Cone pictures and asphericity

Written by Claude Opus 5.5 (Anthropic), October 2026. Self-checked by the writing AI. Public domain (CC0).

The group of the construction is the fundamental group of a two-dimensional complex \(X\): the rose over the alphabet, with a cone attached along every component of the random graph \(\Gamma\). This lesson proves that \(X\) is aspherical in the sense that \(\pi_2(X)=0\), and that every path from \(x_A\) or \(x_B\) to a different vertex has a nontrivial label, whenever \(\Gamma\) admits no reduced spherical arrangement [OpenAI-KZ, Section 5]. A failure of either statement is represented by a map of a sphere or a disk which, outside finitely many disks mapped through the cones, lands in the rose: a cone picture. A picture of least total boundary length is tightened, preimages of midpoints of the rose edges pair its letters, and a surgery along a band shows that in such a picture partners traverse distinct edges. The minimal picture is then a reduced spherical arrangement, which Planar separation and the absence of arrangements excludes.

Signed alphabets, labelled graphs, the rose \(F\) and labels are as in Group algebras and zero divisors; reduced spherical arrangements are as in Definition 1.1 of Spherical arrangements and planar intervals, together with the facts (SURF) stated there. Throughout, \(\Gamma=\Gamma_A\sqcup\Gamma_B\) is an immersed labelled graph over \(T\) with the prescribed types and the prescribed vertices \(x_A,x_B\), for instance one of the graphs of Proposition 3.1 of Planar separation and the absence of arrangements. The hypothesis that it admits no reduced spherical arrangement is used only in Section 6.

1. The cone complex

Let \(o\) be the vertex of the rose \(F\). The labelled graph \(\Gamma\) maps to \(F\) by a map \(\lambda\) sending every vertex to \(o\) and every edge homeomorphically on its interior onto the loop of its letter, so that reading the edge with letter \(t\) traverses the loop in the direction \(t\). For a component \(\Lambda\) of \(\Gamma\) let \[ K_\Lambda=\Lambda\times[0,1]\,/\,\Lambda\times\{1\} \] be its cone, containing \(\Lambda=\Lambda\times\{0\}\). The cone complex \(X\) is obtained from \(F\) and the cones \(K_\Lambda\) of all components by identifying every point \((z,0)\) of a cone with \(\lambda(z)\in F\); write \(\kappa_\Lambda\colon K_\Lambda\to X\) for the resulting maps. It is a finite two-dimensional CW complex: its vertices are \(o\) and the cone points, its edges are the loops of \(F\) and the segments \(z\times[0,1]\) for the vertices \(z\) of \(\Gamma\), and its \(2\)-cells are the triangles \(e\times[0,1]\) for the edges \(e\) of \(\Gamma\). Put \[ G=\pi_1(X,o), \] and let \(\varphi(t)\in G\) be the class of the loop of \(t\), traversed in the direction \(t\); then \(\varphi(\bar t)=\varphi(t)^{-1}\). The label of a path in \(\Gamma\) is the product of the \(\varphi\)-values of its letters, as in Section 3 of Group algebras and zero divisors; it is the class of the loop \(\lambda\circ P\) at \(o\).

By Lemma 5.1 of Torsion-freeness and the zero divisors, \(G\) is generated by the letters with the labels of the closed paths of \(\Gamma\) as relations: it is the group defined by the labelled graph \(\Gamma\) in the sense of graphical small cancellation theory. That theory gives conditions on the labelled graph under which such groups are well behaved: Ollivier gives a combinatorial proof of Gromov's theorem extending small cancellation theory to presentations from labelled graphs [Ollivier], Gruber extends the classical C(6) and C(7) results to graphical presentations [Gruber], and Bereznyuk studies the asphericity of groups defined by graphs [Bereznyuk]. The argument of this lesson uses no small cancellation condition.

A cone is contractible: \(((z,s),u)\mapsto(z,s+u(1-s))\) contracts it to the cone point.

Lemma 1.1 (closed paths). Every closed path in \(\Gamma\) has label \(1\) in \(G\).

Proof. A closed path in a component \(\Lambda\) is a loop in the contractible space \(K_\Lambda\), hence null-homotopic there relative to its base point. Composing with \(\kappa_\Lambda\) shows that the loop \(\lambda\circ P\) is null-homotopic in \(X\). \(\square\)

Lemma 1.2 (contractible targets). Let \(Y\) be contractible. Every map \(S^k\to Y\) extends to the ball \(D^{k+1}\), and two maps from a closed disk to \(Y\) that agree on its boundary are homotopic relative to the boundary.

