Draft:Geometrization in dimension four

  • Comment: draft reads more like an essay or presentation than an encyclopedic article.
    another note, although this isn't related to the decline reason: article might also need a bit of cleanup and copyediting. Darling ☔ (talk · contribs) 05:22, 28 November 2023 (UTC)

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 dimension 3 it is not always possible to assign a geometry to a closed 3-manifold but the resolution of the Geometrization conjecture, proposed by William Thurston (1982), implies that closed 3-manifolds can be decomposed into geometric ``pieces.

Each of these pieces can have one of 8 possible geometries: spherical , Euclidean , hyperbolic , Nil geometry , Sol geometry , , and the products , and .

In dimension four the situation is more complicated. Not every closed 4-manifold can be uniformized by a Lie group or even decomposed into geometrizable pieces. This follows from unsolvability of the homeomorphism problem for 4-manifolds[1]. But, there is still a classification of 4-dimensional geometries due to Richard Filipkiewicz[2]. These fall into 18 distinct geometries and one infinite family. An in depth discussion of the geometries and the manifolds that afford them is given in Hillman's book [3]. The study of complex structures on geometrizable 4-manifolds was initiated by Wall [4]

The Four Dimensional Geometries

edit

The distinction in to the following classes is somewhat arbitrary, the emphasis has been placed on properties of the fundamental group and the uniformizing Lie group. The classification of the geometries is taken from.[2]. The descriptions of the fundamental groups as well as further information on the 4-manifolds that afford them can be found in Hillman's book[3]

Spherical or compact type

edit

Three geometries lie here, the 4-sphere  , the complex projective plane  , and a product of two 2-spheres  . The fundamental group of any such manifold is finite.

Euclidean type

edit

This is the four dimensional Euclidean space  . With isometry group  . The fundamental group of any such manifold is a Bieberbach group. There are 74 homeomorphism classes of manifolds with geometry  , 27 orientable manifolds and 47 non-orientable manifolds.[5]

Nilpotent type

edit

There are two geometries of Nilpotent type   and the reducible geometry  .

The   geometry is a 4-dimensional nilpotent Lie group given as the semi-direct product  , where  . The fundamental group of a closed orientable  -manifold is nilpotent of class 3.

For a closed 4-manifold   admitting a   geometry, there is a finite cover   of   such that  . Here   is the fundamental group of a 3-dimensional nilmanifold. Thus, every such fundamental group is nilpotent of class 2.

Note that one can always take   above to be one of the following groups  , where   is non-zero. These are all fundamental groups of torus bundles over the circle.

Solvable type

edit

There are two unique geometries  , and  . As well as a countably infinite family   where   are integers.

The  -geometry is the Lie group described by the semi-direct product  , where  . The fundamental group of a closed  -manifold is a semidirect product   where   has one real eigenvalue and two conjugate complex eigenvalues. The fundamental group has Hirsh length equal to 4.

The  -geometry is the Lie group described by set of matrices  .

A closed  -manifold   is a mapping torus of a  -manifold. Its fundamental group is a semidirect product  . The fundamental group has Hirsh length equal to 4.

Define  . If   are positive integers such that  , then   has three distinct real roots  .

The  -geometry is the Lie group described by the semi-direct product  , where  . The fundamental group of a closed  -manifold is a semidirect product   where   has three distinct real eigenvalues. The fundamental group has Hirsh length equal to 4.

Isomorphisms between solvable geometries

edit

Note that when   that   has exactly one eigenvalue.

So there is an identification  .

We have that   if the roots   and   satisfy   for some real number  .

A proof of these facts appears in.[6]

Hyperbolic type

edit

There are two geometries here real-hyperbolic 4-space   and the complex hyperbolic plane  . The fundamental groups of closed manifolds here are word hyperbolic groups.

Product of hyperbolic planes

edit

This is the geometry  . Closed manifolds come in two forms here. A  -manifold is reducible if it is finitely covered by a direct product of hyperbolic Riemann surfaces. Otherwise it is irreducible. The irreducible manifolds fundamental groups are arithmetic groups by Margulis' arithmeticity theorem.

The tangent space of the hyperbolic plane

edit

This geometry admits no closed manifolds.

Remaining geometries

edit

The remaining geometries come in two cases:

A product of two 2-dimensional geometries   and  .

A product of a 3-dimensional geometry with  . These are  ,  , and  .

References

edit
  1. ^ Markov, Aleksandr Aleksandrovich (1958). "The insolubility of the problem of homeomorphy". Dokl. Akad. Nauk SSSR. 121: 218–220.
  2. ^ a b Filipkiewicz, Richard (1983). Four dimensional geometries. PhD Thesis (phd). Retrieved 20 November 2023.
  3. ^ a b Hillman, Joseph A (2002). Four-manifolds, geometries and knots. Geometry & Topology Publications, Coventry,Geom. Topol. Monogr., 5. pp. xiv+379 pp.
  4. ^ Wall, C.T.C. (1985). "Geometries and geometric structures in real dimension 4 and complex dimension 2". Geometry and Topology. Lecture Notes in Mathematics. Vol. 1167. Springer, Berlin, Heidelberg. pp. 268–292. doi:10.1007/BFb0075230. ISBN 978-3-540-16053-3. {{cite book}}: |journal= ignored (help)
  5. ^ Brown, H; Bülow, R; Neubüser, J; Wondratschek, H; Zassenhaus, H (1978). Crystallographic groups of four-dimensional space. John Wiley & Sons, New York.
  6. ^ Ma, Jiming; Wang, Zixi (2022). "Distinguishing 4-dimensional geometries via profinite completions". Geometriae Dedicata. 216 (52). arXiv:2011.03784. doi:10.1007/s10711-022-00712-8. S2CID 226281905.