A "simple, detailed" reference for Topological groups, which is sufficient for number-theoretic reasons.

Question

For number-theoretical purposes, I want to learn the infinite Galois theory. During this, I faced some other concepts which I am not comfortable with them, for instance, topological groups.

Why $Gl_n(\mathbb{R})$ is a topological group?

I know the group law on it, which is matrix multiplication. But do not feel comfortable with a topology on it. So I am not with the followed concepts, like continuity. I have to read some examples of topological groups with too many details on its topological aspects because I passed the topology course long ago.

Is there any source about examples of topological groups, with too many details on topological aspects?

Answer 1

First, understand that the space $M_n$ of all $n \times n$ matrices has a natural topology on it: it is the Euclidean topology of $\mathbb{R}^{n^2}$, where we just associate an $n \times n$ matrix with a tuple of $n^2$ points by writing its rows out side-by-side. Now we can give $\operatorname{GL}_n(\mathbb{R})$ the subspace topology inherited from this "matrix-ized" version of $\mathbb{R}^{n^2}$ that I've called $M_n$. In particular, the topology is given by a metric. Even better, the topology is particularly simple because $\operatorname{GL}_n(\mathbb{R})$ is an open subset in $M_n$ (it is the preimage of $\mathbb{R}-\{\mathbf{0}\}$ under the determinant map).

Next, we need to show that matrix multiplication and inversion are continuous. For multiplication it is best to work back in $M_n$. Multiplication of matrices defines a function $M_n \times M_n \to M_n$ as $(A,B) \mapsto AB$, and this is clearly continuous because it is polynomial in the entries of $A, B \in M_n$. Since restrictions of continuous functions are continuous, this multiplication restricts down to a continuous operation $\operatorname{GL}_n(\mathbb{R})\times \operatorname{GL}_n(\mathbb{R}) \to \operatorname{GL}_n(\mathbb{R})$.

For inversion, it is much the same trick. When you write down how inversion acts as a function, $A \mapsto A^{-1}$, you can realize this is given by the inverse of the determinant times the adjugate matrix of $A$. Since the determinant is polynomial in the entries of $A$ and not zero because $A$ is invertible, you can use this to prove inductively that inversion is a continuous function $\operatorname{GL}_n(\mathbb{R}) \to \operatorname{GL}_n(\mathbb{R})$.

For a reference, although it is old, George McCarty's Topology: An Introduction with Application to Topological Groups is an excellent and fast way to learn the basics.

Answer 2

If your final goal is just studying infinite Galois theory, I believe Milne's book on Galois theory is fine(as fas as I could remember, he treats Krull topology in this book). For other purposes such as studying Lie groups or improving your knowledge in commutative algebra, these following books are highly recommended:

• J.Rotman, An Introduction to Algebraic Topology: you could take a glance at some last pages of chapter $3$, The Fundamental Group, where the author defines the notions of topological groups and the so-called $H$-spaces. Roughly, $H$-spaces are topological spaces behaving like topological groups. Next, you may wish to have more insightful treatments about these objects, let turn to chapter $11$, which is also very readable, where Rotman introduces the categorical approach to group objects, $H$-groups, $H'$-groups in a general category. In this point of view, a topological group is simply a group object in $\mathbf{Top}_{*}$ or $\mathbf{hTop}_{*}$.

• M.Atiyah, An Introduction to Commutative Algebra, chapter $10$. In this book, Atiyah gives a short introduction about the motivation of considering topological groups. The method of taking completion arises from purely number theoreic aspect ($p$-adic integers,...) and expressed to be extremely useful in commutative algebra.

• Loring Tu, An Introduction to Manifolds, chapter $4$ about Lie groups and Lie algebras. I strongly believe Tu presents the whole book in a clearest way along with lots of explicit, detailed examples, including $\mathrm{Gl}_n(\mathbb{R})$.It is true that every mathematician should know at least a little theory of Lie groups, as they naturally occur everywhere in mathematics, even from theoretical physics. With this idea in mind, I strongly believe Tu's book is undoubtedly a good choice.

Topological groups are very well-known objects, especially when you are a algebraic inclined students, you should know them like back of you hand. For instance, consider $\mathrm{GL}_n(\mathbb{R})$, this is an example of the so-called Lie groups, which means that the group operators are all $C^{\infty}$. This is even stronger than asserting it to be just a topological group, which just requires the group operators to be continuous. A Lie group (or a topological group) has the advantage of being a infinesimal group, that is, all other elements are infinitesically close to the identity element, or all your problems with group are transfered to an open neighborhood of the identity element. Back to $\mathrm{GL}_n(\mathbb{R})$, as a set, it is just $$\mathrm{GL}_n(\mathbb{R}) = \left \{A = [a_{ij}]\in \mathbb{R}^{n^2} \mid \mathrm{det}(A) \neq 0 \right \}$$ Since the $\mathrm{det}$ map is continuous, $\mathrm{GL}_n$ is an open set, therefore, a submanifold of $\mathbb{R}^{n^2}$. The product of two matrices $A=(a_{ij}),B = (b_{jk})$ reads $$(AB)_{ik} = \sum_{j=1}^n a_{ij}b_{jk}$$ which is obviously $C^{\infty}$ as this is just a polynomial of entrices of $A$ and $B$. Similarly, the entrices of the inverse $A^{-1}$ can be expressed in terms of that of $A$, consequently, the inverse map is also infinitely differentiable.