Proof. Let \(H\colon Y\times[0,1]\to Y\) be a contraction with \(H_0=\mathrm{id}\) and \(H_1\) constant. A map \(h\colon S^k\to Y\) extends by \(rs\mapsto H(h(s),1-r)\) for \(s\in S^k\), \(0\le r\le1\). For two maps \(f_0,f_1\colon D\to Y\) with the same boundary values, the map on the boundary sphere of \(D\times[0,1]\) given by \(f_0\) on \(D\times0\), \(f_1\) on \(D\times1\) and \(f_0(x)\) at \((x,u)\) for \(x\in\partial D\) extends over \(D\times[0,1]\), which is homeomorphic to a ball with that sphere as boundary; the extension is the required homotopy. \(\square\)

2. Topological inputs

We use the following results from earlier lessons of the programme; the labels are used in the proofs below.

(PI1) (fundamental groups of graphs and of \(2\)-complexes; Fundamental groups of graphs and two-complexes, Corollary 2.4 and Theorem 3.3).

  1. Let \(Y^1\) be a connected graph with a spanning tree \(T_0\) and a vertex \(v_0\). For each edge \(e\) not in \(T_0\), oriented from \(u\) to \(w\), let \(c_e\) be the closed path at \(v_0\) that runs in \(T_0\) to \(u\), along \(e\), and in \(T_0\) back to \(v_0\). Then \(\pi_1(Y^1,v_0)\) is free with basis the classes of the \(c_e\).
  2. If \(Y\) is obtained from a path-connected Hausdorff space \(Y^1\) by attaching closed disks along loops, the inclusion induces a surjection \(\pi_1(Y^1,v_0)\to\pi_1(Y,v_0)\) whose kernel is the normal subgroup generated by the attaching loops, joined to \(v_0\) by paths.

(COV) (covering spaces; Universal covers and cellular chains, Theorem 1.1, Corollary 2.3 and Proposition 2.2, and the homotopy lifting property Algebraic Topology, Lecture 9, Corollary 9.4). Let \(Y\) be a connected CW complex and \(y_0\) a vertex. There is a covering map \(p\colon\widetilde Y\to Y\) with \(\widetilde Y\) simply connected, the universal cover, and:

  1. \(\widetilde Y\) is a CW complex whose cells are the lifts of the cells of \(Y\). The group \(\pi_1(Y,y_0)\) acts on \(\widetilde Y\) by cellular homeomorphisms commuting with \(p\), the deck transformations, permuting the lifts of every cell freely and transitively, and transitively on every fibre of \(p\).
  2. Every map from a simply connected, locally path-connected space to \(Y\) lifts to \(\widetilde Y\), and two lifts of a map from a connected space that agree at one point agree everywhere.
  3. Homotopies lift: for \(H\colon Z\times[0,1]\to Y\) and a lift of \(H(\cdot,0)\) there is a lift of \(H\) extending it.
  4. \(p\) induces isomorphisms \(\pi_n(\widetilde Y)\to\pi_n(Y)\) for \(n\ge2\).

(CELL) (cellular homology).

  1. (Universal covers and cellular chains, Section 5 and Proposition 5.2.) Let \(Y\) be a CW complex of dimension at most two whose \(2\)-cells are attached along closed edge paths. Its singular homology \(H_*(Y;\mathbb Z)\) is the homology of the chain complex \(0\to C_2\to C_1\to C_0\to0\), where \(C_k\) is free abelian on the \(k\)-cells, each with a chosen orientation; an edge from \(u\) to \(w\) has boundary \(w-u\), and a \(2\)-cell attached along the closed path \(e_1^{\epsilon_1}\cdots e_r^{\epsilon_r}\) has boundary \(\sum_j\epsilon_je_j\). In particular \(H_k(Y)=0\) for \(k\ge3\) and \(H_2(Y)=\ker(C_2\to C_1)\). With the augmentation \(C_0\to\mathbb Z\) sending every vertex to \(1\), the augmented complex computes the reduced homology \(\widetilde H_*(Y;\mathbb Z)\).
  2. For a triangulated oriented sphere \(S\), the sum of its coherently oriented triangles, as singular simplices, represents the fundamental class \([S]\in H_2(S;\mathbb Z)\). This part is not proved in the programme [Hatcher, Theorem 2.27 and Section 3.3]; it is recorded as an open obligation of this course.

