On coset $n$-valued topological groups on $S^3$ and $\mathbb{R}P^3$
We obtain all coset $n$-valued topological groups on $S^3$ and $\mathbb{R}P^3$, arising from compact Lie groups Sp(1) and SO(3) and there finite groups of automorphisms.
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arXiv subjects
Dmitry V. Gugnin
Publications and source records attributed to Dmitry V. Gugnin.
On Integral Cohomology Ring of Symmetric Products
We prove that the integral cohomology ring modulo torsion $H^*(\mathrm{Sym}^n X;\mathbb{Z})/\mathrm{Tor}$ for the symmetric product of a connected CW-complex $X$ of finite homology type is a functor of $H^*(X;\mathbb{Z})/\mathrm{Tor}$ (see Theorem 1). Moreover, we give an explicit description of this functor. We also consider the important particular case when $X$ is a compact Riemann surface $M^2_g$ of genus $g$. There is a famous theorem of Macdonald of 1962, which gives an explicit description of the integral cohomology ring $H^*(\mathrm{Sym}^n M^2_g;\mathbb{Z})$. The analysis of the original proof by Macdonald shows that it contains three gaps. All these gaps were filled in by Seroul in 1972, and, therefore, he obtained a complete proof of Macdonald's theorem. Nevertheless, in the unstable case $2\le n\le 2g-2$ Macdonald's theorem has a subsection, that needs a slight correction even over $\mathbb{Q}$ (see Theorem 2).
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On Homeomorphism Type of Symmetric Products of Compact Riemann Surfaces with Punctures
Let $M^2_{g,k}$ and $M^2_{g',k'}$ be compact Riemann surfaces with punctures ($g,g'\ge 0$ - genuses, $k,k'\ge 1$ - number of punctures). For any Hausdorff space $X$ the quotient space $\mathrm{Sym}^nX := X^n/S_n$ is the $n$-th symmetric product of $X, \ n\ge 2$. It is well known, that $\mathrm{Sym}^n M^2_{g,k}$ is a smooth quasi-projective variety. Open manifolds $\mathrm{Sym}^n M^2_{g,k}$ and $\mathrm{Sym}^n M^2_{g',k'}$ are homotopy equivalent iff $\ 2g+k=2g'+k'$. Blagojević-Grujić-Živaljević Conjecture (2003). Fix any $n\ge 2$, and two pairs $(g,k)$ and $(g',k')$ with the condition $2g+k=2g'+k'$. If $g\ne g'$, then open manifolds $\mathrm{Sym}^n M^2_{g,k}$ and $\mathrm{Sym}^n M^2_{g',k'}$ are not continuously homeomorphic. The conjecture was proved in 2003 in the paper by P.Blagojević, V.Grujić and R.Živaljević for the case $\mathrm{max}(g,g') \ge \frac{n}{2}$ (this implies the case $n=2$). As far as the author knows, up to this moment there were no results if $\mathrm{max}(g,g') < \frac{n}{2}$. The aim of this paper is to prove the conjecture in full generality.
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