3-manifold

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https://en.wikipedia.org/wiki/3-manifold

An image from inside a 3-torus. All of the cubes in the image are the same cube, since light in the manifold wraps around into closed loops, the effect is that the cube is tiling all of space. This space has finite volume and no boundary.

In mathematics, a 3-manifold is a topological space that locally looks like a three-dimensional Euclidean space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small enough observer, all 3-manifolds look like our universe does to a small enough observer. This is made more precise in the definition below.

Introduction

Definition

A topological space ${\displaystyle M}$ is a 3-manifold if it is a second-countable Hausdorff space and if every point in ${\displaystyle M}$ has a neighbourhood that is homeomorphic to Euclidean 3-space.

Mathematical theory of 3-manifolds

The topological, piecewise-linear, and smooth categories are all equivalent in three dimensions, so little distinction is made in whether we are dealing with say, topological 3-manifolds, or smooth 3-manifolds.

Phenomena in three dimensions can be strikingly different from phenomena in other dimensions, and so there is a prevalence of very specialized techniques that do not generalize to dimensions greater than three. This special role has led to the discovery of close connections to a diversity of other fields, such as knot theory, geometric group theory, hyperbolic geometry, number theory, Teichmüller theory, topological quantum field theory, gauge theory, Floer homology, and partial differential equations. 3-manifold theory is considered a part of low-dimensional topology or geometric topology.

A key idea in the theory is to study a 3-manifold by considering special surfaces embedded in it. One can choose the surface to be nicely placed in the 3-manifold, which leads to the idea of an incompressible surface and the theory of Haken manifolds, or one can choose the complementary pieces to be as nice as possible, leading to structures such as Heegaard splittings, which are useful even in the non-Haken case.

Thurston's contributions to the theory allow one to also consider, in many cases, the additional structure given by a particular Thurston model geometry (of which there are eight). The most prevalent geometry is hyperbolic geometry. Using a geometry in addition to special surfaces is often fruitful.

The fundamental groups of 3-manifolds strongly reflect the geometric and topological information belonging to a 3-manifold. Thus, there is an interplay between group theory and topological methods.

Invariants describing 3-manifolds

3-manifolds are an interesting special case of low-dimensional topology because their topological invariants give a lot of information about their structure in general. If we let ${\displaystyle M}$ be a 3-manifold and ${\displaystyle \pi ={\pi }_1(M)}$ be its fundamental group, then a lot of information can be derived from them. For example, using Poincare duality and the Hurewicz theorem, we have the following homology groups:

${\displaystyle \begin{array}{rlr}{H}_0(M)& =H^3(M)=& \mathbb{Z}\\ H_1(M)& =H^2(M)=& \pi /[\pi ,\pi ]\\ H_2(M)& =H^1(M)=& \text{Hom}(\pi ,\mathbb{Z})\\ H_3(M)& =H^0(M)=& \mathbb{Z}\end{array}}$

where the last two groups are isomorphic to the group homology and cohomology of ${\displaystyle \pi }$, respectively; that is,

${\displaystyle \begin{array}{rl}{H}_1(\pi ;\mathbb{Z})& \cong \pi /[\pi ,\pi ]\\ H^1(\pi ;\mathbb{Z})& \cong \text{Hom}(\pi ,\mathbb{Z})\end{array}}$

From this information a basic homotopy theoretic classification of 3-manifolds can be found. Note from the Postnikov tower there is a canonical map

${\displaystyle q:M\to B\pi }$

If we take the pushforward of the fundamental class ${\displaystyle [M]\in H_3(M)}$ into ${\displaystyle H_3(B\pi )}$ we get an element ${\displaystyle {\zeta }_M=q_{\ast }([M])}$. It turns out the group ${\displaystyle \pi }$ together with the group homology class ${\displaystyle {\zeta }_M\in H_3(\pi ,\mathbb{Z})}$ gives a complete algebraic description of the homotopy type of ${\displaystyle M}$.

Connected sums

One important topological operation is the connected sum of two 3-manifolds ${\displaystyle M_1\mathrm{\#}{M}_2}$. In fact, from general theorems in topology, we find for a three manifold with a connected sum decomposition ${\displaystyle M=M_1\mathrm{\#}\cdots \mathrm{\#}{M}_n}$ the invariants above for ${\displaystyle M}$ can be computed from the ${\displaystyle M_i}$. In particular

${\displaystyle \begin{array}{rl}{H}_1(M)& =H_1(M_1)\oplus \cdots \oplus H_1(M_n)\\ H_2(M)& =H_2(M_1)\oplus \cdots \oplus H_2(M_n)\\ {\pi }_1(M)& ={\pi }_1(M_1)\ast \cdots \ast {\pi }_1(M_n)\end{array}}$

Moreover, a 3-manifold ${\displaystyle M}$ which cannot be described as a connected sum of two 3-manifolds is called prime.

Second homotopy groups

For the case of a 3-manifold given by a connected sum of prime 3-manifolds, it turns out there is a nice description of the second fundamental group as a ${\displaystyle \mathbb{Z}[\pi ]}$-module. For the special case of having each ${\displaystyle {\pi }_1(M_i)}$ is infinite but not cyclic, if we take based embeddings of a 2-sphere

${\displaystyle {\sigma }_i:S^2\to M}$ where ${\displaystyle {\sigma }_i(S^2)\subset M_i-\{B^3\}\subset M}$

then the second fundamental group has the presentation

${\displaystyle {\pi }_2(M)=\frac{\mathbb{Z}[\pi ]\{{\sigma }_1,\dots ,{\sigma }_n\}}{({\sigma }_1+\cdots +{\sigma }_n)}}$

giving a straightforward computation of this group.

Important examples of 3-manifolds

Euclidean 3-space

Main article: Euclidean 3-space

Euclidean 3-space is the most important example of a 3-manifold, as all others are defined in relation to it. This is just the standard 3-dimensional vector space over the real numbers.

