Necks, Caps and Their Topology

Perelman's description of a Ricci flow at high curvature, the canonical neighbourhood theorem (12B.5 The Canonical Neighbourhood Theorem), says that every point where the scalar curvature is large has a neighbourhood that, after rescaling, looks like one of a few models: a long, nearly round cylinder (an $\varepsilon$-neck), a cylinder capped off by a ball or by a punctured projective space (an $\varepsilon$-cap), or a whole compact component of controlled positive curvature. Surgery cuts the manifold along the central spheres of necks, and the components it throws away are entirely covered by such neighbourhoods. Two things have to be known about these models, and both are topological once the geometry is set up:

1. what the regions covered by necks and caps are: tubes $S^2\times(0, 1)$, balls, punctured copies of $\mathbb{RP}^3$, and a short list of closed manifolds;
2. how necks fit together: two overlapping necks line up, and a chain of them is again a tube.

Morgan and Tian devote an appendix, their chapter 19, to these facts, and they use them in every chapter from 9 on. This chapter develops them. It begins with the precise definitions, shows what the geometry of a neck is like (curvature, distances, the spheres), proves the gluing lemma that makes chains of necks into tubes, and classifies the components covered by necks and caps. It ends with the topological effect of surgery, which combines everything with 10A.5 The Prime Decomposition.

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

• define closeness of metrics in the $C^k$ topology, $\varepsilon$-necks, their scale, central spheres and $s$-coordinate, and $(C, \varepsilon)$-caps;
• compute the curvature of the model neck and estimate curvature and distances on an $\varepsilon$-neck;
• prove that gluing two products $S^2\times I$ along a sphere gives a product, and that a chain of overlapping necks is a tube $S^2\times(0, 1)$;
• identify the core of a cap as a ball or $\mathbb{RP}^3$ minus a ball, and identify $\mathbb{RP}^3$ minus a ball with the twisted line bundle over $\mathbb{RP}^2$;
• list the closed manifolds covered by necks and caps, $S^3$, $\mathbb{RP}^3$, $\mathbb{RP}^3\#\mathbb{RP}^3$, $S^2\times S^1$ and the twisted $S^2$-bundle over $S^1$, and prove that the capped ones are on the list;
• describe the topological effect of a surgery, and why the Poincaré conjecture follows from extinction.

6.1Closeness of metrics

All definitions in this chapter compare a metric with a model through a diffeomorphism, as in Cheeger–Gromov convergence (9B.4 Convergence of Manifolds).

Two remarks. Since $\nabla_{g_0}g_0 = 0$, the derivatives $\nabla_{g_0}^\ell g = \nabla_{g_0}^\ell(g - g_0)$ measure the derivatives of the difference. And Perelman always takes $k = [1/\varepsilon]$, the integer part of $\frac1\varepsilon$: a single small number then says both "very close" and "close in very many derivatives". That is convenient, because the conclusions drawn from closeness (about curvature, distances, the shape of the spheres) need a few derivatives each, and with $k = [1/\varepsilon]$ there are always enough once $\varepsilon$ is small.

What closeness buys is continuity of everything built from the metric and finitely many derivatives. The curvature tensor is a polynomial in $g_{ij}$, $g^{ij}$, $\partial g_{ij}$ and $\partial^2g_{ij}$ (9A.4 Curvature and What It Means). So if $g$ is within $\varepsilon$ of $g_0$ in $C^2$, with $\varepsilon$ small, then $|\operatorname{Rm}(g) - \operatorname{Rm}(g_0)|_{g_0} \leq C\varepsilon$, with $C$ depending on bounds for $g_0$'s own curvature and the dimension; and in $C^{k+2}$, the first $k$ derivatives of curvature are close too.

