Articles of topological groups

Reference for Topological Groups

Topological groups were a topic that were covered minimally at my undergraduate institution but it’s a topic that I’m finding a need quite a bit in the number theory I’m reading (class field theory). Is there a recommended source that covers the general theory preferably with lots of examples and exercises?

difference between a $G$ invariant measure on $G/H$ and a haar measure on $G/H$

Let $G$ be a locally compact topological group, and $H$ be a normal subgroup. $G/H$ is a locally-compact topological group as well, and if we assume $H$ to be closed then $G/H$ is hausdorff and therefore admits a haar measure $\mu_{G/H}$. It is also known that if $\Delta_{G}|_{H}=\Delta_{H]$ (the modular functions), then $G/H$ admits a […]

Rank of a cohomology group, Betti numbers.

How is the rank of a cohomology group computed and what does it convey? I am trying to understand the concept behind betti numbers in a simplicial homology. Edited with details: Given a set of nodes/vertices/points X, let $C_{l}(X)$ denote the subsets of $X$ with cardinality $|C_{l}(X)|=l+1$. $\partial_{l}$ and $\delta_{l}$ are bounded linear maps with […]

Computing $\text{Gal}(\mathbb{Q}^{ab}/\mathbb{Q})$

Algebraic class field theory tells us that $\text{Gal}(\mathbb{Q}^{ab}/\mathbb{Q})$ is isomorphic to the group of connected components of the quotient $\mathbb{Q}^{\times}\backslash \mathbb{A}_{\mathbb{Q}}^{\times}\cong \prod_p \mathbb{Z_p}^{\times}\times \mathbb{R}_{>0}$, where $\mathbb{A}_{\mathbb{Q}}$ is the ring of adèles of $\mathbb{Q}$. It’s then said that the group of connected components is given by $\prod_p \mathbb{Z_p}^{\times}$, how can I see this? Thank you very […]

Is there an example of a non-orientable group manifold?

Basically what I’m looking for is a topological group that is also a non-orientable, n-dimensional manifold

Is the orbit space of a Hausdorff space by a compact Hausdorff group Hausdorff?

Let $G$ be a compact Hausdorff group. Let $X$ be a Hausdorff space. Suppose $G$ acts continuously on $X$. Is the orbit space $X/G$ Hausdorff? If not, I would like to know an counter-example. Remark As my answer to this question shows, if $X$ is a locally compact Hausdorff space, $X/G$ is Hausdorff.

Isomorphism from $\widehat{G}$ to $\displaystyle \lim_{\longleftarrow} G/G_n$

Let $G$ be a topological abelian group and let $\widehat{G}$ denote its completion (i.e. equivalence classes of Cauchy sequences). Let $G_n$ be a descending sequence of subgroups, i.e. $G = G_0 \supset G_1 \supset \dots$ such that $G_n$ form a countable neighborhood basis of $0$. Let $\bar{x} \in \widehat{G}$ and let $x$ be its representative […]

Haar measure scalars, what am I doing wrong here?

Let $\mu$ be a Borel measure on a topological space $X$, and let $E \subseteq X$ be Borel. Let $\phi$ be a homeomorphism of $X$, and let $\lambda$ be the measure given by $\lambda(A) = \mu(\phi(A))$. If $f: E \rightarrow \mathbb{C}$ is measurable and integrable, then so is $f \circ \phi: \phi^{-1}E \rightarrow \mathbb{C}$, and […]

Prove that the Pontryagin dual of a locally compact abelian group is also a locally compact abelian group.

Let $ G $ be a locally compact abelian (LCA) group and $ \widehat{G} $ the Pontryagin dual of $ G $, i.e., the set of all continuous homomorphisms $ G \to \mathbb{R} / \mathbb{Z} $. Clearly, $ \widehat{G} $ is an abelian group. The topology on $ \widehat{G} $ is generated by the sub-basic […]

For a topological group $G$ and a subgroup $H$, is it true that $ = \overline{}$? What about algebraic groups?

When discussing with awllower about this question, I begin to think about another one: For a topological group $G$ and a subgroup $H$, is it true that $[\overline{H}, \overline{H}] = \overline{[H,H]}$? where $[H,H]$ denote the derived subgroups of $H$. I think, if I define the map $\phi: G \times G \rightarrow G, (x,y) \mapsto xyx^{-1}y^{-1}$, […]