3-sphere

Main article: 3-sphere

Stereographic projection of the hypersphere's parallels (red), meridians (blue) and hypermeridians (green). Because this projection is conformal, the curves intersect each other orthogonally (in the yellow points) as in 4D. All curves are circles: the curves that intersect <0,0,0,1> have infinite radius (= straight line).

A 3-sphere is a higher-dimensional analogue of a sphere. It consists of the set of points equidistant from a fixed central point in 4-dimensional Euclidean space. Just as an ordinary sphere (or 2-sphere) is a two-dimensional surface that forms the boundary of a ball in three dimensions, a 3-sphere is an object with three dimensions that forms the boundary of a ball in four dimensions. Many examples of 3-manifolds can be constructed by taking quotients of the 3-sphere by a finite group ${\displaystyle \pi }$ acting freely on ${\displaystyle S^3}$ via a map ${\displaystyle \pi \to \text{SO}(4)}$, so ${\displaystyle M=S^3/\pi }$.

Real projective 3-space

Main article: Real projective space

Real projective 3-space, or RP³, is the topological space of lines passing through the origin 0 in R⁴. It is a compact, smooth manifold of dimension 3, and is a special case Gr(1, R⁴) of a Grassmannian space.

RP³ is (diffeomorphic to) SO(3), hence admits a group structure; the covering map S³ → RP³ is a map of groups Spin(3) → SO(3), where Spin(3) is a Lie group that is the universal cover of SO(3).

3-torus

Main article: Torus § n-dimensional torus

The 3-dimensional torus is the product of 3 circles. That is:

${\displaystyle {\mathbf{T}}^3=S^1\times S^1\times S^1.}$

The 3-torus, T³ can be described as a quotient of R³ under integral shifts in any coordinate. That is, the 3-torus is R³ modulo the action of the integer lattice Z³ (with the action being taken as vector addition). Equivalently, the 3-torus is obtained from the 3-dimensional cube by gluing the opposite faces together.

A 3-torus in this sense is an example of a 3-dimensional compact manifold. It is also an example of a compact abelian Lie group. This follows from the fact that the unit circle is a compact abelian Lie group (when identified with the unit complex numbers with multiplication). Group multiplication on the torus is then defined by coordinate-wise multiplication.

Hyperbolic 3-space

Main article: hyperbolic 3-space

A perspective projection of a dodecahedral tessellation in H³.
Four dodecahedra meet at each edge, and eight meet at each vertex, like the cubes of a cubic tessellation in E³

Hyperbolic space is a homogeneous space that can be characterized by a constant negative curvature. It is the model of hyperbolic geometry. It is distinguished from Euclidean spaces with zero curvature that define the Euclidean geometry, and models of elliptic geometry (like the 3-sphere) that have a constant positive curvature. When embedded to a Euclidean space (of a higher dimension), every point of a hyperbolic space is a saddle point. Another distinctive property is the amount of space covered by the 3-ball in hyperbolic 3-space: it increases exponentially with respect to the radius of the ball, rather than polynomially.

Poincaré dodecahedral space

Main article: Homology sphere § Poincaré homology sphere

The Poincaré homology sphere (also known as Poincaré dodecahedral space) is a particular example of a homology sphere. Being a spherical 3-manifold, it is the only homology 3-sphere (besides the 3-sphere itself) with a finite fundamental group. Its fundamental group is known as the binary icosahedral group and has order 120. This shows the Poincaré conjecture cannot be stated in homology terms alone.

In 2003, lack of structure on the largest scales (above 60 degrees) in the cosmic microwave background as observed for one year by the WMAP spacecraft led to the suggestion, by Jean-Pierre Luminet of the Observatoire de Paris and colleagues, that the shape of the universe is a Poincaré sphere. In 2008, astronomers found the best orientation on the sky for the model and confirmed some of the predictions of the model, using three years of observations by the WMAP spacecraft. However, there is no strong support for the correctness of the model, as yet.

Seifert–Weber space

Main article: Seifert–Weber space

In mathematics, Seifert–Weber space (introduced by Herbert Seifert and Constantin Weber) is a closed hyperbolic 3-manifold. It is also known as Seifert–Weber dodecahedral space and hyperbolic dodecahedral space. It is one of the first discovered examples of closed hyperbolic 3-manifolds.

It is constructed by gluing each face of a dodecahedron to its opposite in a way that produces a closed 3-manifold. There are three ways to do this gluing consistently. Opposite faces are misaligned by 1/10 of a turn, so to match them they must be rotated by 1/10, 3/10 or 5/10 turn; a rotation of 3/10 gives the Seifert–Weber space. Rotation of 1/10 gives the Poincaré homology sphere, and rotation by 5/10 gives 3-dimensional real projective space.

With the 3/10-turn gluing pattern, the edges of the original dodecahedron are glued to each other in groups of five. Thus, in the Seifert–Weber space, each edge is surrounded by five pentagonal faces, and the dihedral angle between these pentagons is 72°. This does not match the 117° dihedral angle of a regular dodecahedron in Euclidean space, but in hyperbolic space there exist regular dodecahedra with any dihedral angle between 60° and 117°, and the hyperbolic dodecahedron with dihedral angle 72° may be used to give the Seifert–Weber space a geometric structure as a hyperbolic manifold. It is a quotient space of the order-5 dodecahedral honeycomb, a regular tessellation of hyperbolic 3-space by dodecahedra with this dihedral angle.

Gieseking manifold

Main article: Gieseking manifold

In mathematics, the Gieseking manifold is a cusped hyperbolic 3-manifold of finite volume. It is non-orientable and has the smallest volume among non-compact hyperbolic manifolds, having volume approximately 1.01494161. It was discovered by Hugo Gieseking (1912).

The Gieseking manifold can be constructed by removing the vertices from a tetrahedron, then gluing the faces together in pairs using affine-linear maps. Label the vertices 0, 1, 2, 3. Glue the face with vertices 0,1,2 to the face with vertices 3,1,0 in that order. Glue the face 0,2,3 to the face 3,2,1 in that order. In the hyperbolic structure of the Gieseking manifold, this ideal tetrahedron is the canonical polyhedral decomposition of David B. A. Epstein and Robert C. Penner. Moreover, the angle made by the faces is ${\displaystyle \pi /3}$. The triangulation has one tetrahedron, two faces, one edge and no vertices, so all the edges of the original tetrahedron are glued together.

