Let $G$ be a compact connected Lie group with $pi_1(G)congmathbb{Z}$. We study the homotopy types of gauge groups of principal $G$-bundles over Riemann surfaces. This can be applied to an explicit computation of the homotopy groups of the moduli spaces of stable vector bundles over Riemann surfaces.
We study the homotopy type of the space of the unitary group $operatorname{U}_1(C^ast_u(|mathbb{Z}^n|))$ of the uniform Roe algebra $C^ast_u(|mathbb{Z}^n|)$ of $mathbb{Z}^n$. We show that the stabilizing map $operatorname{U}_1(C^ast_u(|mathbb{Z}^n|))
tooperatorname{U}_infty(C^ast_u(|mathbb{Z}^n|))$ is a homotopy equivalence. Moreover, when $n=1,2$, we determine the homotopy type of $operatorname{U}_1(C^ast_u(|mathbb{Z}^n|))$, which is the product of the unitary group $operatorname{U}_1(C^ast(|mathbb{Z}^n|))$ (having the homotopy type of $operatorname{U}_infty(mathbb{C})$ or $mathbb{Z}times Boperatorname{U}_infty(mathbb{C})$ depending on the parity of $n$) of the Roe algebra $C^ast(|mathbb{Z}^n|)$ and rational Eilenberg--MacLane spaces.
The topological Tverberg theorem states that given any continuous map $fcolonDelta^{(d+1)(r-1)}tomathbb{R}^d$, there are pairwise disjoint faces $sigma_1,ldots,sigma_r$ of $Delta^{(d+1)(r-1)}$ such that $f(sigma_1)capcdotscap f(sigma_r) eemptyset$ wh
enever $r$ is a prime power. We generalize this theorem to a continuous map from a certain CW complex into a Euclidean space.
The index theory for the space of finite propagation unitary operators was developed by Gross, Nesme, Vogts and Werner from the viewpoint of quantum walks in mathematical physics. In particular, they proved that $pi_0$ of the space is determined by t
he index. However, nothing is known about the higher homotopy groups. In this article, we describe the homotopy type of the space of finite propagation unitary operators on the Hilbert space of square summable $mathbb{C}$-valued $mathbb{Z}$-sequences, so we can determine its homotopy groups. We also study the space of (end-)periodic finite propagation unitary operators.
Let $mathrm{TC}_r(X)$ denote the $r$-th topological complexity of a space $X$. In many cases, the generating function $sum_{rge 1}mathrm{TC}_{r+1}(X)x^r$ is a rational function $frac{P(x)}{(1-x)^2}$ where $P(x)$ is a polynomial with $P(1)=mathrm{cat}
(X)$, that is, the asymptotic growth of $mathrm{TC}_r(X)$ with respect to $r$ is $mathrm{cat}(X)$. In this paper, we introduce a lower bound $mathrm{MTC}_r(X)$ of $mathrm{TC}_r(X)$ for a rational space $X$, and estimate the growth of $mathrm{MTC}_r(X)$.
A generalised Postnikov tower for a space $X$ is a tower of principal fibrations with fibres generalised Eilenberg-MacLane spaces, whose inverse limit is weakly homotopy equivalent to $X$. In this paper we give a characterisation of a polyhedral prod
uct $Z_K(X,A)$ whose universal cover either admits a generalised Postnikov tower of finite length, or is a homotopy retract of a space admitting such a tower. We also include $p$-local and ration
Polyhedral products were defined by Bahri, Bendersky, Cohen and Gitler, to be spaces obtained as unions of certain product spaces indexed by the simplices of an abstract simplicial complex. In this paper we give a very general homotopy theoretic cons
truction of polyhedral products over arbitrary pointed posets. We show that under certain restrictions on the poset $calp$, that include all known cases, the cohomology of the resulting spaces can be computed as an inverse limit over $calp$ of the cohomology of the building blocks. This motivates the definition of an analogous algebraic construction - the polyhedral tensor product. We show that for a large family of posets, the cohomology of the polyhedral product is given by the polyhedral tensor product. We then restrict attention to polyhedral posets, a family of posets that include face posets of simplicial complexes, and simplicial posets, as well as many others. We define the Stanley-Reisner ring of a polyhedral poset and show that, like in the classical cases, these rings occur as the cohomology of certain polyhedral products over the poset in question. For any pointed poset $calp$ we construct a simplicial poset $s(calp)$, and show that if $calp$ is a polyhedral poset then polyhedral products over $calp$ coincide up to homotopy with the corresponding polyhedral products over $s(calp)$.
The $p$-local homotopy types of gauge groups of principal $G$-bundles over $S^4$ are classified when $G$ is a compact connected exceptional Lie group without $p$-torsion in homology except for $(G,p)=(mathrm{E}_7,5)$.
There is a product decomposition of a compact connected Lie group $G$ at the prime $p$, called the mod $p$ decomposition, when $G$ has no $p$-torsion in homology. Then in studying the multiplicative structure of the $p$-localization of $G$, the Samel
son products of the factor space inclusions of the mod $p$ decomposition are fundamental. This paper determines (non-)triviality of these fundamental Samelson products in the $p$-localized exceptional Lie groups when the factor spaces are of rank $le 2$, that is, $G$ is quasi-$p$-regular.
The first aim of this paper is to study the $p$-local higher homotopy commutativity of Lie groups in the sense of Sugawara. The second aim is to apply this result to the $p$-local higher homotopy commutativity of gauge groups. Although the higher hom
otopy commutativity of Lie groups in the sense of Williams is already known, the higher homotopy commutativity in the sense of Sugawara is necessary for this application. The third aim is to resolve the $5$-local higher homotopy non-commutativity problem of the exceptional Lie group $mathrm{G}_2$, which has been open for a long time.