(PL) (simplicial approximation; parts 1 and 2 are Cell complexes and homotopy equivalences, Propositions 5.1 and 5.2).

  1. A finite CW complex of dimension two whose \(2\)-cells are attached along closed edge paths of a graph has a triangulation in which the graph, with its edges subdivided, is a subcomplex.
  2. A map \(f\colon K\to M\) from a finite simplicial complex to a simplicial complex is homotopic to a map \(g\), simplicial with respect to an iterated barycentric subdivision of \(K\), such that for every \(x\) the point \(g(x)\) lies in the closed simplex of \(M\) containing \(f(x)\) in its interior; the homotopy moves \(f(x)\) to \(g(x)\) along a segment of that simplex. In particular, points mapped into a subcomplex stay in it.
  3. Let \(W\) be a compact surface and \(g\colon W\to F\) a map that runs along every boundary circle of \(W\) as a closed edge path: the circle is divided into arcs mapped homeomorphically onto loops of \(F\), with endpoints at \(o\). Then \(g\) is homotopic relative to \(\partial W\) to a map \(g'\) such that for the midpoint \(m_\epsilon\) of every loop \(\epsilon\) of \(F\), the preimage \(g'^{-1}(m_\epsilon)\) is a compact one-dimensional submanifold of \(W\), meeting \(\partial W\) exactly in its own boundary points, and \(g'\) maps the two sides of this submanifold near each of its points to the two sides of \(m_\epsilon\) in \(\epsilon\).

Part 3 is obtained from part 2 by making \(g\) simplicial on a triangulation of \(W\) for which the boundary map is simplicial: the preimage of the midpoint of an edge of the target then consists of one segment in each triangle mapped onto that edge, and these segments join across the edges of the triangulation. The existence of such a triangulation of \(W\), and the form of part 2 relative to \(\partial W\) that this argument uses, are not proved in the programme [Hatcher, Theorem 2C.1 and its proof]; part 3 is recorded, with (CELL 2), as an open obligation of this course.

3. Cone pictures

Let \(\Sigma\) be an oriented sphere or a closed oriented disk.

Definition 3.1 (cone pictures). A cone picture on \(\Sigma\) is a map \(f\colon\Sigma\to X\) together with pairwise disjoint closed disks \(D_1,\ldots,D_k\) in the interior of \(\Sigma\), the inner disks, such that:

  1. \(f\) maps the complement of the interiors of the inner disks into \(F\);
  2. for each \(i\) there are a component \(\Lambda_i\) of \(\Gamma\), a closed path \(p_i\) in \(\Lambda_i\), the boundary path, and a map \(f_i\colon D_i\to K_{\Lambda_i}\) with \(f=\kappa_{\Lambda_i}\circ f_i\) on \(D_i\), such that \(f_i\) runs along \(\partial D_i\) as the closed path \(p_i\): the circle is divided into \(|p_i|\) consecutive arcs mapped homeomorphically onto the edges of \(p_i\), or mapped to one vertex if \(|p_i|=0\);
  3. if \(\Sigma\) is a disk, there is a path \(P\) in \(\Gamma\), the outer path, such that \(f\) runs along \(\partial\Sigma\), from a marked point once around, as \(\lambda\circ P\).

Its length is \(\sum_i|p_i|\), plus \(|P|\) if \(\Sigma\) is a disk.

Let \(p\colon\widetilde X\to X\) be the universal cover. A map \(f\colon S^2\to X\) is essential if for a lift \(\tilde f\colon S^2\to\widetilde X\), which exists by (COV 2), the class \(\tilde f_*[S^2]\in H_2(\widetilde X;\mathbb Z)\) is nonzero. This does not depend on the lift: two lifts differ by a deck transformation, by (COV 1) and (COV 2), and a deck transformation is a homeomorphism. Homotopic maps are essential or not together, since a homotopy lifts by (COV 3) and homotopic maps induce the same map on homology.

Lemma 3.2 (essential spheres and \(\pi_2\)). If no map \(S^2\to X\) is essential, then \(\pi_2(X)=0\).

