Papakyriakopoulos' Sphere Theorem and Loop Theorem

Theorem6.2Loop Theorem and Sphere Theorem

(Loop Theorem, Papakyriakopoulos 1957) Let $M$ be a 3-manifold with boundary component $S$ . If the induced map $\pi_1(S) \to \pi_1(M)$ is not injective (i.e., there is a loop in $S$ that is null-homotopic in $M$ but not in $S$ ), then there exists a properly embedded disk $D \subset M$ with $\partial D \subset S$ such that $[\partial D] \neq 0$ in $\pi_1(S)$ .

(Sphere Theorem) If $M$ is a compact orientable 3-manifold with $\pi_2(M) \neq 0$ , then there exists an embedded 2-sphere $S^2 \hookrightarrow M$ representing a nontrivial element of $\pi_2(M)$ .

Proof (Key Ideas)

Proof

Proof of the Loop Theorem (following Papakyriakopoulos' tower construction, simplified by Stallings).

Step 1: Singular disk. The hypothesis gives a map $f: (D^2, \partial D^2) \to (M, S)$ with $f|_{\partial D^2}$ non-null-homotopic in $S$ . The map $f$ may have self-intersections.

Step 2: Tower construction. Consider the singularities of $f$ (the set where $f$ is not injective). The idea is to "unstack" self-intersections by passing to covering spaces.

Define the tower: $M_0 = M$ , and inductively, if the image of the disk in $M_i$ has self-intersections, let $\tilde{M}_{i+1} \to M_i$ be an appropriate covering space where the disk lifts with fewer self-intersections. Precisely: if $f_i: D^2 \to M_i$ has a self-intersection, there exists a loop in $M_i$ along which $f_i$ intersects itself. Take the covering corresponding to the subgroup that kills this loop.

Step 3: Tower terminates. The tower $M_0 \leftarrow M_1 \leftarrow \cdots$ terminates after finitely many steps because each covering increases the topology of the boundary (measured by its genus or complexity). Since we are working with compact manifolds, this cannot continue indefinitely.

Step 4: Embedded disk at the top. At the top of the tower $M_k$ , the lifted disk is embedded (no more self-intersections to resolve). We have an embedded disk $D_k \subset M_k$ with $\partial D_k$ non-trivial on $\partial M_k$ .

Step 5: Push down. Project $D_k$ back down through the tower. At each stage, the projection may re-introduce intersections, but a cut-and-paste argument (surgery on the disk) allows us to find an embedded disk at each level while preserving the non-triviality of the boundary curve. After finitely many such operations, we obtain an embedded disk $D \subset M$ with $[\partial D] \neq 0$ in $\pi_1(S)$ .

Sphere Theorem. Similar tower argument: a non-trivial element of $\pi_2(M)$ is represented by a map $f: S^2 \to M$ . The tower construction eliminates self-intersections, yielding an embedded $S^2$ representing a non-trivial class. An additional argument using equivariant techniques ensures the embedded sphere represents the original non-trivial element (not just some non-trivial element). $\square$

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ExampleApplications to Knot Theory

Asphericity of knot complements: For a knot $K \subset S^3$ , the sphere theorem implies that $\pi_2(S^3\setminus K) = 0$ (any embedded sphere in the complement bounds a ball, since $S^3\setminus K$ is irreducible). Thus knot complements are aspherical ( $K(\pi,1)$ spaces), and the knot group determines all the homotopy theory. Incompressibility of Seifert surfaces: The loop theorem implies that a minimal-genus Seifert surface is $\pi_1$ -injective (incompressible) in the knot complement.

RemarkDehn's Lemma and the Disk Theorem

Dehn's lemma (proved by Papakyriakopoulos 1957, correcting Dehn's flawed 1910 proof) states: if $f: D^2 \to M^3$ is a map with $f|_{\partial D^2}$ an embedding and $f$ an embedding near $\partial D^2$ , then $f|_{\partial D^2}$ extends to an embedding of $D^2$ . This is closely related to the loop theorem and was historically the first of these results to be proved. Together, the loop theorem, sphere theorem, and Dehn's lemma are the foundational tools of classical 3-manifold topology, enabling the theory of incompressible surfaces and Haken manifolds.