TOPOLOGY
Ciprian Manolescu UCLA
DHC Ceremony, Babes-Bolyai University July 11, 2018
What is the shape of the Earth?
• We don’t know, so here’s an easier question:
What is the shape of the universe?
• Near each point, the Earth looks like flat space:
• We say it is a two-dimensional manifold.
Actually, the Earth is a geoid:
But in topology we do not distinguish between objects that can be deformed into each other without breaking them
Geoid = Sphere
For topologists, a coffee mug is the same as a donut:
However, the sphere is topologically different from the donut, and from the flat (Euclidean) space.
Classification of two dimensional manifolds:
Euclidean space
. . . . . .
. . .
What is the shape of the universe?
Finite or infinite?
Near each point, it looks like three dimensional Euclidean space
We say it’s a three dimensional manifold.
• Physicists propose various answers:
1. Infinite (flat)
2. Poincare dodecahedral space (Luminet et al 2003)
3. Picard horn (Aurich et al 2004) etc.
• The job of mathematicians is to classify all possible manifolds.
Higher dimensions
String theory: the universe may have hidden dimensions Topology: classify manifolds of any dimension n
n=0,1,2,3,4,5,6…
A manifold of dimension n is a space such that near each point we can move in n directions (n degrees of freedom) n=0: point .
n=1: line circle
• n=2:
Euclidean space
. . .
. . .
. . .
Thurston (1982) proposed a classification scheme in dimension three
. . .
Thurston Geometrization Conjecture
• The classification was showed to be true by Perelman (2003)
• This implies the Poincaré conjecture, one of the seven Clay Millennium Problems ($1 million)
Higher dimensions
n=4, 5, 6, … Examples:
n-dimensional Euclidean space n-dimensional sphere
…
Theorem: One cannot classify manifolds of dimension 4 or higher.
Triangulations
Manifolds of dimension 0, 1, 2, 3 can be triangulated:
This is not true for manifolds of dimension 4, 5, 6, … !
Higher dimensions
• Theorem: One cannot classify manifolds of dimension 4 or higher.
• Instead, we can focus on simply-connected manifolds, those on which we can contract every loop:
Simply connected
Not simply connected
Classification of n-dimensional simply- connected manifolds:
Doable in dimensions n=5,6,7,… (1960’s) Unknown in dimension n=4
Four-dimensional topology is the hardest!
Smooth structures
If you can deform two shapes into each other without breaking them, can you deform them without making corners?
If you can’t, we say that they represent different smooth structures on the same manifold.
Smooth structures
• In dimensions n=0,1,2,3, every manifold has a unique smooth structure
• The first exotic smooth structures were found by Milnor (1956) on the 7-dimensional sphere
• n-dimensional Euclidean space has:
a unique smooth structure if n=0,1,2,3, 5,6,7,8,9, ….
infinitely many smooth structures if n=4
(cf. Donaldson, Gompf 1980’s)
n-dimensional sphere
(cf. Kervaire-Milnor 1963)dimension # structures
1 1
2 1
3 1
4 ?
5 1
6 1
7 28
8 2
9 8
10 6
dimension # structures
11 992
12 1
13 3
14 2
15 16256
16 2
17 16
18 16
19 523264
20 24
An open question in topology
The smooth four-dimensional Poincaré conjecture:
Is there a unique smooth structure on the 4-dimensional sphere?
?
Earth sphere: http://www.freepik.com/free-vector/big-crystal-earth-sphere_677399.htm Tangent space: http://rqgravity.net/BasicsOfCurvature
Two-dimensional Euclidean space: http://spaceguard.iasf-roma.inaf.it/NScience/neo/dictionary/newton.htm
Geoid: Image courtesy of the University of Texas Center for Space Research and NASA.
http://celebrating200years.noaa.gov/foundations/gravity_surveys/ggm01_americas.html Coffee mug turning into doughnut: http://en.wikipedia.org/wiki/Topology
Compact two-dimensional manifolds: http://mathworld.wolfram.com/CompactManifold.html
Three-manifold three-torus: An image from inside a 3-torus, generated by Jeff Weeks' CurvedSpaces software. http://en.wikipedia.org/wiki/3- manifold#mediaviewer/File:3-Manifold_3-Torus.png
Poincare dodecahedral space: View from inside PDS along an arbitrary direction, calculated by the CurvedSpaces program, with multiple images of the Earth (from Jeff Weeks).
http://www.science20.com/news_releases/poincare_dodecahedral_space_model_gains_support_to_explain_the_shape_of_space
Grigori Perelman, solver of Poincaré conjecture, gives a lecture on his solution at New York’s Weaver Hall in 2003. Photograph: Frances M Roberts. http://www.theguardian.com/books/2011/mar/27/perfect-rigour-grigori-perelman-review
A function (blue) and a piecewise linear approximation to it (red). http://en.wikipedia.org/wiki/Piecewise_linear_function Torus: triangulated by the marching method: http://en.wikipedia.org/wiki/Surface_triangulation
A sphere is simply connected because every loop can be contracted (on the surface) to a point.
http://en.wikipedia.org/wiki/Simply_connected_space
The torus is not simply-connected: http://inperc.com/wiki/index.php?title=File:Torus.JPG
