Let G be a compact connected Lie group and p: E → ΣA be a principal G-bundle with a characteristic map α: A → G, where A = ΣA₀ for some A₀. Let $K_i\to F_{i-1}\hookrightarrow F_i|1\le i\le m$ with F₀ = ∗, F₁ = ΣK₁ and Fₘ ≃ G be a cone-decomposition of G of length m and F’₁ = ΣK’₁ ⊂ F₁ with K’₁ ⊂ K₁ which satisfy $F_i{F}^{\text{'}}₁\subset F_{i+1}$ up to homotopy for all i. Then cat(E) ≤ m + 1, under suitable conditions, which is used to determine cat(SO(10)). A similar result was obtained by Kono and the first author (2007) to determine cat(Spin(9)), but that result could not...

We introduce and study the concept of characteristic for metrics. It turns out that metrizable spaces endowed with an L*-operator which admit a metric of characteristic zero play an important role in the theory of fixed points. We prove the existence of such spaces among infinite-dimensional linear topological spaces.

We prove that for any positive integers $n_1,n_2,...,n_k$ there exists a real flag manifold $F(1,...,1,n_1,n_2,...,n_k)$ with cup-length equal to its dimension. Additionally, we give a necessary condition that an arbitrary real flag manifold needs to satisfy in order to have cup-length equal to its dimension.

Let $f,g:M_1\to M_2$ be maps where $M_1$ and $M_2$ are connected triangulable oriented n-manifolds so that the set of coincidences $C_{f,g}=x\in M_1|f\left(x\right)=g\left(x\right)$ is compact in $M_1$. We define a Nielsen equivalence relation on $C_{f,g}$ and assign the coincidence index to each Nielsen coincidence class. In this note, we show that, for n ≥ 3, if $M_2={\tilde{M}}_2/K$ where ${\tilde{M}}_2$ is a connected simply connected topological group and K is a discrete subgroup then all the Nielsen coincidence classes of f and g have the same coincidence index. In particular, when $M_1$ and $M_2$ are compact, f...

We prove that for every n ∈ ℕ the space M(K(x 1, …, x n) of ℝ-places of the field K(x 1, …, x n) of rational functions of n variables with coefficients in a totally Archimedean field K has the topological covering dimension dimM(K(x 1, …, x n)) ≤ n. For n = 2 the space M(K(x 1, x 2)) has covering and integral dimensions dimM(K(x 1, x 2)) = dimℤ M(K(x 1, x 2)) = 2 and the cohomological dimension dimG M(K(x 1, x 2)) = 1 for any Abelian 2-divisible coefficient group G.

A metric space (X,ϱ) satisfies the disjoint (0,n)-cells property provided for each point x ∈ X, any map f of the n-cell $B^n$ into X and for each ε > 0 there exist a point y ∈ X and a map $g:B^n\to X$ such that ϱ(x,y) < ε, $\widehat{\varrho }(f,g)<\epsilon$ and $y\notin g\left(B^n\right)$. It is proved that each homogeneous locally compact ANR of dimension >2 has the disjoint (0,2)-cells property. If dimX = n > 0, X has the disjoint (0,n-1)-cells property and X is a locally compact $L{C}^{n-1}$-space then local homologies satisfy $H_k(X,X-x)=0$ for k < n and Hn(X,X-x) ≠ 0.