Some important classes of 3-manifolds

• Graph manifold
• Haken manifold
• Homology spheres
• Hyperbolic 3-manifold
• I-bundles
• Knot and link complements
• Lens space
• Seifert fiber spaces, Circle bundles
• Spherical 3-manifold
• Surface bundles over the circle
• Torus bundle

Hyperbolic link complements

Borromean rings are a hyperbolic link.

A hyperbolic link is a link in the 3-sphere with complement that has a complete Riemannian metric of constant negative curvature, i.e. has a hyperbolic geometry. A hyperbolic knot is a hyperbolic link with one component.

The following examples are particularly well-known and studied.

• Figure eight knot
• Whitehead link
• Borromean rings

The classes are not necessarily mutually exclusive.

Some important structures on 3-manifolds

Contact geometry

Main article: Contact geometry

Contact geometry is the study of a geometric structure on smooth manifolds given by a hyperplane distribution in the tangent bundle and specified by a one-form, both of which satisfy a 'maximum non-degeneracy' condition called 'complete non-integrability'. From the Frobenius theorem, one recognizes the condition as the opposite of the condition that the distribution be determined by a codimension one foliation on the manifold ('complete integrability').

Contact geometry is in many ways an odd-dimensional counterpart of symplectic geometry, which belongs to the even-dimensional world. Both contact and symplectic geometry are motivated by the mathematical formalism of classical mechanics, where one can consider either the even-dimensional phase space of a mechanical system or the odd-dimensional extended phase space that includes the time variable.

Haken manifold

Main article: Haken manifold

A Haken manifold is a compact, P²-irreducible 3-manifold that is sufficiently large, meaning that it contains a properly embedded two-sided incompressible surface. Sometimes one considers only orientable Haken manifolds, in which case a Haken manifold is a compact, orientable, irreducible 3-manifold that contains an orientable, incompressible surface.

A 3-manifold finitely covered by a Haken manifold is said to be virtually Haken. The Virtually Haken conjecture asserts that every compact, irreducible 3-manifold with infinite fundamental group is virtually Haken.

Haken manifolds were introduced by Wolfgang Haken. Haken proved that Haken manifolds have a hierarchy, where they can be split up into 3-balls along incompressible surfaces. Haken also showed that there was a finite procedure to find an incompressible surface if the 3-manifold had one. Jaco and Oertel gave an algorithm to determine if a 3-manifold was Haken.

Essential lamination

An essential lamination is a lamination where every leaf is incompressible and end incompressible, if the complementary regions of the lamination are irreducible, and if there are no spherical leaves.

Essential laminations generalize the incompressible surfaces found in Haken manifolds.

Heegaard splitting

Main article: Heegaard splitting

A Heegaard splitting is a decomposition of a compact oriented 3-manifold that results from dividing it into two handlebodies.

Every closed, orientable three-manifold may be so obtained; this follows from deep results on the triangulability of three-manifolds due to Moise. This contrasts strongly with higher-dimensional manifolds which need not admit smooth or piecewise linear structures. Assuming smoothness the existence of a Heegaard splitting also follows from the work of Smale about handle decompositions from Morse theory.

Taut foliation

Main article: Taut foliation

A taut foliation is a codimension 1 foliation of a 3-manifold with the property that there is a single transverse circle intersecting every leaf. By transverse circle, is meant a closed loop that is always transverse to the tangent field of the foliation. Equivalently, by a result of Dennis Sullivan, a codimension 1 foliation is taut if there exists a Riemannian metric that makes each leaf a minimal surface.

Taut foliations were brought to prominence by the work of William Thurston and David Gabai.

Foundational results

Some results are named as conjectures as a result of historical artifacts.

We begin with the purely topological:

Moise's theorem

Main article: Moise's theorem

In geometric topology, Moise's theorem, proved by Edwin E. Moise in, states that any topological 3-manifold has an essentially unique piecewise-linear structure and smooth structure.

As corollary, every compact 3-manifold has a Heegaard splitting.

Prime decomposition theorem

Main article: prime decomposition (3-manifold)

The prime decomposition theorem for 3-manifolds states that every compact, orientable 3-manifold is the connected sum of a unique (up to homeomorphism) collection of prime 3-manifolds.

A manifold is prime if it cannot be presented as a connected sum of more than one manifold, none of which is the sphere of the same dimension.

Kneser–Haken finiteness

Kneser-Haken finiteness says that for each 3-manifold, there is a constant C such that any collection of surfaces of cardinality greater than C must contain parallel elements.

Loop and Sphere theorems

Main articles: Loop theorem and Sphere theorem (3-manifolds)

The loop theorem is a generalization of Dehn's lemma and should more properly be called the "disk theorem". It was first proven by Christos Papakyriakopoulos in 1956, along with Dehn's lemma and the Sphere theorem.

A simple and useful version of the loop theorem states that if there is a map

${\displaystyle f:(D^2,\mathrm{\partial }{D}^2)\to (M,\mathrm{\partial }M)}$

with ${\displaystyle f|\mathrm{\partial }{D}^2}$ not nullhomotopic in ${\displaystyle \mathrm{\partial }M}$, then there is an embedding with the same property.

The sphere theorem of Papakyriakopoulos (1957) gives conditions for elements of the second homotopy group of a 3-manifold to be represented by embedded spheres.

One example is the following:

Let ${\displaystyle M}$ be an orientable 3-manifold such that ${\displaystyle {\pi }_2(M)}$ is not the trivial group. Then there exists a non-zero element of ${\displaystyle {\pi }_2(M)}$ having a representative that is an embedding ${\displaystyle S^2\to M}$.

Annulus and Torus theorems

The annulus theorem states that if a pair of disjoint simple closed curves on the boundary of a three manifold are freely homotopic then they cobound a properly embedded annulus. This should not be confused with the high dimensional theorem of the same name.

