The Sphere and Simple Connectedness¶

Summary

The $n$-sphere $S^n$ is the model manifold of topology and the central protagonist of the Poincaré Conjecture. Through loops and homotopy one defines the fundamental group $\pi_1(M)$; if it vanishes, $M$ is called simply connected. Precisely this property is supposed to single out the 3-sphere among all closed 3-manifolds.

1. The $n$-sphere¶

The $n$-sphere is defined as the unit sphere in $\mathbb{R}^{n+1}$:

\[ S^n = \{x \in \mathbb{R}^{n+1} : \|x\| = 1\}. \]

It is a closed, smooth $n$-manifold. A convenient equivalent description comes from stereographic projection: remove a single point – say the north pole – and the rest is mapped homeomorphically onto $\mathbb{R}^n$. The sphere is therefore the one-point compactification of $\mathbb{R}^n$:

\[ S^n \cong \mathbb{R}^n \cup \{\infty\}. \]

In low dimensions this gives familiar pictures. $S^0$ consists of two points, $S^1$ is the circle, $S^2$ the ordinary sphere, $S^3$ sits in $\mathbb{R}^4$ and can be visualised as two glued solid balls $D^3 \cup_{S^2} D^3$. This very 3-sphere $S^3$ is the object whose topological uniqueness Poincaré conjectured in 1904.

2. Paths and loops¶

A path in a topological space $X$ is a continuous map $\gamma \colon [0,1] \to X$. If start and end agree, that is $\gamma(0) = \gamma(1) = x_0$, one calls it a loop with base point $x_0$. Intuitively: a connected thread that begins and ends at a fixed point.

Loops can be combined. Given two loops $\alpha, \beta$ at the same base point, the concatenation $\alpha \cdot \beta$ first traces $\alpha$ and then $\beta$ (each at double speed). The constant loop $c_{x_0}$ stays at $x_0$, and the inverse $\bar\alpha(t) = \alpha(1-t)$ runs $\alpha$ backwards. After identifying homotopic loops, this structure becomes a group.

3. Homotopy¶

Two continuous maps $f, g \colon X \to Y$ are homotopic if one can be continuously deformed into the other. Formally:

A homotopy between $f$ and $g$ is a continuous map $H \colon X \times [0,1] \to Y$ with $H(\cdot, 0) = f$ and $H(\cdot, 1) = g$.

For loops one additionally requires that the base point is held fixed during the deformation (base-point-preserving homotopy). Intuitively: the loop may stretch, shrink and bend arbitrarily, the thread must not be cut, and the seam at the base point stays sewn.

The homotopy relation is an equivalence relation on loops. We write $[\alpha]$ for the class of all loops homotopic to $\alpha$.

4. The fundamental group¶

On the set of homotopy classes the concatenation induces a well-defined operation. This yields:

The fundamental group of $X$ at base point $x_0$ is $$\pi_1(X, x_0) = \{[\alpha] : \alpha \text{ a loop at } x_0\}$$ with operation $[\alpha][\beta] = [\alpha \cdot \beta]$, identity $[c_{x_0}]$ and inverse $[\alpha]^{-1} = [\bar\alpha]$.

For path-connected spaces $\pi_1(X, x_0)$ is independent of the base point up to isomorphism; one then writes $\pi_1(X)$. The fundamental group is a topological invariant: homeomorphic spaces have isomorphic fundamental groups.

"The fundamental group $\pi_1(X)$ is the most basic and most useful of the algebraic invariants associated to a topological space." — Allen Hatcher, Algebraic Topology (2002), p. 25

5. Simple connectedness¶

If the fundamental group vanishes – that is, every loop can be contracted continuously to a point – the space is called simply connected:

\[ X \text{ simply connected} \;:\Longleftrightarrow\; X \text{ path-connected and } \pi_1(X) = 0. \]

Visual criterion: a rubber band placed anywhere on $X$ can be pulled to a point without leaving $X$ and without breaking.

6. Fundamental groups of some spaces¶

Three examples worth remembering for the Poincaré story:

The sphere $S^n$ for $n \geq 2$. Every loop on the sphere can be contracted to a point: $\pi_1(S^n) = 0$. Intuitively, slide the rubber band around the surface to a pole. So $S^n$ is simply connected for $n \geq 2$.

The circle $S^1$. A loop that winds $k$ times around the circle cannot be contracted without crossing through the other side. We have

\[ \pi_1(S^1) \cong \mathbb{Z}, \]

with the winding number giving the isomorphism. $S^1$ is not simply connected.

The torus $T^2 = S^1 \times S^1$. There are two independent generating loops, one around each $S^1$-factor. The fundamental group is abelian:

\[ \pi_1(T^2) \cong \mathbb{Z} \oplus \mathbb{Z}. \]

In dimension 3 the 3-torus $T^3$ has $\pi_1(T^3) \cong \mathbb{Z}^3$, and the lens spaces $L(p, q)$ have finite fundamental groups $\mathbb{Z}/p$.

7. Higher homotopy groups¶

Loops are maps $S^1 \to X$. Replacing $S^1$ by $S^k$ yields the higher homotopy groups $\pi_k(X)$. For the sphere itself one has $\pi_n(S^n) \cong \mathbb{Z}$, classified by the mapping degree. The higher $\pi_k(S^n)$ for $k > n$ are highly nontrivial – their study was one of the driving topics of algebraic topology in the twentieth century.

For the Poincaré Conjecture only $\pi_1$ matters: $S^n$ is simply connected for $n \geq 2$, that is, $\pi_1(S^n) = 0$. "Simply connected" is precisely $\pi_1 = 0$.

8. Relation to the conjecture¶

The Poincaré Conjecture in dimension 3 states:

\[ M^3 \text{ closed and simply connected} \;\Longrightarrow\; M^3 \cong S^3. \]

This makes plain why the sphere and its fundamental-group profile are central: $\pi_1(S^3) = 0$ is the characterising property that is supposed to single it out among all closed 3-manifolds. In all other dimensions $n \geq 5$ the analogous statement was proved by Smale (1961), in dimension 4 by Freedman (1982). In dimension 3 it was Perelman who succeeded.

"The Poincaré Conjecture says that $S^3$ is the only closed 3-manifold whose fundamental group is trivial." — John W. Morgan, Gang Tian, Ricci Flow and the Poincaré Conjecture (2007), p. 3

9. Outlook¶

With manifold, sphere, homotopy and fundamental group in place, all the ingredients are there to phrase the conjecture precisely. The next article traces its history – from Poincaré's Analysis Situs (1904) through the higher-dimensional resolutions to Hamilton's programme.

Article	Topic
04 – What Is the Poincaré Conjecture?	Original formulation 1904, higher dimensions
05 – Thurston's Geometrization Conjecture	Thurston's bigger picture

Sources¶

Allen Hatcher: Algebraic Topology, Cambridge University Press (2002), Chapter 1
John M. Lee: Introduction to Topological Manifolds, 2nd ed., Springer (2011), Chapter 7
John W. Morgan, Gang Tian: Ricci Flow and the Poincaré Conjecture, Clay Mathematics Monographs 3, AMS (2007), Chapter 1
Henri Poincaré: Cinquième complément à l'Analysis Situs, Rendiconti del Circolo Matematico di Palermo 18 (1904), 45–110