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We investigate the classical Alexandroff-Borsuk problem in the category of non-triangulable manifolds: Given an $n$-dimensional compact non-triangulable manifold $M^n$ and $varepsilon > 0$, does there exist an $varepsilon$-map of $M^n$ onto an $n$-dimensional finite polyhedron which induces a homotopy equivalence?
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We calculate the singular homology and v{C}ech cohomology groups of the Harmonic archipelago. As a corollary, we prove that this space is not homotopy equivalent to the Griffiths space. This is interesting in view of Edas proof that the first singular homology groups of these spaces are isomorphic.
Thickenings of a metric space capture local geometric properties of the space. Here we exhibit applications of lower bounding the topology of thickenings of the circle and more generally the sphere. We explain interconnections with the geometry of circle actions on Euclidean space, the structure of zeros of trigonometric polynomials, and theorems of Borsuk-Ulam type. We use the combinatorial and geometric structure of the convex hull of orbits of circle actions on Euclidean space to give geometric proofs of the homotopy type of metric thickenings of the circle. Homotopical connectivity bounds of thickenings of the sphere allow us to prove that a weighted average of function values of odd maps $S^n to mathbb{R}^{n+2}$ on a small diameter set is zero. We prove an additional generalization of the Borsuk-Ulam theorem for odd maps $S^{2n-1} to mathbb{R}^{2kn+2n-1}$. We prove such results for odd maps from the circle to any Euclidean space with optimal quantitative bounds. This in turn implies that any raked homogeneous trigonometric polynomial has a zero on a subset of the circle of a specific diameter; these results are optimal.
We show that the Snake on a square $SC(S^1)$ is homotopy equivalent to the space $AC(S^1)$ which was investigated in the previous work by Eda, Karimov and Repovvs. We also introduce related constructions $CSC(-)$ and $CAC(-)$ and investigate homotopical differences between these four constructions. Finally, we explicitly describe the second homology group of the Hawaiian tori wedge.
We specify a result of Yokoi cite{yo} by proving that if $G$ is an abelian group and $X$ is a homogeneous metric $ANR$ compactum with $dim_GX=n$ and $check{H}^n(X;G) eq 0$, then $X$ is an $(n,G)$-bubble. This implies that any such space $X$ has the following properties: $check{H}^{n-1}(A;G) eq 0$ for every closed separator $A$ of $X$, and $X$ is an Alexandroff manifold with respect to the class $D^{n-2}_G$ of all spaces of dimension $dim_Gleq n-2$. We also prove that if $X$ is a homogeneous metric continuum with $check{H}^n(X;G) eq 0$, then $check{H}^{n-1}(C;G) eq 0$ for any partition $C$ of $X$ such that $dim_GCleq n-1$. The last provides a partial answer to a question of Kallipoliti and Papasoglu cite{kp}.
A compact space $X$ is said to be minimal if there exists a map $f:Xto X$ such that the forward orbit of any point is dense in $X$. We consider rigid minimal spaces, motivated by recent results of Downarowicz, Snoha, and Tywoniuk [J. Dyn. Diff. Eq., 2016] on spaces with cyclic group of homeomorphisms generated by a minimal homeomorphism, and results of the first author, Clark and Oprocha [ Adv. Math., 2018] on spaces in which the square of every homeomorphism is a power of the same minimal homeomorphism. We show that the two classes do not coincide, which gives rise to a new class of spaces that admit minimal homeomorphisms, but no minimal maps. We modify the latter class of examples to show for the first time the existence of minimal spaces with degenerate homeomorphism groups. Finally, we give a method of constructing decomposable compact and connected spaces with cyclic group of homeomorphisms, generated by a minimal homeomorphism, answering a question in Downarowicz et al.
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