The torus theorem is as follows: Let M be a compact, irreducible 3-manifold with nonempty boundary. If M admits an essential map of a torus, then M admits an essential embedding of either a torus or an annulus

JSJ decomposition

Main article: JSJ decomposition

The JSJ decomposition, also known as the toral decomposition, is a topological construct given by the following theorem:

Irreducible orientable closed (i.e., compact and without boundary) 3-manifolds have a unique (up to isotopy) minimal collection of disjointly embedded incompressible tori such that each component of the 3-manifold obtained by cutting along the tori is either atoroidal or Seifert-fibered.

The acronym JSJ is for William Jaco, Peter Shalen, and Klaus Johannson. The first two worked together, and the third worked independently.

Scott core theorem

Main article: Scott core theorem

The Scott core theorem is a theorem about the finite presentability of fundamental groups of 3-manifolds due to G. Peter Scott. The precise statement is as follows:

Given a 3-manifold (not necessarily compact) with finitely generated fundamental group, there is a compact three-dimensional submanifold, called the compact core or Scott core, such that its inclusion map induces an isomorphism on fundamental groups. In particular, this means a finitely generated 3-manifold group is finitely presentable.

A simplified proof is given in, and a stronger uniqueness statement is proven in.

Lickorish–Wallace theorem

Main article: Lickorish–Wallace theorem

The Lickorish–Wallace theorem states that any closed, orientable, connected 3-manifold may be obtained by performing Dehn surgery on a framed link in the 3-sphere with ${\displaystyle \pm 1}$ surgery coefficients. Furthermore, each component of the link can be assumed to be unknotted.

Waldhausen's theorems on topological rigidity

Friedhelm Waldhausen's theorems on topological rigidity say that certain 3-manifolds (such as those with an incompressible surface) are homeomorphic if there is an isomorphism of fundamental groups which respects the boundary.

Waldhausen conjecture on Heegaard splittings

Waldhausen conjectured that every closed orientable 3-manifold has only finitely many Heegaard splittings (up to homeomorphism) of any given genus.

Smith conjecture

Main article: Smith conjecture

The Smith conjecture (now proven) states that if f is a diffeomorphism of the 3-sphere of finite order, then the fixed point set of f cannot be a nontrivial knot.

Cyclic surgery theorem

Main article: Cyclic surgery theorem

The cyclic surgery theorem states that, for a compact, connected, orientable, irreducible three-manifold M whose boundary is a torus T, if M is not a Seifert-fibered space and r,s are slopes on T such that their Dehn fillings have cyclic fundamental group, then the distance between r and s (the minimal number of times that two simple closed curves in T representing r and s must intersect) is at most 1. Consequently, there are at most three Dehn fillings of M with cyclic fundamental group.

Thurston's hyperbolic Dehn surgery theorem and the Jørgensen–Thurston theorem

Main article: Hyperbolic Dehn surgery

Thurston's hyperbolic Dehn surgery theorem states: ${\displaystyle M(u_1,u_2,\dots ,u_n)}$ is hyperbolic as long as a finite set of exceptional slopes ${\displaystyle E_i}$ is avoided for the i-th cusp for each i. In addition, ${\displaystyle M(u_1,u_2,\dots ,u_n)}$ converges to M in H as all ${\displaystyle p_i^2+q_i^2\to \mathrm{\infty }}$ for all ${\displaystyle p_i/q_i}$ corresponding to non-empty Dehn fillings ${\displaystyle u_i}$.

This theorem is due to William Thurston and fundamental to the theory of hyperbolic 3-manifolds. It shows that nontrivial limits exist in H. Troels Jorgensen's study of the geometric topology further shows that all nontrivial limits arise by Dehn filling as in the theorem.

Another important result by Thurston is that volume decreases under hyperbolic Dehn filling. In fact, the theorem states that volume decreases under topological Dehn filling, assuming of course that the Dehn-filled manifold is hyperbolic. The proof relies on basic properties of the Gromov norm.

Jørgensen also showed that the volume function on this space is a continuous, proper function. Thus by the previous results, nontrivial limits in H are taken to nontrivial limits in the set of volumes. In fact, one can further conclude, as did Thurston, that the set of volumes of finite volume hyperbolic 3-manifolds has ordinal type ${\displaystyle {\omega }^{\omega }}$. This result is known as the Thurston-Jørgensen theorem. Further work characterizing this set was done by Gromov.

Also, Gabai, Meyerhoff & Milley showed that the Weeks manifold has the smallest volume of any closed orientable hyperbolic 3-manifold.

Thurston's hyperbolization theorem for Haken manifolds

Main article: Hyperbolization theorem

One form of Thurston's geometrization theorem states: If M is an compact irreducible atoroidal Haken manifold whose boundary has zero Euler characteristic, then the interior of M has a complete hyperbolic structure of finite volume.

The Mostow rigidity theorem implies that if a manifold of dimension at least 3 has a hyperbolic structure of finite volume, then it is essentially unique.

The conditions that the manifold M should be irreducible and atoroidal are necessary, as hyperbolic manifolds have these properties. However the condition that the manifold be Haken is unnecessarily strong. Thurston's hyperbolization conjecture states that a closed irreducible atoroidal 3-manifold with infinite fundamental group is hyperbolic, and this follows from Perelman's proof of the Thurston geometrization conjecture.

Tameness conjecture, also called the Marden conjecture or tame ends conjecture

Main article: Tameness conjecture

The tameness theorem states that every complete hyperbolic 3-manifold with finitely generated fundamental group is topologically tame, in other words homeomorphic to the interior of a compact 3-manifold.

The tameness theorem was conjectured by Marden. It was proved by Agol and, independently, by Danny Calegari and David Gabai. It is one of the fundamental properties of geometrically infinite hyperbolic 3-manifolds, together with the density theorem for Kleinian groups and the ending lamination theorem. It also implies the Ahlfors measure conjecture.

