All Questions

Previously, I ask for comments/suggestions on setting up the calculation in Quotient space, homogeneous space, and higher homotopy groups. There, however, I was looking for whatever methods and tools ...

Let $ G $ be a Lie group with finitely many connected components, $ K $ a maximal compact subgroup, and $ \Gamma $ a torsion free cocompact lattice. Then $$ \Gamma \backslash G/K $$ is an aspherical ...

Let $ U_1 \to E \to B $ be a $ U_1 $ principal bundle. Suppose that $ B $ is homogenous (admits a transitive action by a Lie group) and compact. Then must it be the case that $ E $, the total space of ...

Let $ M $ be a connected manifold. Let $ \pi_1(M) $ be the fundamental group of $ M $. Suppose there exists a compact group $ K $ that acts transitively on $ M $. Then $ \pi_1(M) $ must have a finite ...

Preparation and my input: For the quotient space $G/H$, knowing the homotopy groups of $G$ and $H$ one can determine homotopy groups from the long exact sequence $$ ... \to \pi_n(H) \to \pi_n(G) ...

Consider braids on $n$ strands and pick $n$ distinct labels $1, \dots, n$. There is a groupoid $\mathcal P_n$ whose objects are tuples $(l_1, \dots, l_n)$ of labels and whose morphisms are braids, ...