Proof. Let \(g\colon S^2\to\widetilde X\) be a map. It is a lift of \(p\circ g\), which is not essential, so \(g_*[S^2]=0\). Since \(\widetilde X\) is simply connected, the Hurewicz theorem with \(q=2\) gives \(\pi_2(\widetilde X)=0\), and \(\pi_2(X)\cong\pi_2(\widetilde X)\) by (COV 4). \(\square\)

Lemma 3.3 (pictures; [OpenAI-KZ, Lemma 5.2]).

  1. Every essential map \(S^2\to X\) is homotopic to an essential map that is part of a cone picture.
  2. If a path \(P\) in \(\Gamma\) has label \(1\) in \(G\), there is a cone picture on a disk with outer path \(P\).

Proof. A model with disks. For every component \(\Lambda\) choose a spanning tree and a base vertex, and let the \(c_e\) be the closed paths of (PI1 1). Let \(Y_\Lambda\) be \(\Lambda\) with a closed disk attached along every \(c_e\). By (PI1), \(Y_\Lambda\) is simply connected. By (CELL 1), \(H_2(Y_\Lambda)\) is the kernel of the boundary map on the \(2\)-cells; the boundary of the disk of \(e\) is \(e\) plus a combination of tree edges, and these boundaries are linearly independent, since each contains its own edge \(e\) outside the tree. So \(H_2(Y_\Lambda)=0\), and \(\pi_2(Y_\Lambda)=0\) by the Hurewicz theorem. Consequently every map \(S^k\to Y_\Lambda\) with \(k\le2\) extends to \(D^{k+1}\).

Comparison of the cone with the model. The cone \(K_\Lambda\) arises from \(\Lambda\) by attaching the cone point, the segments over the vertices and the triangles over the edges. Extending the identity of \(\Lambda\) cell by cell, using the extension property of \(Y_\Lambda\), gives a map \(\phi\colon K_\Lambda\to Y_\Lambda\) that is the identity on \(\Lambda\); extending the identity over the disks of \(Y_\Lambda\), using Lemma 1.2, gives \(\psi\colon Y_\Lambda\to K_\Lambda\) that is the identity on \(\Lambda\). The composites \(\psi\phi\) and \(\phi\psi\) are homotopic to the identities relative to \(\Lambda\): a homotopy is a map on the product with \([0,1]\), prescribed on the two ends and on \(\Lambda\times[0,1]\), and it extends over the remaining product cells, of dimension at most three, by Lemma 1.2 for \(K_\Lambda\) and by the extension property for \(Y_\Lambda\).

Let \(X'\) be \(F\) with a disk attached along \(\lambda\circ c_e\) for every component \(\Lambda\) and every \(e\). The maps \(\phi\) and \(\psi\) are the identity on \(\Lambda\), so together with the identity of \(F\) they define maps \(\Phi\colon X\to X'\) and \(\Psi\colon X'\to X\), and the homotopies combine to homotopies \(\Psi\Phi\simeq\mathrm{id}\) and \(\Phi\Psi\simeq\mathrm{id}\) relative to \(F\).

Transversality. Let \(f\colon\Sigma\to X\) be the given map: an essential map of a sphere, or a map of a disk running along the boundary as \(\lambda\circ P\), which exists because the loop \(\lambda\circ P\) is null-homotopic. Triangulate \(X'\) by (PL 1) and apply (PL 2) to \(\Phi\circ f\). In the disk case the boundary stays in \(F\) during the homotopy, and we add to \(\Sigma\) an outer collar mapped by the homotopy on the boundary, so that the new disk again runs along its boundary as \(\lambda\circ P\). We obtain a map \(g\), simplicial with respect to a triangulation of \(\Sigma\), homotopic to \(\Phi\circ f\) in the sphere case.

In each attached disk \(r\) of \(X'\) choose a point \(p_r\) in the interior of a triangle inside the open disk. Its preimage under \(g\) consists of one interior point of each triangle of \(\Sigma\) mapped by \(g\) onto that triangle; triangles mapped to lower-dimensional simplices miss \(p_r\). Small closed disks around these points, inside their triangles, are mapped homeomorphically onto a small disk \(U_r\) around \(p_r\). Let \(\rho\colon X'\to X'\) be the identity on \(F\) and, on each attached disk, the radial map that stretches \(U_r\) onto the whole disk and pushes the rest of it onto its boundary; it is homotopic to the identity relative to \(F\). Then \(\rho\circ g\) maps the complement of the small disks into \(F\), and each small disk onto an attached disk, homeomorphically on the interior, running along its boundary as \(\lambda\circ c_e\) or its inverse.