6.2The model neck and $\varepsilon$-necks

Let $h_0$ be the round metric on $S^2$ with scalar curvature $1$. A round sphere of radius $\rho$ has Gauss curvature $\frac{1}{\rho^2}$ and scalar curvature $\frac{2}{\rho^2}$, so $h_0$ is the sphere of radius $\sqrt2$, with Gauss curvature $\frac12$, area $8\pi$ and intrinsic diameter $\sqrt2\pi$. The model neck is $S^2\times\mathbb{R}$ with the product metric $h_0 + ds^2$. Its curvature, by the product formulas of 9A.5 Computing Curvature:

• the sectional curvature of the plane tangent to the sphere is $\frac12$; planes containing $\partial_s$ have curvature $0$;
• the Ricci tensor is $\frac12h_0\oplus0$, with eigenvalues $\frac12, \frac12, 0$; the scalar curvature is $1$;
• in Hamilton's normalisation (11A.2 How Curvature Evolves) the curvature operator has eigenvalues $1, 0, 0$.

It is the shrinking cylinder of 11A.1 The Equation and Its First Solutions at the moment when its scalar curvature is $1$.

The definition comes with a vocabulary, used constantly in Morgan–Tian.

• The scale of $N$ is $r_N = R(x)^{-1/2}$. Since the rescaled metric is close to the model, the spheres $\varphi(S^2\times\{s\})$ have area close to $8\pi r_N^2$ and intrinsic diameter close to $\sqrt2\pi r_N$: the scale is the size of the neck's cross-section, up to the factor $\sqrt2$.
• The $s$-coordinate is $s_N = \operatorname{pr}_2\circ\varphi^{-1} : N \to (-\varepsilon^{-1}, \varepsilon^{-1})$. Its level sets are the 2-spheres of the neck, and $S_N = \varphi(S^2\times\{0\})$ is the central sphere. In the metric $g$, the $s$-coordinate measures length along the neck in units of $r_N$: the neck has length about $2\varepsilon^{-1}r_N$.
• The ends, halves, middle third and so on are defined through $s_N$: the positive half is $s_N^{-1}(0, \varepsilon^{-1})$, the middle third is $s_N^{-1}(-\frac{1}{3\varepsilon}, \frac{1}{3\varepsilon})$. Reversing the direction (composing $\varphi$ with $(\omega, s) \mapsto (\omega, -s)$) gives the opposite neck structure.

The definition is scale-invariant: if $N$ is an $\varepsilon$-neck for $g$, it is one for $\lambda g$, with the same $\varphi$, since $R_{\lambda g}(x)\lambda g = R_g(x)g$. A neck is a statement about shape, not size, and the size is recorded separately by $r_N$. Small $\varepsilon$ asks for a long neck, of length $2\varepsilon^{-1}$ times its scale, that is close to round in many derivatives.

6.3The geometry of an $\varepsilon$-neck

Throughout, "for $\varepsilon$ small enough" means: there is $\varepsilon_0 > 0$, depending on the stated tolerances, such that the statement holds for all $\varepsilon \leq \varepsilon_0$. By scale invariance we may rescale so that $R(x) = 1$.

The proposition is how Perelman and Morgan–Tian recognise necks intrinsically: at a point of a neck the curvature operator has one large eigenvalue and two small ones, and the plane of the large one is tangent to the cross-sectional sphere. This plane field is canonical (it does not depend on the choice of $\varphi$), and it is what lets two overlapping necks be compared.

Distances in a neck are nearly those of the model. On the model, $d\big((\omega, s), (\omega', s')\big)^2 = d_{h_0}(\omega, \omega')^2 + (s - s')^2$, with $d_{h_0} \leq \sqrt2\pi$; so for points far apart along the neck, $d \approx |s - s'|$ with an error at most $\frac{\pi^2}{|s - s'|}$. On an $\varepsilon$-neck the metric is within a factor $1 \pm C\varepsilon$ of the model, so lengths of curves inside $N$ are distorted by at most that factor. With a little care about curves that leave $N$ (they must cross the ends, which costs length), this gives:

6.4Gluing products

The topology of chains of necks rests on one simple lemma and one standard fact about spheres in products.

Notice that the lemma needs no knowledge of the diffeomorphism group of $S^2$: the twist $f_0$ is absorbed into one side. (A deeper fact, Smale's theorem that the orientation-preserving diffeomorphisms of $S^2$ form a group that deformation retracts onto $SO(3)$, is what one uses to see that such gluings can be made compatible with families; Morgan and Tian use it in chapter 12. Here it is not needed.)

