We construct a Peano continuum $X$ such that: (i) $X$ is a one-point compactification of a polyhedron; (ii) $X$ is weakly homotopy equivalent to a point; (iii) $X$ is noncontractible; and (iv) $X$ is (co)homologically locally connected. We also prove that all classical homology groups (singular, Čech, and Borel-Moore), all classical cohomology groups (singular and Čech), and all finite-dimensional Hawaiian groups of $X$ are trivial.
COBISS.SI-ID: 15382873
We show that $Y$ has an eventual H-space exponent at $p$ if and only if the space ${\text{map}}_\ast(S^m[p^{-1}],Y)$ of pointed maps $S^m[p^{-1}] \to Y$ has the homotopy type of a CW complex for some (and hence all big enough) $m$. This makes it possible to interpret the question of eventual H-space exponents in terms of phantom phenomena of mapping spaces.
COBISS.SI-ID: 15638105
We begin with a study of some extensions of the concept of idempotent lifting and prove the generalizations of some classical lifting theorems. Then we describe the method of induced liftings, which allows us to transfer liftings from a ring to its subrings. Using this method we are able to show that under certain assumptions a subring of an exchange ring is also an exchange ring, and to prove that a finite algebra over a commutative local ring is semi-perfect, provided it can be suitably represented in an exchange ring.
COBISS.SI-ID: 15627865
We extend the definition of Bockstein basis $\sigma(G)$ to nilpotent groups $G$. The Bockstein First Theorem says that all compact spaces are Bockstein spaces. Here are the main results of the paper: Theorem 0.1. Let $X$ be a Bockstein space. If $G$ is nilpotent, then $\dim_G(X) \le 1$ if and only if $\sup \{ \dim_H(X)\vert H \in \sigma(G)\} \le 1$. Theorem 0.2. $X$ is a Bockstein space if and only if $\dim_{Z(l)}(X) = \dim_{\hat{Z}(l)}(X)$ for all subsets $l$ of prime numbers.
COBISS.SI-ID: 15493977
The importance of small loops in the covering space theory was pointed out by Brodskiy, Dydak, Labuz, and Mitra. We construct a small loop space using the Harmonic Archipelago. Furthermore, we define the small loop group of a space and study its impact on covering spaces, in particular its contribution to the fundamental group of the universal covering space.
COBISS.SI-ID: 15308633