Back to the cones. The map \(\Psi\circ\rho\circ g\) is homotopic to \(\Psi\circ\Phi\circ f\simeq f\) in the sphere case, so it is essential, and in the disk case it runs along the boundary as \(\lambda\circ P\). On a small disk over the disk of \(e\) in \(Y_\Lambda\) it factors as \(\kappa_\Lambda\circ\psi\) composed with the map onto that disk, so it factors through \(K_\Lambda\) and runs along the boundary as the closed path \(c_e\) or its inverse. Outside the small disks it maps into \(F\). This is a cone picture. \(\square\)

4. Minimal failure pictures

Suppose that a conclusion of Proposition 6.1 below fails. In the sphere case, some map \(S^2\to X\) is essential, and \(\mathcal P\) is the set of cone pictures on spheres whose maps are essential; the homotopy class may vary. In the disk case, some path from \(x\in\{x_A,x_B\}\) to a vertex \(y\ne x\) has label \(1\); we fix such \(x,y\), and \(\mathcal P\) is the set of cone pictures on disks whose outer path runs from \(x\) to \(y\). By Lemma 3.3, \(\mathcal P\) is not empty. Choose a picture in \(\mathcal P\) of least length.

Lemma 4.1 (tightening). The chosen picture can be changed, without changing its length and staying in \(\mathcal P\), so that every boundary path is closed, cyclically immersed and of positive length, the outer path in the disk case is immersed and of positive length, and there is at least one inner disk.

Proof. If a boundary path \(p_i\) has a backtrack, possibly across its end and start, removing the two edges gives a closed path \(p_i'\) of length \(|p_i|-2\), freely homotopic to \(p_i\) in \(\Lambda_i\). Shrink \(D_i\) to a smaller concentric disk \(D_i'\), map the annulus between them into \(\Lambda_i\) by this free homotopy, and \(D_i'\) into \(K_{\Lambda_i}\) by the cone on \(p_i'\). The new map on \(D_i\) factors through \(K_{\Lambda_i}\) and agrees with the old one on \(\partial D_i\), so by Lemma 1.2 it is homotopic to it relative to \(\partial D_i\); the annulus maps into \(\Lambda_i\), hence by \(\lambda\) into \(F\). This is a picture in \(\mathcal P\) of smaller length, which is impossible. In the same way, removing a backtrack of the outer path by a collar mapped into \(F\) keeps its ends \(x,y\) and shortens the picture. So the boundary paths are cyclically immersed and the outer path is immersed; the outer path has positive length since \(x\ne y\).

If \(|p_i|=0\), the map \(f_i\) is constant on \(\partial D_i\), at a vertex \(z\); replace it by the constant map at \(z\), homotopic relative to \(\partial D_i\) by Lemma 1.2, and drop \(D_i\) from the inner disks. The length does not change.

Suppose there is no inner disk. In the sphere case, a lift of \(f\) maps into the preimage of \(F\) in \(\widetilde X\), a CW complex of dimension one by (COV 1), whose second homology vanishes by (CELL 1); so \(f\) is not essential. In the disk case, the loop \(\lambda\circ P\) is null-homotopic in \(F\). But \(P\) is immersed and of positive length, so its letters form a nonempty word without cancelling turns, which is a nontrivial element of the free group \(\pi_1(F,o)\) by (PI1 1), applied to \(F\) with the tree \(\{o\}\). Both cases are impossible. \(\square\)

From now on the picture is as in Lemma 4.1. In the disk case, glue an exterior disk \(D_{\rm ext}\) to \(\Sigma\) along its boundary, which gives a sphere \(S\); no map on \(D_{\rm ext}\) is needed. Orient \(S\) so that the boundary of \(D_{\rm ext}\), oriented by \(D_{\rm ext}\), reads \(P\) from \(x\) to \(y\), and mark the break on it at the common image of the start and end of \(P\). Reversing the orientation of \(S\) only reverses the boundary paths of the inner disks. In the sphere case put \(S=\Sigma\). The disks of the picture are the inner disks and, in the disk case, \(D_{\rm ext}\); each carries its path, read in its boundary orientation, divided into occurrences as in Definition 1.1 of Spherical arrangements and planar intervals.