The standard fact is the product region theorem: an embedded 2-sphere $\Sigma$ in $S^2\times(0, 1)$ that is isotopic to a sphere $S^2\times\{s\}$ cuts $S^2\times(0, 1)$ into two pieces, each diffeomorphic to $S^2\times$ a half-open interval, with $\Sigma$ as the boundary. Equivalently, two disjoint isotopic spheres in a tube cobound a region diffeomorphic to $S^2\times[0, 1]$. (This is the version of Alexander's theorem (10A.5 The Prime Decomposition) for $S^2\times\mathbb{R}$; it follows from it by capping the tube with two balls to get $S^3$ and applying Alexander's theorem on each side. Statement here; Hatcher's notes, chapter 1.)

6.5Overlapping necks and chains

Two $\varepsilon$-necks that meet substantially are almost aligned. Precisely (Morgan–Tian, Proposition 19.11; statement here, with the idea): if $N$ and $N'$ are $\varepsilon$-necks centred at $x$ and $x'$ that intersect, and $\varepsilon$ is small enough, then

• their scales agree to within a factor $1 \pm \alpha$: $1 - \alpha < \frac{R(x)}{R(x')} < 1 + \alpha$;
• at a point $y$ well inside both, the spheres of $N$ and of $N'$ through $y$ are nearly tangent, and, after possibly reversing one neck, $\partial_{s_N}$ and $\partial_{s_{N'}}$ nearly agree;
• the spheres of either neck that lie in the middle of the other are isotopic, inside the other, to its spheres.

The idea: by Proposition 6.3, at a point $y \in N\cap N'$ the scalar curvature is close to $R(x)$ (from $N$) and to $R(x')$ (from $N'$), which gives the first statement; the plane of largest sectional curvature is canonical and close to the tangent planes of the spheres of both necks, which gives the second; and a sphere of $N$ in the middle of $N'$ is then a nearly round sphere of nearly the right size whose tangent planes are nearly those of $N'$'s spheres, so it is a graph over a sphere of $N'$ and isotopic to it (Morgan–Tian, Corollary 19.3).

A frontier remark used in the arguments: the union $U$ of a chain has frontier only at its two extreme ends (the negative end of the first neck and the positive end of the last), so a connected set that meets both $U$ and its complement must reach one of these extreme ends. And an infinite chain has no frontier at its infinite end.

6.6Manifolds covered by necks

The manifolds $S^2\times\mathbb{R}$, $S^2\times S^1$ and $S^2\tilde\times S^1$ all carry metrics in which every point is the centre of an $\varepsilon$-neck: long round cylinders, closed up or not. In a Ricci flow they appear as components that are entirely necks, and they are thrown away by surgery when their curvature is large.

6.7Caps

The neck is oriented so that its negative end meets the core. Like necks, caps are scale-invariant notions. They arise in 12B.3 The Structure of κ-Solutions as the ends of noncompact $\kappa$-solutions: by the soul theorem (9B.5 Splitting and Soul Theorems) a noncompact $\kappa$-solution with positive curvature is diffeomorphic to $\mathbb{R}^3$, and seen from far away it looks like a half-infinite neck with a cap at the soul. The other possible core comes from quotients.

In a Ricci flow, $(S^2\times\mathbb{R})/\iota$ with the quotient of the round cylinder metric is the model for the second type of cap: a neck closed off at one end not by a ball but by an $\mathbb{RP}^2$, the "cross-cap". It is a quotient of the shrinking cylinder, so it is an ancient solution, and it is one of the $\kappa$-solutions of 12B.1 κ-Solutions.

6.8Components covered by necks and caps

The structural part, that the possibilities are only these, is Morgan and Tian's Proposition 19.21; its proof extends chains of necks as in Proposition 6.9, uses the frontier remarks to see what happens when a chain runs into the core of a cap, and rules out infinite nested sequences of caps by the curvature and diameter controls of a $(C, \varepsilon)$-cap. The identification of the closed components, which is the part used at the end of the Poincaré conjecture, follows from the gluing lemma and can be done completely.

