Topological Spaces and Quotients

The Poincaré conjecture is a statement about topology: about the properties of a space that survive any continuous deformation, stretching and bending but not tearing or gluing. A sphere and the surface of a cube are the same to a topologist; a sphere and a doughnut are not. To say this precisely one needs spaces whose only structure is "nearness", with no distances: topological spaces. Book 2B did almost everything for metric spaces, and most of it carries over word for word once the open sets, rather than the metric, are taken as the basic notion.

What is genuinely new is the ability to glue. Take a square and identify its opposite sides, and you get a torus, a space that is not given to you as a subset of anything. Gluing, formally the quotient topology, is how the spaces of the Poincaré conjecture are built: lens spaces and the Poincaré homology sphere are quotients of the 3-sphere (7A.6 Covering Spaces), three-manifolds are glued from pieces (10A.2 Building Three-Manifolds), and Perelman's surgery cuts a manifold and glues in caps (12B.4 Surgery). This chapter sets up topological spaces and then concentrates on quotients.

By the end of this chapter you will be able to:

• state the axioms of a topology and recognise continuity, homeomorphisms and bases;
• explain the Hausdorff and second countability conditions, and why each appears in the definition of a manifold;
• form subspaces and products;
• define the quotient topology, use its characteristic property, and build the torus, the Klein bottle, the projective plane and the Möbius band from a square;
• recognise group quotients such as $\mathbb{R}/\mathbb{Z}$ and $\mathbb{R}^n/\mathbb{Z}^n$.

1.1A world that wraps around

1.2Topological spaces

These are exactly the properties of open sets in a metric space that 2B.1 Metric Spaces proved, taken as axioms. Every metric space is a topological space, with the metric topology. But there are topologies that come from no metric:

• the discrete topology, in which every set is open (this does come from the metric $d(x, y) = 1$ for $x \neq y$);
• the trivial topology $\{\varnothing, X\}$, in which no two points can be separated;
• on an infinite set, the cofinite topology, whose open sets are $\varnothing$ and the complements of finite sets. Any two non-empty open sets intersect.

Most notions of Book 2B are defined from open sets alone, so they carry over. A neighbourhood of $x$ is an open set containing $x$. The closure $\overline A$ is the smallest closed set containing $A$, and the interior the largest open set inside it. A sequence $x_k \to x$ if every neighbourhood of $x$ contains all but finitely many $x_k$. A collection $\mathcal B$ of open sets is a basis if every open set is a union of members of $\mathcal B$: the open balls are a basis for a metric topology.

For metric spaces this agrees with the $\varepsilon$–$\delta$ definition (2B.1 Metric Spaces). The preimage formulation works because preimages respect unions, intersections and complements (2A.2 Sets, Functions and Equivalence). A topological property is one preserved by homeomorphisms: compactness, connectedness, the number of path components, and (in 7A.4 The Fundamental Group) the fundamental group. Distance, angle, straightness and curvature are not topological.

1.3Hausdorff and second countable

General topological spaces can be very strange. Two conditions exclude the strangeness that matters for geometry.

Metric spaces are Hausdorff: balls of radius $\frac12d(x, y)$ around $x$ and $y$ are disjoint. In a Hausdorff space, limits of sequences are unique (Exercise 1.7), finite sets are closed, and, as 7A.2 Compactness and Compactification shows, compact sets are closed. The cofinite topology on an infinite set is not Hausdorff, and in it the sequence $1, 2, 3, \dots$ in $\mathbb{N}$ converges to every point.

A more geometric failure is the line with two origins: take two copies of $\mathbb{R}$ and glue them together at every point except $0$ (Figure 1.1). Every point has a neighbourhood homeomorphic to an open interval, so the space looks locally exactly like a line. But the two origins $0_a$ and $0_b$ cannot be separated: every neighbourhood of $0_a$ contains points $x \neq 0$ near $0$, and so does every neighbourhood of $0_b$, and those points have been identified. A sequence $\frac1k$ converges to both origins. This is why "locally Euclidean" is not enough to define a manifold: the Hausdorff condition is added to the definition (7A.3 Manifolds and Surfaces).