Pairing arcs. Let \(W\) be the complement in \(S\) of the interiors of the disks. It maps into \(F\) and runs along each boundary circle as a closed edge path. By (PL 3), after a homotopy relative to \(\partial W\), the preimage of the midpoint \(m_\epsilon\) of each loop \(\epsilon\) is a compact one-dimensional submanifold. Each occurrence of the letters of \(\epsilon\) is mapped homeomorphically onto \(\epsilon\), so it contains exactly one point over \(m_\epsilon\), in its interior, and this point is an endpoint of exactly one component of the preimage. Discard the closed components. The remaining components are pairwise disjoint arcs, with interiors outside the disks, and each occurrence contains exactly one arc end, in its interior.

Partners carry inverse letters. Co-orient an arc \(\alpha\) over \(\epsilon\) by the direction \(t\) of \(\epsilon\). A thin neighbourhood of \(\alpha\) is a rectangle with \(\alpha\) as its middle line, meeting \(\partial W\) in its two short sides, and the boundary orientation of \(W\) runs along these two sides in opposite directions across \(\alpha\). So, read in the boundary orientation of \(W\), the two ends of \(\alpha\) carry \(t\) and \(\bar t\). The boundary orientation of each disk is opposite to that of \(W\) along their common circle, by (SURF 4), which replaces both letters by their inverses.

Every condition of a reduced spherical arrangement now holds, except possibly that partners traverse distinct edges of \(\Gamma\).

5. Band surgery

Lemma 5.1 (band surgery; [OpenAI-KZ, Section 5]). In a picture of least length as above, partners traverse distinct edges of \(\Gamma\).

Proof. Suppose an arc \(\alpha\) joins two occurrences \(o,o'\) that traverse the same edge \(e\) of a component \(\Lambda\). Their letters are inverse, so they traverse \(e\) in opposite directions; orient \(e\) from \(u\) to \(v\) so that \(o\) reads \(e\) and \(o'\) reads \(e^{-1}\).

The band. The arc \(\alpha\) maps to \(m_\epsilon\), where \(\epsilon\) is the loop of the letter of \(e\), so a thin closed band \(B\supset\alpha\), meeting \(\partial W\) in short arcs inside \(o\) and \(o'\), maps into the open loop \(\epsilon^\circ\). The map \(\lambda\) restricts to a homeomorphism from the open edge \(e^\circ\) onto \(\epsilon^\circ\), so \(f|_B\) lifts to a map \(\beta\colon B\to e^\circ\subset\Lambda\). On the two short arcs, the boundary paths of the disks containing \(o\) and \(o'\) also map into \(e^\circ\) and compose with \(\lambda\) to \(f\); so they agree with \(\beta\) there. Consequently, if \(o\) and \(o'\) lie on inner disks, these disks and \(B\) together are mapped through \(K_\Lambda\).

Cancellation. Attaching \(B\) to the disks containing \(o\) and \(o'\) produces new boundary circles. Along them, each side of \(B\) is preceded by a part of one of the occurrences \(o,o'\) and followed by a part of the other, and in \(\Lambda\) these three pieces form a path that leaves an end \(w\in\{u,v\}\) of \(e\) into \(e^\circ\), stays in \(e^\circ\), and returns to \(w\). Such a path lies in \(e\) minus its other end, a contractible set, so it contracts to the constant path at \(w\) relative to its ends. Doing this at both sides of \(B\) removes the two occurrences \(o,o'\) and adds no letters; the contractions do not pass through the break. Write paths with the endpoints forced by the concatenations; the four cases are:

  1. Two inner disks, with boundary paths \(ea\) and \(e^{-1}b\), where \(a\colon v\to u\) and \(b\colon u\to v\). The union of the two disks and \(B\) is a disk whose boundary contracts to the closed path \(ab\).
  2. An inner disk and the exterior disk, with paths \(e^{-1}b\) and \(P=aec\), where \(a\colon x\to u\), \(b\colon u\to v\), \(c\colon v\to y\). The union of the two disks and \(B\) is a disk whose boundary contracts to the path \(abc\) from \(x\) to \(y\), with the break kept.
  3. One inner disk twice, with boundary path \(eae^{-1}b\), where \(a\colon v\to v\) and \(b\colon u\to u\). The union of the disk and \(B\) is an annulus, since \(S\) is orientable, and its two boundary circles contract to the closed paths \(a\) and \(b\).
  4. The exterior disk twice, with \(P=aebe^{-1}c\), where \(a\colon x\to u\), \(b\colon v\to v\), \(c\colon u\to y\). The union of \(D_{\rm ext}\) and \(B\) is an annulus whose boundary circles contract to \(ac\), which carries the break, and to \(b\).