Why only these. The list in Theorem 6.12 is short because of what caps are allowed to be: a ball or a punctured $\mathbb{RP}^3$. That restriction is not a definition of convenience but a theorem about $\kappa$-solutions (12B.3 The Structure of κ-Solutions): the only ways a three-dimensional $\kappa$-solution can end, seen at the scale of its curvature, are a capped neck with core a ball (positive curvature, soul theorem) or the cross-cap quotient above. Everything else that is compact and highly curved is a whole component of positive curvature, a $C$-component diffeomorphic to $S^3$ or $\mathbb{RP}^3$, or an $\varepsilon$-round component, which is a spherical space form (10A.2 Finite Rotation Groups and Spherical Space Forms).

6.9The topological effect of surgery

Combine this with 10A.5 The Prime Decomposition. At a surgery time, the flow has a region where the curvature stays bounded and $\varepsilon$-horns, tubes along which the spheres shrink towards the singular set. In each horn that matters, a sphere deep inside, the central sphere of a strong $\delta$-neck, is cut, and the piece beyond it is replaced by a cap from the standard solution (12B.9 Surgery). Topologically:

• each cut is a surgery along a 2-sphere in an $S^2\times I$, which by 10A.5 The Prime Decomposition undoes a connected sum or removes an $S^2\times S^1$ summand, and may split off copies of $S^3$;
• each component thrown away entirely is covered by $\varepsilon$-necks and $(C, \varepsilon)$-caps, or is a $C$-component or an $\varepsilon$-round component, and so is diffeomorphic to $S^3$, $\mathbb{RP}^3$, $\mathbb{RP}^3\#\mathbb{RP}^3$, $S^2\times S^1$, $S^2\tilde\times S^1$, or a spherical space form.

So the manifold before surgery is a connected sum of the manifold after surgery with copies of these, and of copies of $S^2\times S^1$ (or $S^2\tilde\times S^1$) from nonseparating cuts. If the flow eventually becomes extinct, the original manifold is a connected sum of spherical space forms and $S^2$-bundles over $S^1$, which is Morgan and Tian's deduction of their main theorem from finite extinction. For a simply connected manifold, every summand must be simply connected, and the only simply connected entry on the list is $S^3$ (10A.5 The Prime Decomposition, corollary on surgery on simply connected manifolds).

6.10History

Hamilton introduced necks into the study of singularities in his 1995 survey and used them to do surgery on four-manifolds in 1997. Perelman's definitions of $\varepsilon$-necks, caps and canonical neighbourhoods are in §12 of his first preprint and throughout the second (2002–03). The careful development of overlapping necks, chains and the components covered by canonical neighbourhoods is in Kleiner and Lott's notes and in Morgan and Tian's appendix (2006–07). Smale proved that the diffeomorphism group of $S^2$ deformation retracts onto $O(3)$ in 1959.

6.11Exercises

Solution

Planes tangent to the sphere have curvature $\frac{1}{\rho^2}$, planes containing $\partial_s$ have curvature $0$ (9A.5 Computing Curvature). $\operatorname{Ric} = \frac{1}{\rho^2}g_{S^2(\rho)}\oplus0$ (each sphere direction has Ricci curvature $\frac{1}{\rho^2} + 0$), and $R = \frac{2}{\rho^2}$, which is $1$ iff $\rho = \sqrt2$. Under the flow the sphere factor evolves by $\partial_t(\rho^2) = -2\cdot1$ (11A.1 The Equation and Its First Solutions), so $\rho^2 = 2 - 2t$ from $\rho(0) = \sqrt2$, and $R = \frac{2}{\rho^2} = \frac{1}{1 - t}$. At time $t$, multiplying the metric by $R = \frac{1}{1 - t}$ gives the cylinder of radius $\sqrt2$ exactly, so for every $\varepsilon$ the map $(\omega, s) \mapsto (\omega, \sqrt{1 - t}\,(s + s_0))$ (recentred at any point) is an $\varepsilon$-neck structure with zero error.