$\mathbb{R}^n$ is second countable: balls with rational centres and rational radii form a countable basis. So is every subspace of $\mathbb{R}^n$. A disjoint union of uncountably many lines is locally Euclidean and Hausdorff but not second countable. The condition matters because it makes manifolds embeddable in some $\mathbb{R}^N$ and gives them partitions of unity, the tool for gluing local constructions such as Riemannian metrics into global ones (8A.2 Partitions of Unity). Without it, a "manifold" could be too big to carry a metric.

1.4Subspaces and products

If $A \subseteq X$, the subspace topology on $A$ consists of the sets $U \cap A$ with $U$ open in $X$. For subsets of metric spaces this is the metric topology of the restricted metric. The inclusion $A \hookrightarrow X$ is continuous, and a map into $A$ is continuous exactly when it is continuous into $X$.

If $X$ and $Y$ are topological spaces, the product topology on $X \times Y$ has as a basis the products $U \times V$ of open sets. For $\mathbb{R}^m \times \mathbb{R}^n$ it is the usual topology of $\mathbb{R}^{m+n}$. A map into $X \times Y$ is continuous exactly when its two components are. Products give the cylinder $S^1\times\mathbb{R}$, the torus $S^1\times S^1$, and, in dimension three, $S^2\times S^1$, one of the manifolds the Poincaré conjecture must exclude (7A.9 The Poincaré Conjecture, Precisely).

1.5Quotient spaces

Now the new construction. Let $X$ be a topological space and $\sim$ an equivalence relation on it (2A.2 Sets, Functions and Equivalence), and let $X/{\sim}$ be the set of equivalence classes, with the projection $\pi : X \to X/{\sim}$ sending each point to its class.

It is the largest topology for which $\pi$ is continuous. A set of equivalence classes is open when the union of all the points in those classes is open in $X$. The fundamental fact about quotients is how to map out of them.

Example: the circle. On $[0, 1]$, identify $0 \sim 1$ and nothing else. The map $f(t) = (\cos2\pi t, \sin2\pi t)$ is continuous, takes the same value at $0$ and $1$, and so induces a continuous bijection $\bar f : [0, 1]/{\sim} \to S^1$. It is a homeomorphism: its inverse is continuous, which is easiest to see with the compactness lemma of 7A.2 Compactness and Compactification (a continuous bijection from a compact space to a Hausdorff space is a homeomorphism). So $[0, 1]/(0 \sim 1) \cong S^1$.

Gluing a square. Let $X = [0, 1]^2$. Different identifications of its edges give different spaces (Figure 1.2):

• identify $(0, y) \sim (1, y)$ only: a cylinder $S^1\times[0, 1]$;
• identify $(0, y) \sim (1, 1 - y)$ only (one pair of edges glued with a half twist): the Möbius band;
• identify $(0, y) \sim (1, y)$ and $(x, 0) \sim (x, 1)$: the torus $T^2 \cong S^1\times S^1$, the Asteroids screen;
• identify $(0, y) \sim (1, y)$ and $(x, 0) \sim (1 - x, 1)$: the Klein bottle, which cannot be embedded in $\mathbb{R}^3$ without crossing itself;
• identify $(0, y) \sim (1, 1 - y)$ and $(x, 0) \sim (1 - x, 1)$: the real projective plane $\mathbb{R}P^2$.

Collapsing a subspace. If $A \subseteq X$, the space $X/A$ identifies all of $A$ to a single point. For the closed disc and its boundary, $D^n/\partial D^n \cong S^n$ (Exercise 1.9): crumple the disc up and pinch its whole boundary to one point, like closing a drawstring bag. This is one of the standard descriptions of the $n$-sphere.

