Topological Manifolds

This post was machine-translated from the Korean original by Marvin (via Kimi). It may contain errors or awkward phrasing — the Korean original is the source of truth.

We study homology, cohomology, and other concepts essential to geometry in this category. These notions are defined for general topological spaces, but for them to behave well the space must satisfy additional properties; a space meeting all such requirements is precisely the topological manifold defined below. In this post we examine the properties and examples of topological manifolds, and in the next post we will illustrate what homology is through rough examples. These two posts outline the broad direction of this category; the substantive content begins with the third post.

Definition of a Topological Manifold

A topological manifold is defined as follows.

Definition 1 A topological space \(M\) is said to be locally Euclidean of dimension \(m\) if for every \(x\in M\) there exists an open neighborhood \(U\) of \(x\) such that \(U\) is homeomorphic to an open subset of \(\mathbb{R}^m\).

Definition 2 A space that is second countable, Hausdorff, and locally Euclidean of dimension \(m\) is called a topological manifold of dimension \(m\).

For convenience, we call a topological manifold of dimension \(m\) an \(m\)-manifold. Although not treated in Definition 1, we sometimes replace \(\mathbb{R}^m\) in the above definition by the half-space

\[\mathbb{H}^m=\left\{(x_1,\ldots,x_m)\in \mathbb{R}^m\mid x_m\geq 0\right\}\]

and consider a manifold with boundary.

Examples of Topological Manifolds

We have already learned in topology many ways to construct new spaces from given ones. Therefore, once topological manifolds are given, it is natural to ask whether the resulting spaces remain topological manifolds.

Example 3 (Open submanifold) An open subspace \(U\) of an \(m\)-manifold \(M\) is again an \(m\)-manifold. Indeed, if \(\mathcal{B}\) is a base for \(M\), then the collection

\[\mathcal{B}_U=\left\{B\cap U\mid B\in \mathcal{B}\right\}\]

is a base for \(U\), so \(U\) is second-countable; a subspace of a Hausdorff space is always Hausdorff (§Hausdorff Spaces); and if \(x\in U\) is arbitrary, then by the assumption that \(M\) is locally Euclidean we can choose an open neighborhood \(V\) of \(x\) in \(M\) such that \(V\) is homeomorphic to an open subset of \(\mathbb{R}^m\), whence \(U\cap V\) is an open neighborhood of \(x\) in \(U\) homeomorphic to an open subset of \(\mathbb{R}^m\).

Similarly, the set in §Hausdorff Spaces, ⁋Corollary 7 also yields an example of a topological manifold as follows.

Example 4 For an open subset \(U\subseteq \mathbb{R}^n\) and a continuous function \(f:U\rightarrow\mathbb{R}^k\), the graph of \(f\)

\[\Gamma(f)=\left\{(x,f(x))\mid x\in U\right\}\subset \mathbb{R}^m\times \mathbb{R}^k\]

is an \(m\)-manifold. Indeed, this is because the two continuous maps

\[x\mapsto (x,f(x)),\qquad (x,f(x))\mapsto x\]

are inverses of each other, so \(\Gamma(f)\) and \(U\) are homeomorphic.

By §Hausdorff Spaces, ⁋Corollary 7, \(\Gamma(f)\) is a closed subset of \(\mathbb{R}^{m+k}\), so this gives an example of a somewhat different character from Example 3 (Open submanifold).

On the other hand, the following also holds for the product topology.

Example 5 (Product manifold) Let two topological manifolds \(M_1\), \(M_2\) be given, of dimensions \(m_1\) and \(m_2\) respectively. Then \(M_1\times M_2\) is an \((m_1+m_2)\)-manifold. This is because if \(\mathcal{B}_i\) is a base for \(M_i\), then the collection

\[\mathcal{B}=\left\{B_1\times B_2\mid B_i\in \mathcal{B}_i\right\}\]

is a basis for \(M_1\times M_2\), so \(M_1\times M_2\) is second countable; the product of Hausdorff spaces is Hausdorff (§Hausdorff Spaces, ⁋Proposition 8); and for any \((x_1,x_2)\in M_1\times M_2\), if \(U_i\) is a Euclidean neighborhood of \(x_i\) in \(M_i\), then \(U_1\times U_2\) is a Euclidean neighborhood of \((x_1,x_2)\) in \(M_1\times M_2\).

Finally, the general construction we examine is the quotient space. However, as we saw in §Hausdorff Spaces, an arbitrary quotient of a Hausdorff space need not be Hausdorff. Moreover, there is no guarantee that the quotient of a Euclidean space is Euclidean, so to show that a quotient space is a topological manifold one must verify the Hausdorff and locally Euclidean conditions separately. On the other hand, second countability follows from the locally Euclidean condition.

Proposition 6 For a quotient map \(X \rightarrow X/R\), suppose \(X\) is second-countable and \(X/R\) is locally Euclidean. Then \(X/R\) is second countable.

Proof

Since \(X/R\) is locally Euclidean, we can cover \(X/R\) by Euclidean neighborhoods \((U_i)_{i\in I}\), and the collection of their preimages \((\pi^{-1}(U_i))_{i\in I}\) covers \(X\). Now any second-countable space is Lindelöf (§Compactness and Convergence of Filters, ⁋Proposition 12), so there exists a countable subset \(J\subset I\) such that \((\pi^{-1}(U_i))_{i\in J}\) is a countable open cover of \(X\), and hence the corresponding \((U_i)_{i\in J}\) form a countable cover of \(X/R\). But each of these is a Euclidean neighborhood, hence has a countable base, and since there are countably many of them, their union is a countable base for \(X/R\).

Viewed solely within the flow of this category, our interest might be sufficiently confined to topological manifolds; however, especially when dealing with the multiplicative structure of cohomology, it is more convenient to keep in mind the notion of integration on smooth manifolds.