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Let $G$ and $H$ be two topological groups. Assume that $\phi:\pi_{1}(G) \to \pi_{1}(H)$ is a group homomorphism. Is there a continuous function $f:G\to H$ such that $f_{*}=\phi$?

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 …

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 …

Let $\mathcal{A}$ be the category of $C^*$ algebras. For a group $G$ let $\tilde{G}$ be the space of all conjugacy classes of $G$. What is a precise description of a maximal ,or in some sense w …

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 …

Let $G$ be the group of all self-homeomorphisms $f$ of $\mathbb{R}^n$ which satisfy $$f(x+m)=f(x)+m,\quad \forall m\in \mathbb{Z}^n.$$ In other words, $G$ is the group of all equivariant self-homeomo …

Let we have a complex of abelian topological or lie groups $$\ldots \to G_{n}\to G_{n+1}\to \ldots$$ such that the image of $G_{n}$ is a closed subgroup of $G_{n+1}$. Then we have a complex of fu …

Let $G$ be a topological group. We define an equivalence relation on $G$ as follows: For $a,b\in G$ we set $a\sim b$ if the following two maps are topologically conjugate: $$x\mapsto ax,\qquad x\maps …

Let $G$ be the set of all homeomorphisms $f$ of $\mathbb{R}^2$ such that both $f$ and $f^{-1}$ are polynomial maps. We equip $G$ with the compact open topology and the obvious group …

Regarding the last part of your question: Let $f_{0}$ and $f_{1}$ be two orientation preserving homeomorphism in $H_{+}(S^{1})$. Lift $f_{0}$ and $f_{1}$ to homeomorphisms $F_{0}$ and $F_{1}$ on $\ma …

Motivation: It is obvious that for a finite index subgroup $H$ of a group $G$, there exists a normal subgroup $K$ of $G$, $K\subset H$, with $|G/K|<\infty$. Our question: Let $G$ be a topological gro …

Is there a discrete non-abelian group whose dual in a reasonable sense is isomorphic to the solenoid constructed via a sequence of quaternions $S^3$ instead of a sequence of circles? The motivation c …

What is an example of a torsion free discrete abelian group $G$ whose dual space $\hat{G}$ is not a path connected space? The Motivation: The motivation comes from the idempotent conjecture of Kapla …