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In the usual singular homology of a topological space $X$, one consider the free abelian group generated by all continuous maps from the standard simplex $\Delta^{n}$ to $X$. Now we can r …

Edit: I revise the question based on the comment conversations Let $\mathcal{F}$ be the set of all equivalence classes of finite groups under the "Isomorphism" equivalence relation. We define …

Let $X=\{f\in \mathbb{C}[z]\mid |z| \neq 1 \implies f(z) \neq 0\} $. The motivation for consideration of such an $X$ is the the concept of Lee-Yang polynomials. With the standard multiplication, $X …

Many years ago I found in google the notation "Holomorph of group". It is the semi direct product of $G$ with $Aut(G)$. Why is the term "Holomorph" used here, while it is usually used for complex anal …

Let $A$ be a unital $C^*$-algebra which is equipped with a faithful trace $T$. In particular we may consider $A=C^*_{\text{red}} (G)$ for some discrete group $G$. We consider the following differentia …

Let $\mathcal{FG}$ be the category of finite groups. Let $S$ be a full subcategory of $\mathcal{FG}$. Assume that $G\in \mathcal{FG}$ and $P\in S$ is a subgroup of $G$. We say that $P$ is $S$-maximal …

On the objects of the category of groups we define the mapping $G\mapsto \operatorname{Hol}(G)$, the holomorph $G\rtimes \operatorname{Aut}(G)$ of $G$. Can we extend this mapping to a functor on this …

Is there an orientation preserving smooth diffeomorphism $f$ on a compact manifold $M$ such that for every $n\in \mathbb{N}$, there is a smooth diffeomorphism $g:M \to M$, as $n$th root of …

1)Let $G$ be a countable discrete group. Can $G$ be embbeded in a locally connected Lie group? 2)let $G$ be a countable discrete group with a prescribed generating set and corresponding word metr …

Let $G$ be the group of all permutations of $\mathbb{N}$. If I am not mistaken, this group is denoted by $S^{\infty}$. Is there a precise locally compact topology on $G$ such that $G$ would b …

Is there a locally compact group $G$ such that the canonical map from $C^{*}(G)$ to $C^{*}_{red} G$ is not isomorphism, hence $G$ is not amenable but these two $C^{*}$ algebras are isomorphic …

Edit: According to the comment of Todd Trimble, I revise the question. What are some examples of functors $F$ on the category of Abelian groups or category of rings which satisfy $$F(G\times H)\cong …

Revision: According to comment of Wojowu we give a complete revise for this post. A group $G$ is a pr-group if all projections of $C^*_{\text{red}} G$ are contained in its dense subalgebra $\mathbb{ …

The following paper and its references contains some algebraic consequences of vector bundle theory. Vakhtang Lomadze, Applications of vector bundles to factorization of rational matrices, Line …

Let $G$ be a group. We define $$\tilde{G}=\{\phi:G \to \mathbb{T}\mid \phi(gh){\phi(g)}^{-1}{\phi(h)}^{-1}\in Tor(\mathbb{T})\}$$ where $Tor(\mathbb{T})$ is the group of torsion elements of the unit c …