Ending lamination conjecture

Main article: Ending lamination conjecture

The ending lamination theorem, originally conjectured by William Thurston and later proven by Jeffrey Brock, Richard Canary, and Yair Minsky, states that hyperbolic 3-manifolds with finitely generated fundamental groups are determined by their topology together with certain "end invariants", which are geodesic laminations on some surfaces in the boundary of the manifold.

Poincaré conjecture

Main article: Poincaré conjecture

The 3-sphere is an especially important 3-manifold because of the now-proven Poincaré conjecture. Originally conjectured by Henri Poincaré, the theorem concerns a space that locally looks like ordinary three-dimensional space but is connected, finite in size, and lacks any boundary (a closed 3-manifold). The Poincaré conjecture claims that if such a space has the additional property that each loop in the space can be continuously tightened to a point, then it is necessarily a three-dimensional sphere. An analogous result has been known in higher dimensions for some time.

After nearly a century of effort by mathematicians, Grigori Perelman presented a proof of the conjecture in three papers made available in 2002 and 2003 on arXiv. The proof followed on from the program of Richard S. Hamilton to use the Ricci flow to attack the problem. Perelman introduced a modification of the standard Ricci flow, called Ricci flow with surgery to systematically excise singular regions as they develop, in a controlled way. Several teams of mathematicians have verified that Perelman's proof is correct.

Thurston's geometrization conjecture

Main article: Thurston's geometrization conjecture

Thurston's geometrization conjecture states that certain three-dimensional topological spaces each have a unique geometric structure that can be associated with them. It is an analogue of the uniformization theorem for two-dimensional surfaces, which states that every simply connected Riemann surface can be given one of three geometries (Euclidean, spherical, or hyperbolic). In three dimensions, it is not always possible to assign a single geometry to a whole topological space. Instead, the geometrization conjecture states that every closed 3-manifold can be decomposed in a canonical way into pieces that each have one of eight types of geometric structure. The conjecture was proposed by William Thurston (1982), and implies several other conjectures, such as the Poincaré conjecture and Thurston's elliptization conjecture.

Thurston's hyperbolization theorem implies that Haken manifolds satisfy the geometrization conjecture. Thurston announced a proof in the 1980s and since then several complete proofs have appeared in print.

Grigori Perelman sketched a proof of the full geometrization conjecture in 2003 using Ricci flow with surgery. There are now several different manuscripts (see below) with details of the proof. The Poincaré conjecture and the spherical space form conjecture are corollaries of the geometrization conjecture, although there are shorter proofs of the former that do not lead to the geometrization conjecture.

Virtually fibered conjecture and Virtually Haken conjecture

Main articles: Virtually fibered conjecture and Virtually Haken conjecture

The virtually fibered conjecture, formulated by American mathematician William Thurston, states that every closed, irreducible, atoroidal 3-manifold with infinite fundamental group has a finite cover which is a surface bundle over the circle.

The virtually Haken conjecture states that every compact, orientable, irreducible three-dimensional manifold with infinite fundamental group is virtually Haken. That is, it has a finite cover (a covering space with a finite-to-one covering map) that is a Haken manifold.

In a posting on the ArXiv on 25 Aug 2009, Daniel Wise implicitly implied (by referring to a then unpublished longer manuscript) that he had proven the Virtually fibered conjecture for the case where the 3-manifold is closed, hyperbolic, and Haken. This was followed by a survey article in Electronic Research Announcements in Mathematical Sciences. Several more preprints have followed, including the aforementioned longer manuscript by Wise. In March 2012, during a conference at Institut Henri Poincaré in Paris, Ian Agol announced he could prove the virtually Haken conjecture for closed hyperbolic 3-manifolds. The proof built on results of Kahn and Markovic in their proof of the Surface subgroup conjecture and results of Wise in proving the Malnormal Special Quotient Theorem and results of Bergeron and Wise for the cubulation of groups. Taken together with Wise's results, this implies the virtually fibered conjecture for all closed hyperbolic 3-manifolds.

Simple loop conjecture

If ${\displaystyle f:S\to T}$ is a map of closed connected surfaces such that ${\displaystyle f_{\star }:{\pi }_1(S)\to {\pi }_1(T)}$ is not injective, then there exists a non-contractible simple closed curve ${\displaystyle \alpha \subset S}$ such that ${\displaystyle f{|}_a}$ is homotopically trivial. This conjecture was proven by David Gabai.

Surface subgroup conjecture

Main article: Surface subgroup conjecture

The surface subgroup conjecture of Friedhelm Waldhausen states that the fundamental group of every closed, irreducible 3-manifold with infinite fundamental group has a surface subgroup. By "surface subgroup" we mean the fundamental group of a closed surface not the 2-sphere. This problem is listed as Problem 3.75 in Robion Kirby's problem list.

Assuming the geometrization conjecture, the only open case was that of closed hyperbolic 3-manifolds. A proof of this case was announced in the Summer of 2009 by Jeremy Kahn and Vladimir Markovic and outlined in a talk August 4, 2009 at the FRG (Focused Research Group) Conference hosted by the University of Utah. A preprint appeared on the arxiv in October 2009. Their paper was published in the Annals of Mathematics in 2012. In June 2012, Kahn and Markovic were given the Clay Research Awards by the Clay Mathematics Institute at a ceremony in Oxford.

Important conjectures

Cabling conjecture

The cabling conjecture states that if Dehn surgery on a knot in the 3-sphere yields a reducible 3-manifold, then that knot is a ${\displaystyle (p,q)}$-cable on some other knot, and the surgery must have been performed using the slope ${\displaystyle pq}$.

Lubotzky–Sarnak conjecture

The fundamental group of any finite volume hyperbolic n-manifold does not have Property τ.

Photo-oxidation of polymers

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Photo-oxidation_of_polymers

Comparison of rope which has been degraded by weathering to fresh rope. Note the fraying and discolouration.

This plastic bucket has been used as an open-air flowerpot for some years. Photodegradation has made it brittle, causing part of it to break off while the bucket was moved.