The cases with \(e\) and \(e^{-1}\) exchanged are the same after exchanging \(u\) and \(v\). In each case, the new boundary paths have total length smaller by \(2\) than the old ones.

New pictures. In case 1, replace the two inner disks by one inner disk, the union with \(B\), shrunk inside a collar: the collar maps into \(\Lambda\) by the contraction, and the smaller disk into \(K_\Lambda\) by the cone on \(ab\). The new map on the union agrees with the old one on its boundary and both factor through \(K_\Lambda\), so they are homotopic relative to the boundary by Lemma 1.2. The new picture is in \(\mathcal P\) and shorter.

In case 2, the union of \(D_{\rm ext}\), the inner disk and \(B\) is a disk, so by (SURF 5) of Spherical arrangements and planar intervals the closure of its complement in \(S\) is a disk. With the restricted map and a collar mapped into \(F\) by the contraction, it is a disk picture with outer path \(abc\) from \(x\) to \(y\) and the remaining inner disks: in \(\mathcal P\) and shorter.

In case 4, by (SURF 5), the closure of \(S\) minus the annulus consists of two disks, one bounded by each circle. The disk \(\Delta\) bounded by the circle that carries the break lies in \(\Sigma\); with a collar it is a disk picture with outer path \(ac\) from \(x\) to \(y\), in \(\mathcal P\) and shorter by at least \(|b|+2\).

In case 3, let \(A\) be the annulus and \(\Delta_a,\Delta_b\) the two disks of (SURF 5) that form the closure of \(S\setminus A\), bounded by the circles \(C_a,C_b\) that contract to \(a\) and \(b\). On \(A\) the map is \(\kappa_\Lambda\circ\gamma\) for a map \(\gamma\colon A\to K_\Lambda\). Glue a cap, a disk, to \(\Delta_a\) along \(C_a\), mapped by \(h_a\colon{\rm cap}_a\to K_\Lambda\): a collar into \(\Lambda\) by the contraction and the rest by the cone on \(a\); its boundary values are those of \(\gamma\) on \(C_a\). This gives a sphere \(S_a=\Delta_a\cup{\rm cap}_a\) with a cone picture of length the old inner lengths inside \(\Delta_a\) plus \(|a|\); in the same way \(S_b\). The lengths of the two new pictures add up to the old length minus \(2\). In the disk case, \(D_{\rm ext}\) lies in one of \(\Delta_a,\Delta_b\), say \(\Delta_a\); removing the interior of \(D_{\rm ext}\) from \(S_a\) gives a disk picture with outer path \(P\), in \(\mathcal P\) and shorter. In the sphere case it remains to show that \(S_a\) or \(S_b\) is essential.

Essentiality in case 3. Let \(\tilde f\) be a lift of the essential map \(f\colon S\to X\). The map \(\kappa_\Lambda\) lifts to \(\tilde\kappa\colon K_\Lambda\to\widetilde X\), by (COV 2), since the cone is simply connected; choose the lift so that \(\tilde\kappa\circ\gamma\) agrees with \(\tilde f\) at one point of \(A\). Both are lifts of \(f|_A\), and \(A\) is connected, so they agree on \(A\). Define lifts \(\tilde f_a\) of the map on \(S_a\) by \(\tilde f\) on \(\Delta_a\) and \(\tilde\kappa\circ h_a\) on the cap, which agree on \(C_a\), and \(\tilde f_b\) in the same way. Triangulate everything compatibly and orient \(S_a,S_b\) by \(\Delta_a,\Delta_b\). By (CELL 2), in singular chains, \[ \begin{aligned} &\tilde f_*[S]-\tilde f_{a*}[S_a]-\tilde f_{b*}[S_b]\\ &\quad=\tilde\kappa_*\bigl(\gamma_*[A]-h_{a*}[{\rm cap}_a]-h_{b*}[{\rm cap}_b]\bigr). \end{aligned} \] The chain in the bracket is a cycle in \(K_\Lambda\): along \(C_a\) the boundary of \([A]\) is the boundary of \([{\rm cap}_a]\), both being minus the boundary of \([\Delta_a]\), and \(\gamma=h_a\) there; the same holds along \(C_b\). Since \(H_2(K_\Lambda)=0\), the cycle is a boundary, and so is its image under \(\tilde\kappa\). Hence \[ \tilde f_*[S]=\tilde f_{a*}[S_a]+\tilde f_{b*}[S_b]\quad\text{in }H_2(\widetilde X;\mathbb Z), \] and since the left side is nonzero, one of the maps on \(S_a\) and \(S_b\) is essential. That picture is in \(\mathcal P\) and shorter than the old one.

