Manifolds¶
Summary
A manifold is a space that locally looks like Euclidean space \(\mathbb{R}^n\) but may be globally complicated. The notion is the proper generalisation of "curve" and "surface" to arbitrary dimensions and is the kind of object the Poincaré Conjecture talks about.
1. Locally Euclidean – the basic idea¶
Up close, a circle looks like a piece of a line, and a sphere looks like a piece of a plane. This local agreement with Euclidean space makes such objects tractable: in the small one may compute as in \(\mathbb{R}^n\), while the topology controls global behaviour.
The formal definition:
A Hausdorff space \(M\) is an \(n\)-dimensional topological manifold if every point \(p \in M\) has an open neighbourhood \(U\) that is homeomorphic to an open subset of \(\mathbb{R}^n\).
Such a homeomorphism \(\varphi \colon U \to \varphi(U) \subseteq \mathbb{R}^n\) is called a chart, a family of charts covering all of \(M\) an atlas. The terminology comes directly from cartography: the surface of the Earth – a 2-manifold – is described by finitely many flat charts.
"A manifold is a topological space that locally looks like Euclidean space." — John M. Lee, Introduction to Smooth Manifolds (2013), p. 1
2. Examples in low dimensions¶
Dimension 1. The only connected closed 1-manifolds without boundary are the circle \(S^1\) and the line \(\mathbb{R}\). Other examples: open and half-open intervals.
Dimension 2. Closed surfaces are classified by two pieces of data: the genus \(g\) (number of handles) and orientability. Orientable representatives are the sphere \(S^2\) (\(g = 0\)), the torus \(T^2 = S^1 \times S^1\) (\(g = 1\)), the genus-2 surface, and so on. Non-orientable examples are the projective plane \(\mathbb{RP}^2\) and the Klein bottle.
Dimension 3. Classification here is far harder and is the subject of the Geometrization Conjecture (see Article 05). The simplest examples are the 3-sphere \(S^3\), the 3-torus \(T^3 = S^1 \times S^1 \times S^1\), and \(S^2 \times S^1\).
3. Closed, open, with boundary¶
Three adjectives appear in every discussion of manifolds:
Closed. A manifold is closed if it is compact and has no boundary. Sphere, torus and projective plane are closed; the open plane \(\mathbb{R}^2\) is not. Important: in topology "closed" is not the opposite of "open" as for subsets.
Open. In the context of manifolds, open usually means "non-compact and without boundary" – e.g. \(\mathbb{R}^n\) or an open half-plane.
With boundary. An \(n\)-manifold with boundary additionally allows charts whose image lies in the closed half-space \(\{x \in \mathbb{R}^n : x_n \geq 0\}\). The boundary \(\partial M\) is itself a closed \((n-1)\)-manifold. Example: the closed ball \(D^3\) is a 3-manifold with boundary \(\partial D^3 = S^2\).
The Poincaré Conjecture concerns closed 3-manifolds.
4. Smooth manifolds¶
On a topological manifold one can define notions like "continuity" and "homeomorphism", but not yet differentiation. For that one needs an extra structure:
An atlas is smooth (\(C^\infty\)) if all transition maps \(\varphi_j \circ \varphi_i^{-1}\) – homeomorphisms between open subsets of \(\mathbb{R}^n\) – are infinitely differentiable. A smooth manifold is a topological manifold equipped with a maximal smooth atlas.
On a smooth manifold one can speak of smooth functions \(f \colon M \to \mathbb{R}\), smooth maps \(M \to N\), and diffeomorphisms – homeomorphisms whose inverses are also smooth. Diffeomorphism is the natural equivalence for smooth manifolds.
The relation between topological and smooth classification is subtle:
- In dimension \(\leq 3\) every topological manifold has a unique smooth structure (Moise 1952).
- In dimension 4 there exist versions of \(\mathbb{R}^4\) that are homeomorphic but not diffeomorphic to standard \(\mathbb{R}^4\) – a discovery of Donaldson (1983) and Freedman.
- In dimension 7 Milnor (1956) found seven pairwise non-diffeomorphic smooth structures on the topological 7-sphere – the famous exotic spheres.
For the Poincaré Conjecture in dimension 3 this distinction is irrelevant: topological and smooth coincide. Perelman's proof nevertheless works in the smooth category, because the Ricci flow is a differential equation for smooth Riemannian metrics.
5. Riemannian manifolds¶
A smooth manifold supports differentiation but not yet measurement. A Riemannian metric \(g\) assigns to each point \(p \in M\) an inner product \(g_p\) on the tangent space \(T_pM\), depending smoothly on \(p\). With it one can define lengths of curves, angles, volumes, and – via the curvature tensor – the local geometry.
A Riemannian manifold \((M, g)\) is the central stage of Act 2 of the Poincaré storyline: Hamilton's Ricci flow \(\partial_t g_{ij} = -2 R_{ij}\) is precisely an evolution equation for such metrics (see Act 2).
Important for understanding the conjecture: which Riemannian metric one chooses on a manifold is topologically irrelevant – the conjecture is a statement about the manifold, not about the metric. In the proof, however, a metric is chosen on purpose, its evolution under the Ricci flow is controlled, and a topological conclusion is drawn at the end.
6. Examples for the storyline¶
Three manifolds appear repeatedly in the Poincaré discussion:
The 3-sphere \(S^3\). Definable as \(\{x \in \mathbb{R}^4 : \|x\| = 1\}\) or, dually, as the one-point compactification of \(\mathbb{R}^3\). It is the only closed simply connected 3-manifold – this is precisely the Poincaré Conjecture.
The 3-torus \(T^3\). The product \(S^1 \times S^1 \times S^1\), intuitively a cube with opposite faces identified. Closed but not simply connected; its fundamental group is \(\mathbb{Z}^3\).
Lens spaces \(L(p,q)\). Quotients of the 3-sphere \(S^3\) under a free action of a finite cyclic group \(\mathbb{Z}/p\). They provide examples of closed 3-manifolds with finite fundamental group – topologically non-trivial, in the sense of the conjecture "not simply connected".
7. What comes next¶
The next article introduces the notions of loop, homotopy and fundamental group and examines the \(n\)-dimensional sphere \(S^n\) in greater detail. Only then can one say precisely what simply connected means and why this property singles out the 3-sphere among all closed 3-manifolds.
| Article | Topic |
|---|---|
| 03 – The Sphere and Simple Connectedness | \(S^n\), fundamental group, homotopy |
| 04 – What Is the Poincaré Conjecture? | Original formulation 1904 |
Background knowledge
Tangent spaces, tensors and curvature are prepared informally in the Background Knowledge block "Geometry and Analysis (Advanced)". For this article the picture "locally like \(\mathbb{R}^n\)" suffices.
Sources¶
- John M. Lee: Introduction to Topological Manifolds, 2nd ed., Springer (2011)
- John M. Lee: Introduction to Smooth Manifolds, 2nd ed., Springer (2013)
- John Milnor: On manifolds homeomorphic to the 7-sphere, Annals of Mathematics 64 (1956), 399–405
- Edwin E. Moise: Affine structures in 3-manifolds. V. The triangulation theorem and Hauptvermutung, Annals of Mathematics 56 (1952), 96–114