In polymer chemistry photo-oxidation (sometimes: oxidative photodegradation) is the degradation of a polymer surface due to the combined action of light and oxygen. It is the most significant factor in the weathering of plastics. Photo-oxidation causes the polymer chains to break (chain scission), resulting in the material becoming increasingly brittle. This leads to mechanical failure and, at an advanced stage, the formation of microplastics. In textiles the process is called phototendering.

Technologies have been developed to both accelerate and inhibit this process. For example, plastic building components like doors, window frames and gutters are expected to last for decades, requiring the use of advanced UV-polymer stabilizers. Conversely, single-use plastics can be treated with biodegradable additives to accelerate their fragmentation. Many pigments and dyes can similarly have effects due to their ability to absorb UV-energy.

Susceptible polymers

2015 Global plastic production by polymer type:
PP: polypropylene, PE: polyethylene, PVC: Polyvinyl chloride, PS: Polystyrene, PET: Polyethylene terephthalate

Susceptibility to photo-oxidation varies depending on the chemical structure of the polymer. Some materials have excellent stability, such as fluoropolymers, polyimides, silicones and certain acrylate polymers. However, global polymer production is dominated by a range of commodity plastics which account for the majority of plastic waste. Of these polyethylene terephthalate (PET) has only moderate UV resistance and the others, which include polystyrene, polyvinyl chloride (PVC) and polyolefins like polypropylene (PP) and polyethylene (PE) are all highly susceptible.

Photo-oxidation is a form of photodegradation and begins with formation of free radicals on the polymer chain, which then react with oxygen in chain reactions. For many polymers the general autoxidation mechanism is a reasonable approximation of the underlying chemistry. The process is autocatalytic, generating increasing numbers of radicals and reactive oxygen species. These reactions result in changes to the molecular weight (and molecular weight distribution) of the polymer and as a consequence the material becomes more brittle. The process can be divided into four stages:

Initiation the process of generating the initial free radical.

Propagation the conversion of one active species to another

Chain branching steps which end with more than one active species being produced. The photolysis of hydroperoxides is the main example.

Termination steps in which active species are removed, for instance by radical disproportionation

Photo-oxidation can occur simultaneously with other processes like thermal degradation, and each of these can accelerate the other.

Polyolefins

Polyolefins such as polyethylene and polypropylene are susceptible to photo-oxidation and around 70% of light stabilizers produced world-wide are used in their protection, despite them representing only around 50% of global plastic production. Aliphatic hydrocarbons can only adsorb high energy UV-rays with a wavelength below ~250 nm, however the Earth’s atmosphere and ozone layer screen out such rays, with the normal minimum wavelength being 280–290 nm. The bulk of the polymer is therefore photo-inert and degradation is instead attributed to the presence of various impurities, which are introduced during the manufacturing or processing stages. These include hydroperoxide and carbonyl groups, as well as metal salts such as catalyst residues.

All of these species act as photoinitiators. The organic hydroperoxide and carbonyl groups are able to absorb UV light above 290 nm whereupon they undergo photolysis to generate radicals. Metal impurities act as photocatalysts, although such reactions can be complex. It has also been suggested that polymer-O₂ charge-transfer complexes are involved. Initiation generates radical-carbons on the polymer chain, sometimes called macroradicals (P•).

The cyclic mechanism of autoxidation

Chain initiation

${\displaystyle \text{Polymer}⟶\text{P}\bullet +\text{P}\bullet }$

Chain propagation

${\displaystyle \text{P}\bullet +{\text{O}}_2^{}⟶\text{POO}\bullet }$

${\displaystyle \text{POO}\bullet +\text{PH}⟶\text{POOH}+\text{P}\bullet }$

Chain branching

${\displaystyle \text{POOH}⟶\text{PO}\bullet +\text{OH}\bullet }$

${\displaystyle \text{PH}+\text{OH}\bullet ⟶\text{P}\bullet +{\text{H}}_2^{}\text{O}}$

${\displaystyle \text{PO}\bullet ⟶\text{Chain}\text{scission}\text{reactions}}$

Termination

${\displaystyle \text{POO}\bullet +\text{POO}\bullet ⟶\text{cross}\text{linking}\text{reaction}\text{to}\text{non}-\text{radical}\text{product}}$

${\displaystyle \text{POO}\bullet +\text{P}\bullet ⟶\text{cross}\text{linking}\text{reaction}\text{to}\text{non}-\text{radical}\text{product}}$

${\displaystyle \text{P}\bullet +\text{P}\bullet ⟶\text{cross}\text{linking}\text{reaction}\text{to}\text{non}-\text{radical}\text{product}}$

Classically the carbon-centred macroradicals (P•) rapidly react with oxygen to form hydroperoxyl radicals (POO•), which in turn abstract an H atom from the polymer chain to give a hydroperoxide (POOH) and a fresh macroradical. Hydroperoxides readily undergo photolysis to give an alkoxyl macroradical radical (PO•) and a hydroxyl radical (HO•), both of which may go on to form new polymer radicals via hydrogen abstraction. Non-classical alternatives to these steps have been proposed. The alkoxyl radical may also undergo beta scission, generating a acyl-ketone and macroradical. This is considered to be the main cause of chain breaking in polypropylene.

Secondary hydroperoxides can also undergo an intramolecular reaction to give a ketone group, although this is limited to polyethylene.

The ketones generated by these processes are themselves photo-active, although much more weakly. At ambient temperatures they undergo Type II Norrish reactions with chain scission. They may also absorb UV-energy, which they can then transfer to O₂, causing it to enter its highly reactive singlet state. Singlet oxygen is a potent oxidising agent can go on to form cause further degradation.

Polystyrene

Propagration steps in the degradation of polystyrene

For polystyrene the complete mechanism of photo-oxidation is still a matter of debate, as different pathways may operate concurrently and vary according to the wavelength of the incident light. Regardless, there is agreement on the major steps.