Group quotients. If a group $G$ acts on $X$ by homeomorphisms, the orbits form an equivalence relation, and the quotient is written $X/G$. Examples:

• $\mathbb{R}/\mathbb{Z} \cong S^1$, the integers acting by translation;
• $\mathbb{R}^n/\mathbb{Z}^n \cong T^n$, the $n$-torus;
• $S^n/\{\pm1\} = \mathbb{R}P^n$, real projective space: antipodal points identified, or equivalently the space of lines through the origin in $\mathbb{R}^{n+1}$.

In dimension three, $\mathbb{R}P^3$ is the space of rotations of $\mathbb{R}^3$ (7A.6 Covering Spaces), and quotients $S^3/\Gamma$ by finite groups acting freely are the spherical space forms, which include the lens spaces and the Poincaré homology sphere. They are what the Ricci flow leaves behind when it shrinks a three-manifold with finite fundamental group to round pieces (12C.1 Reading Off the Topology).

A warning. Quotients can be badly behaved. The quotient $\mathbb{R}/\mathbb{Q}$ (identify $x \sim y$ when $x - y$ is rational) has the trivial topology: every non-empty open set of $\mathbb{R}$ meets every class (Exercise 1.10). Even the line with two origins is a quotient of two lines. Whether a quotient is Hausdorff has to be checked; for quotients of manifolds by groups, the condition that guarantees it is a properly discontinuous action (7A.6 Covering Spaces).

1.6History

Felix Hausdorff's Grundzüge der Mengenlehre (1914) introduced topological spaces through neighbourhood axioms that included the separation condition now named after him; the open-set axioms in their modern form became standard in the 1920s and 30s, in the work of Kuratowski, Alexandroff and others. The Möbius band was described independently by August Möbius and Johann Listing in 1858, and Felix Klein described his bottle in 1882. Henri Poincaré's Analysis Situs (1895) and its supplements built three-manifolds by gluing the faces of polyhedra, and it is there that the questions of this book begin.

1.7Exercises

Solution

The preimage of the image of $(-1, 1)\times\{a\}$ is $(-1, 1)\times\{a\}\cup\big((-1, 1)\setminus\{0\}\big)\times\{b\}$, which is open, so the image is open; the projection restricted to $(-1, 1)\times\{a\}$ is a continuous injective open map onto it, hence a homeomorphism. A neighbourhood of $0_a$ has open preimage containing $(0, a)$, hence containing $(-\varepsilon, \varepsilon)\times\{a\}$; similarly for $0_b$ with some $\varepsilon'$. Both contain the image of $(\frac12\min(\varepsilon, \varepsilon'), \cdot)$.

Solution

An open set contains an interval $(a, b)$, and $x + \mathbb{Q}$ meets $(a, b)$ because $\mathbb{Q}$ is dense. So the saturation of any non-empty open set is all of $\mathbb{R}$, and the only open sets of the quotient are $\varnothing$ and everything. Not Hausdorff: it has more than one point but no two disjoint non-empty open sets.

Solution

Longitudes differing by $360°$ describe the same meridian, so the set of meridians is $\mathbb{R}/360\mathbb{Z}$, and a path crossing the antimeridian is continuous there. If $g : S^1 \to \mathbb{R}$ were continuous and injective, let $m$ and $M$ be its minimum and maximum, attained at $p \neq q$. The two arcs of $S^1$ from $p$ to $q$ are each mapped onto $[m, M]$ by the intermediate value theorem, so some value in $(m, M)$ is taken on both arcs, contradicting injectivity.

Solution

(a) $\phi^p = \operatorname{id}$ since $\zeta^p = 1$, and $\phi$ preserves $|z|^2 + |w|^2$. (b) $\phi^k(z, w) = (\zeta^kz, \zeta^{kq}w)$ for $0 < k < p$. A fixed point needs $\zeta^kz = z$ and $\zeta^{kq}w = w$. Since $\zeta^k \neq 1$, $z = 0$, so $|w| = 1$ and we need $\zeta^{kq} = 1$, i.e. $p \mid kq$. If $\gcd(p, q) = 1$ this forces $p \mid k$, impossible; if $d = \gcd(p, q) > 1$, take $k = p/d$, and the points $(0, w)$ are fixed.