothings of a smooth $G$-manifold in the sense of Lashof \cite{LashofStableGSmoothing} whenever the fixed set has nonzero second rational cohomology.

math.GT↗

arXiv2020

A spectral sequence for Dehn fillings

We study how the cohomology of a type $F_\infty$ relatively hyperbolic group pair $(G,\mathcal{P})$ changes under Dehn fillings (i.e. quotients of group pairs). For sufficiently long Dehn fillings where the quotient pair $(\bar{G},\bar{\mathcal{P}})$ is of type $F_\infty$, we show that there is a spectral sequence relating the cohomology groups $H^i(G,\mathcal{P};\mathbb{Z} G)$ and $H^i\left(\bar{G},\bar{\mathcal{P}};\mathbb{Z}\bar{G}\right)$. As a consequence, we show that essential cohomological dimension does not increase under these Dehn fillings.

math.GR↗