Pure polystyrene should not be able to absorb light with a wavelength below ~280 nm and initiation is explained though photo-labile impurities (hydroperoxides) and charge transfer complexes, all of which are able to absorb normal sunlight. Charge-transfer complexes of oxygen and polystyrene phenyl groups absorb light to form singlet oxygen, which acts as a radical initiator.  Carbonyl impurities in the polymer (c.f. acetophenone) also absorb light in the near ultraviolet range (300 to 400 nm), forming excited ketones able to abstract hydrogen atoms directly from the polymer. Hyroperoxide undergoes photolysis to form hydroxyl and alkoxyl radicals.

These initiation steps generate macroradicals at tertiary sites, as these are more stabilised. The propagation steps are essentially identical to those seen for polyolefins; with oxidation, hydrogen abstraction and photolysis leading to beta scission reactions and increasing numbers of radicals. These steps account for the majority of chain-breaking, however in a minor pathway the hydroperoxide reacts directly with polymer to form a ketone group (acetophenone) and a terminal alkene without the formation of additional radicals.

Polystyrene is observed to yellow during photo-oxidation, which is attributed to the formation of polyenes from these terminal alkenes.

Polyvinyl chloride (PVC)

Pure organochlorides like polyvinyl chloride (PVC) do not absorb any light above 220 nm. The initiation of photo-oxidation is instead caused by various irregularities in the polymer chain, such as structural defects as well as hydroperoxides, carbonyl groups, and double bonds. Hydroperoxides formed during processing are the most important initiator to begin with, however their concentration decreases during photo-oxidation whereas carbonyl concentration increases, as such carbonyls may become the primary initiator over time.

Propagation steps involve the hydroperoxyl radical, which can abstract hydrogen from both hydrocarbon (-CH₂-) and organochloride (-CH₂Cl-) sites in the polymer at comparable rates. Radicals formed at hydrocarbon sites rapidly convert to alkenes with loss of radical chlorine. This forms allylic hydrogens (shown in red) which are more susceptible to hydrogen abstraction leading to the formation of polyenes in zipper-like reactions.

When the polyenes contain at least eight conjugated double bonds they become coloured, leading to yellowing and eventual browning of the material. This is off-set slightly by longer polyenes being photobleached with atmospheric oxygen, however PVC does eventually discolour unless polymer stabilisers are present. Reactions at organochloride sites proceed via the usual hydroperoxyl and hydroperoxide before photolysis yields the α-chloro-alkoxyl radical. This species can undergo various reactions to give carbonyls, peroxide cross-links and beta scission products.

Photo-oxidation of PVC. Fate of the α-chloro-alkoxyl radical (clockwise from top): Beta scission to give either an acid chloride or ketone. Dimerization to give a peroxide cross-link. Hydrogen abstraction followed by loss of HCl to form a ketone.

Poly(ethylene terephthalate) - (PET)

Unlike most other commodity plastics polyethylene terephthalate (PET) is able to absorb the near ultraviolet rays in sunlight. Absorption begins at 360 nm, becoming stronger below 320 nm and is very significant below 300 nm. Despite this PET has better resistance to photo-oxidation than other commodity plastics, this is due to a poor quantum yield or the absorption. The degradation chemistry is complicated due to simultaneous photodissociation (i.e. not involving oxygen) and photo-oxidation reactions of both the aromatic and aliphatic parts of the molecule. Chain scission is the dominant process, with chain branching and the formation of coloured impurities being less common. Carbon monoxide, carbon dioxide, and carboxylic acids are the main products. The photo-oxidation of other linear polyesters such as polybutylene terephthalate and polyethylene naphthalate proceeds similarly.

Photodissociation involves the formation of an excited terephthalic acid unit which undergoes Norrish reactions. The type I reaction dominates, which cause chain scission at the carbonyl unit to give a range of products.

Type II Norrish reactions are less common but give rise to acetaldehyde by way of vinyl alcohol esters. This has an exceedingly low odour and taste threshold and can cause an off-taste in bottled water.

Radicals formed by photolysis may initiate the photo-oxidation in PET. Photo-oxidation of the aromatic terephthalic acid core results in its step-wise oxidation to 2,5-dihydroxyterephthalic acid. The photo-oxidation process at aliphatic sites is similar to that seen for polyolefins, with the formation of hydroperoxide species eventually leading to beta-scission of the polymer chain.

Secondary factors

Environment

Perhaps surprisingly, the effect of temperature is often greater than the effect of UV exposure. This can be seen in terms of the Arrhenius equation, which shows that reaction rates have an exponential dependence on temperature. By comparison the dependence of degradation rate on UV exposure and the availability of oxygen is broadly linear. As the oceans are cooler than land plastic pollution in the marine environment degrades more slowly. Materials buried in landfill do not degrade by photo-oxidation at all, though they may gradually decay by other processes.

Mechanical stress can effect the rate of photo-oxidation and may also accelerate the physical breakup of plastic objects. Stress can be caused by mechanical load (tensile and shear stresses) or even by temperature cycling, particularly in composite systems consisting of materials with differing temperature coefficients of expansion. Similarly, sudden rainfall can cause thermal stress.

Effects of dyes and other additives

Dyes and pigments are used in polymer materials to provide colour, however they can also effect the rate of photo-oxidation. Many absorb UV rays and in so doing protect the polymer, however absorption can cause the dyes to enter an excited state where they may attack the polymer or transfer energy to O₂ to form damaging singlet oxygen. Cu-phthalocyanine is an example, it strongly absorbs UV light however the excited Cu-phthalocyanine may act as a photoinitiator by abstracting hydrogen atoms from the polymer. Its interactions may become even more complicated when other additives are present. Fillers such as carbon black can screen out UV light, effectively stabilisers the polymer, whereas flame retardants tend to cause increased levels of photo-oxidation.

Additives to enhance degradation

Biodegradable additives may be added to polymers to accelerate their degradation. In the case of photo-oxidation OXO-biodegradation additives are used. These are transition metal salts such as iron (Fe), manganese (Mn), and cobalt (Co). Fe complexes increase the rate of photooxidation by promoting the homolysis of hydroperoxides via Fenton reactions.