In every case the minimality of the length is contradicted. \(\square\)

The minimization runs over all essential spheres, not over one homotopy class: in case 3 the essential class splits as a sum of two classes, and only one of them need be nonzero.

6. Asphericity and root separation

Proposition 6.1 (asphericity and root separation; [OpenAI-KZ, Proposition 5.1]). Suppose that \(\Gamma\) admits no reduced spherical arrangement. Then \(\pi_2(X)=0\), and every path in \(\Gamma\) from \(x_A\) or \(x_B\) to a different vertex has a nontrivial label in \(G\).

Proof. If a conclusion fails, the class \(\mathcal P\) of Section 4 is not empty; by Lemma 3.2, the failure of \(\pi_2(X)=0\) gives an essential map. Take a picture in \(\mathcal P\) of least length, tighten it by Lemma 4.1, and construct the pairing arcs. By Lemma 5.1 partners traverse distinct edges. The disks, their paths with the break in the disk case, and the arcs now form a reduced spherical arrangement over \(\Gamma\): there is at least one inner disk, whose path is closed, cyclically immersed and of positive length, and the exterior path, if present, is immersed, of positive length, and starts at \(x_A\) or \(x_B\). This contradicts the hypothesis. \(\square\)

Together with Proposition 3.1 of Planar separation and the absence of arrangements, this gives, for all large \(n\), graphs of the prescribed types whose cone complex has \(\pi_2(X)=0\) and whose roots are separated. Torsion-freeness and the zero divisors derives torsion-freeness from this and proves the theorem.

7. Exercises

7.1. Let \(\Lambda\) be a cycle of length \(\ell\ge3\). Describe the spanning tree, the single closed path \(c_e\) and the space \(Y_\Lambda\) of the proof of Lemma 3.3, and show directly that \(Y_\Lambda\) is a disk.

7.2. Show that a nonempty immersed path in \(\Gamma\) has a nontrivial label in \(\pi_1(F,o)\), but may have label \(1\) in \(G\).

7.3. In case 2 of the band surgery, take \(a\), \(b\), \(c\) to be single edges. Write out the old and the new paths and check that the length drops by \(2\).

7.4. Explain why, in case 4 of the band surgery, the new outer path \(ac\) has positive length, although \(a\) and \(c\) may both be empty.

7.5. Show that in case 3 of the band surgery both paths \(a\) and \(b\) are nonempty.

8. Solutions

7.1. A spanning tree is the cycle minus one edge \(e\); \(c_e\) runs once around the cycle. \(Y_\Lambda\) is the cycle, a circle, with one disk attached along a map running once around it, a homeomorphism; so \(Y_\Lambda\) is a disk.

7.2. The letters of an immersed path form a word without cancelling turns, which is a nontrivial element of the free group \(\pi_1(F,o)\) by (PI1 1). A closed immersed path, for instance once around a cycle of \(\Gamma\), has label \(1\) in \(G\) by Lemma 1.1.

7.3. Let \(a=g_1\colon x\to u\), \(b=g_2\colon u\to v\), \(c=g_3\colon v\to y\). The old paths are \(g_1eg_3\) and \(e^{-1}g_2\), of total length \(5\); the new outer path is \(g_1g_2g_3\), of length \(3\).

7.4. The new outer path runs from \(x\) to \(y\), and \(x\ne y\); a path of length \(0\) would have equal ends.

7.5. The boundary path \(eae^{-1}b\) is cyclically immersed by Lemma 4.1. If \(a\) were empty, \(e\) would be followed by \(e^{-1}\), a backtrack; if \(b\) were empty, \(e^{-1}\) would be followed, across the end and start, by \(e\).

References