Let $\pi$ be a discrete group, and let $G$ be a compact-connected Lie group. Then, there is a map $\Theta \colon \mathrm {Hom}(\pi,G)_0\to \mathrm {map}_*(B\pi,BG)_0$ between the null components of the spaces of homomorphisms and based maps, which sends a homomorphism to the induced map between classifying spaces. Atiyah and Bott studied this map for $\pi$ a surface group, and showed that it is surjective in rational cohomology. In this paper, we prove that the map $\Theta$ is surjective in rational cohomology for $\pi =\mathbb {Z}^m$ and the classical group $G$ except for $SO(2n)$, and that it is not surjective for $\pi =\mathbb {Z}^m$ with $m\ge 3$ and $G=SO(2n)$ with $n\ge 4$. As an application, we consider the surjectivity of the map $\Theta$ in rational cohomology for $\pi$ a finitely generated nilpotent group. We also consider the dimension of the cokernel of the map $\Theta$ in rational homotopy groups for $\pi =\mathbb {Z}^m$ and the classical groups $G$ except for $SO(2n)$.

Let G be a compact connected Lie group, and let $\operatorname {Hom}({\mathbb {Z}}^m,G)$ be the space of pairwise commuting m-tuples in G. We study the problem of which primes $p \operatorname {Hom}({\mathbb {Z}}^m,G)_1$, the connected component of $\operatorname {Hom}({\mathbb {Z}}^m,G)$ containing the element $(1,\ldots ,1)$, has p-torsion in homology. We will prove that $\operatorname {Hom}({\mathbb {Z}}^m,G)_1$ for $m\ge 2$ has p-torsion in homology if and only if p divides the order of the Weyl group of G for $G=SU(n)$ and some exceptional groups. We will also compute the top homology of $\operatorname {Hom}({\mathbb {Z}}^m,G)_1$ and show that $\operatorname {Hom}({\mathbb {Z}}^m,G)_1$ always has 2-torsion in homology whenever G is simply-connected and simple. Our computation is based on a new homotopy decomposition of $\operatorname {Hom}({\mathbb {Z}}^m,G)_1$, which is of independent interest and enables us to connect torsion in homology to the combinatorics of the Weyl group.

Let $G$ be a compact connected simple Lie group of type $(n_{1},\,\ldots,\,n_{l})$, where $n_{1}<\cdots < n_{l}$. Let $\mathcal {G}_k$ be the gauge group of the principal $G$-bundle over $S^{4}$ corresponding to $k\in \pi _3(G)\cong \mathbb {Z}$. We calculate the mod-$p$ homology of the classifying space $B\mathcal {G}_k$ provided that $n_{l}< p-1$.

Ganea proved that the loop space of $\mathbb{C} P^n$ is homotopy commutative if and only if $n=3$ . We generalize this result to that the loop spaces of all irreducible Hermitian symmetric spaces but $\mathbb{C} P^3$ are not homotopy commutative. The computation also applies to determining the homotopy nilpotency class of the loop spaces of generalized flag manifolds $G/T$ for a maximal torus T of a compact, connected Lie group G.

We determine the (non-)triviality of Samelson products of inclusions of factors of the mod p decomposition of $G_{(p)}$ for $(G,p)=(E_7,5),(E_7,7),(E_8,7)$. This completes the determination of the (non-)triviality of those Samelson products in p-localized exceptional Lie groups when G has p-torsion-free homology.

For certain pairs of Lie groups (G, H) and primes p, Harris showed a relation of the p-localized homotopy groups of G and H. This is reinterpreted as a p-local homotopy equivalence G ≃ (p)H × G/H, and so there is a projection G(p) → H(p). We show how much this projection preserves the higher homotopy associativity.

A Lie group is called p-regular if it has the p-local homotopy type of a product of spheres. (Non)triviality of the Samelson products of the inclusions of the factor spheres into p-regular SO(2n(p) is determined, which completes the list of (non)triviality of such Samelson products in p-regular simple Lie groups. As an application, we determine the homotopy normality of the inclusion SO(2n − 1) → SO(2n) in the sense of James at any prime p.

The method for computing the $p$-localization of the group $\left[ X,\,\text{U}\left( n \right) \right]$, by Hamanaka in 2004, is revised. As an application, an explicit description of the self-homotopy group of $\text{Sp}\left( 3 \right)$ localized at $p\,\ge \,5$ is given and the homotopy nilpotency of $\text{Sp}\left( 3 \right)$ localized at $p\,\ge \,5$ is determined.

The aim of this paper is to show that the p-local homotopy type of the gauge group of a principal bundle over an even-dimensional sphere is completely determined by the divisibility of the classifying map by p. In particular, for gauge groups of principal SU(n)-bundles over S2d for 2 ≤ d ≤ p − 1 and n ≤ 2p − 1, we give a concrete classification of their p-local homotopy types.

Selenium (Se) is an important element for assessing the safety of high-level waste disposal. Se is redox-sensitive, and its oxidation state varies from -2 to 6 depending on the redox conditions and pH of the solution. Large quantities of ferrous ions formed in bentonite due to corrosion of carbon steel overpack after the closure of a repository are expected to maintain a reducing environment near the repository. Therefore, the migration behavior of Se in the presence of Fe in bentonite was investigated by electrochemical experiments. Na2SeO3 solution was used as tracer solution. Dry density range of bentonite was from 0.8 to 1.4 ×103 kg/m3.

Results indicated that Se was strongly retained by the processes such as precipitation reaction with ferrous ions in bentonite. Se K-edge X-ray absorption near-edge structure (XANES) measurements were performed at the BL-11 beamline at SAGA Light Source, and the results revealed that the oxidation state of Se in the bentonite remained Se(IV).

Certain generalized Samelson products of SO(3) are calculated and applications to the homotopy of gauge groups are given.

The present paper describes the first reported case, to our knowledge, of a myoepithelioma arising in an accessory parotid gland. Because pre-operative fine-needle aspiration cytological findings and operative findings suggested that this tumour was malignant, the decision was made to remove the tumour surgically along with the buccal branches of the facial nerve. The resected nerve was reconstructed by nerve transplantation, using the great auricular nerve. Subsequently the tumour was found to be benign. One year after surgery, the patient had excellent facial nerve function and so far there has been no evidence of recurrence.