Tse Leung So | Searcharxiv

arXiv2018

Homotopy types of gauge groups over non-simply-connected closed 4-manifolds

Let $G$ be a simply-connected simple compact Lie group and let $M$ be an orientable smooth closed 4-manifold. In this paper we calculate the homotopy type of the suspension of $M$ and the homotopy types of the gauge groups of principal $G$-bundles over $M$ when $π_1(M)$ is: (1)~$\mathbb{Z}^{*m}$, (2)~$\mathbb{Z}/p^r\mathbb{Z}$, or (3)~$\mathbb{Z}^{*m}*(*^n_{j=1}\mathbb{Z}/p_j^{r_j}\mathbb{Z})$, where $p$ and the $p_j$'s are odd primes.

math.AT↗

arXiv2018

The odd primary order of the commutator on low rank Lie groups

Let $G$ be a simply-connected, compact, simple Lie group of low rank relative to a fixed prime $p$. After localization at $p$, there is a space $A$ which "generates" $G$ in a certain sense. Assuming $G$ satisfies a homotopy nilpotency condition relative to $p$, we show that the Samelson product $\langle Id_G, Id_G\rangle$ of the identity of $G$ equals the order of the Samelson product $\langle\imath,\imath\rangle$ of the inclusion $\imath:A\to G$. Applying this result, we calculate the orders of $\langle Id_G,Id_G\rangle$ for all $p$-regular Lie groups and give bounds on the orders of $\langle Id_G,Id_G\rangle$ for certain quasi-$p$-regular Lie groups.

math.AT↗