Solution

The length is about $2\varepsilon^{-1}r_N = 200r_N$, and the cross-section, a nearly round sphere of radius $\sqrt2r_N$, has intrinsic diameter about $\sqrt2\pi r_N \approx 4.44r_N$. So the neck is about $\frac{200}{4.44} \approx 45$ times longer than it is wide. The closeness is in $C^{[1/\varepsilon]} = C^{100}$: one hundred derivatives.

Solution

Glue the first two: by the lemma, $S^2\times(0, 1]\cup_fS^2\times[1, 2] \cong S^2\times(0, 2]$, by a diffeomorphism that is the identity on $S^2\times[1, 2]$, so the second gluing sphere and $f'$ are unchanged; then glue on the third. In the cyclic case there is no free end to absorb the twist into: the result is the mapping torus of $f$, which is $S^2\times S^1$ when $f$ is isotopic to the identity and the nonorientable $S^2\tilde\times S^1$ when $f$ reverses orientation (for instance $f$ the antipodal map).

Solution

$Z$ is locally isometric to the cylinder, so its curvature is that of the cylinder, constant and bounded. A ball of radius $1$ in $Z$ centred at $[p]$ is the image of the ball of radius $1$ around a lift $p$ in the cylinder, under the 2-to-1 projection; so its volume is at least half that of a cylinder ball, which is a fixed positive number $v_0$ (the cylinder is homogeneous). So $Z$ is $\kappa$-noncollapsed at scale $1$ with $\kappa \geq \frac{v_0}{2}$, and as an ancient solution (the quotient of the shrinking cylinder) it is a $\kappa$-solution. The cigar times a line has ends that are collapsed at large scales (9B.3 Collapsing and Noncollapsing), which Perelman's noncollapsing excludes; the cross-cap has no collapsed end.

Solution

The union is the connected sum by definition, and the connected sum does not depend on the gluing (orientation-preserving gluings are isotopic; an orientation-reversing one can be corrected by an orientation-reversing self-diffeomorphism of $\mathbb{RP}^3$, which exists, induced by a reflection of $S^3$ commuting with $x \mapsto -x$). By van Kampen (7A.5 Computing π₁), $\pi_1 = \mathbb{Z}/2 * \mathbb{Z}/2 = \langle a, b \mid a^2 = b^2 = 1\rangle$, and $ab$ has infinite order (the words $(ab)^n$ are reduced and distinct), so the group is infinite. A spherical space form has finite fundamental group, so $\mathbb{RP}^3\#\mathbb{RP}^3$ is not one. It is covered by necks and caps: a long round cylinder with a cross-cap at each end. Its universal cover is $S^2\times\mathbb{R}$ (10A.4 Homotopy Groups, Degree and the Hurewicz Theorem).

Solution

Every surgery is along a sphere, and in a simply connected manifold every sphere separates and every piece stays simply connected (10A.5 The Prime Decomposition); so $M$ is the connected sum of all the components discarded along the way. Each discarded component is simply connected (it is a summand). On the list of possible components, $S^3$, $\mathbb{RP}^3$, $\mathbb{RP}^3\#\mathbb{RP}^3$, $S^2\times S^1$, $S^2\tilde\times S^1$ and the spherical space forms $S^3/\Gamma$, the only simply connected ones are $S^3$ (and $S^3/\Gamma$ with $\Gamma = 1$, which is $S^3$): $\pi_1(\mathbb{RP}^3) = \mathbb{Z}/2$, $\pi_1(\mathbb{RP}^3\#\mathbb{RP}^3)$ is infinite, $\pi_1(S^2\times S^1) = \pi_1(S^2\tilde\times S^1) = \mathbb{Z}$. So every summand is $S^3$ and $M \cong S^3\#\cdots\#S^3 \cong S^3$. Simple connectivity was used to make every surgery separating (no $S^2\times S^1$ summands appear from cuts) and to exclude every entry but $S^3$.