The use of such additives has been controversial due to concerns that treated plastics do not fully biodegrade and instead result in the accelerated formation of microplastics. Oxo-plastics would be difficult to distinguish from untreated plastic but their inclusion during plastic recycling can create a destabilised product with fewer potential uses, potentially jeopardising the business case for recycling any plastic. OXO-biodegradation additives were banned in the EU in 2019

Prevention

See also: Polymer stabilizers

 

Bisoctrizole: A phenolic benzotriazole based UV absorber used to protect polymers

 

Active principle of the ultraviolet absorption via a photochromic transition

UV attack by sunlight can be ameliorated or prevented by adding anti-UV polymer stabilizers, usually prior to shaping the product by injection moulding. UV stabilizers in plastics usually act by absorbing the UV radiation preferentially, and dissipating the energy as low-level heat. The chemicals used are similar to those in sunscreen products, which protect skin from UV attack. They are used frequently in plastics, including cosmetics and films. Different UV stabilizers are utilized depending upon the substrate, intended functional life, and sensitivity to UV degradation. UV stabilizers, such as benzophenones, work by absorbing the UV radiation and preventing the formation of free radicals. Depending upon substitution, the UV absorption spectrum is changed to match the application. Concentrations normally range from 0.05% to 2%, with some applications up to 5%.

Frequently, glass can be a better alternative to polymers when it comes to UV degradation. Most of the commonly used glass types are highly resistant to UV radiation. Explosion protection lamps for oil rigs for example can be made either from polymer or glass. Here, the UV radiation and rough weathers belabor the polymer so much, that the material has to be replaced frequently.

Poly(ethylene-naphthalate) (PEN) can be protected by applying a zinc oxide coating, which acts as protective film reducing the diffusion of oxygen. Zinc oxide can also be used on polycarbonate (PC) to decrease the oxidation and photo-yellowing rate caused by solar radiation.

Analysis

Weather testing of polymers

Main article: Weather testing of polymers

An accelerated weathering tester, a type of environmental chamber. It exposes materials to alternating cycles of UV light and moisture at elevated temperatures (at T≈60 °C for example), simulating the effects of sunlight, and dew and rain. This is used to test the yellowing of coatings (such as white paints).

The photo-oxidation of polymers can be investigated by either natural or accelerated weather testing. Such testing is important in determining the expected service-life of plastic items as well as the fate of waste plastic.

In natural weather testing, polymer samples are directly exposed to open weather for a continuous period of time, while accelerated weather testing uses a specialized test chamber which simulates weathering by sending a controlled amount of UV light and water at a sample. A test chamber may be advantageous in that the exact weathering conditions can be controlled, and the UV or moisture conditions can be made more intense than in natural weathering. Thus, degradation is accelerated and the test is less time-consuming.

Through weather testing, the impact of photooxidative processes on the mechanical properties and lifetimes of polymer samples can be determined. For example, the tensile behavior can be elucidated through measuring the stress–strain curve for a specimen. This stress–strain curve is created by applying a tensile stress (which is measured as the force per area applied to a sample face) and measuring the corresponding strain (the fractional change in length). Stress is usually applied until the material fractures, and from this stress–strain curve, mechanical properties such as the Young’s modulus can be determined. Overall, weathering weakens the sample, and as it becomes more brittle, it fractures more easily. This is observed as a decrease in the yield strain, fracture strain, and toughness, as well as an increase in the Young’s modulus and break stress (the stress at which the material fractures).

Aside from measuring the impact of degradation on mechanical properties, the degradation rate of plastic samples can also be quantified by measuring the change in mass of a sample over time, as microplastic fragments can break off from the bulk material as degradation progresses and the material becomes more brittle through chain-scission. Thus, the percentage change in mass is often measured in experiments to quantify degradation.

Mathematical models can also be created to predict the change in mass of a polymer sample over the weathering process. Because mass loss occurs at the surface of the polymer sample, the degradation rate is dependent on surface area. Thus, a model for the dependence of degradation on surface area can be made by assuming that the rate of change in mass ${\displaystyle -\frac{\mathrm{d}m}{\mathrm{d}t}}$ resulting from degradation is directly proportional to the surface area SA of the specimen:

${\displaystyle -\frac{\mathrm{d}m}{\mathrm{d}t}=k_d\rho SA}$

Here, ${\displaystyle \rho }$ is the density and k_d is known as the specific surface degradation rate (SSDR), which changes depending on the polymer sample’s chemical composition and weathering environment. Furthermore, for a microplastic sample, SA is often approximated as the surface area of a cylinder or sphere. Such an equation can be solved to determine the mass of a polymer sample as a function of time.

Detection

IR spectrum showing carbonyl absorption due to UV degradation of polyethylene

Degradation can be detected before serious cracks are seen in a product by using infrared spectroscopy, which is able to detect chemical species formed by photo-oxidation. In particular, peroxy-species and carbonyl groups have distinct absorption bands.

In the example shown at left, carbonyl groups were easily detected by IR spectroscopy from a cast thin film. The product was a road cone made by rotational moulding in LDPE, which had cracked prematurely in service. Many similar cones also failed because an anti-UV additive had not been used during processing. Other plastic products which failed included polypropylene mancabs used at roadworks which cracked after service of only a few months.

Different polymer samples are visualized using a scanning electron microscope (SEM) before and after weathering. Included polymers are low-density polyethylene (LDPE), polypropylene (PP), polystyrene (PS), polyamide 66 (PA66), styrene butadiene rubber (SBR), and high-density polyethylene (HDPE).

The effects of degradation can also be characterized through scanning electron microscopy (SEM). For example, through SEM, defects like cracks and pits can be directly visualized, as shown at right. These samples were exposed to 840 hours of exposure to UV light and moisture using a test chamber. Crack formation is often associated with degradation, such that materials that do not display significant cracking behavior, such as HDPE in the right example, are more likely to be stable against photooxidation compared to other materials like LDPE and PP. However, some plastics that have undergone photooxidation may also appear smoother in an SEM image, with some defects like grooves having disappeared afterwards. This is seen in polystyrene in the right example.