1. MSRI – Manifolds
Carlo H. Séquin

2. A A
2-Manifolds
Structures made out of thin sheets of material, such as paper, sheet metal, or plastic.
Thin mathematical surfaces, where every interior point has a disk-like neighborhood, and every rim-point has a half-disk neighborhood.
Many abstract geometrical sculptures are of this kind:
Eva Hild Charles O. Perry Brent Collins
They can be classified topologically by three numbers:
B: the number of borders (1D rim-lines),
S: single- / double- sided (orientability),
Genus, G, or: Euler Characteristic, X, (connectivity).

3. Classification of 2-ManifoldsB B
Classification of 2-Manifolds
We need to determine three numbers:
B: number of borders (1D rim-lines)
S: the number of sides
G: the genus (connectivity)
2-sided = orientable
1-sided = non-orientable 2
Run your finger along rims;
Count number of loops.
Can you paint surface with 2 colors? – or:
Is there a path from one side to the other?
How many non-intersecting closed loops can be drawn on the surface without dividing the surface into different countries ?
This is the hardest task,
so it helps to have a few tricks …

4. Finding the Genus of a 2-ManifoldC C
Finding the Genus of a 2-Manifold
Handle Bodies: 2-sided surfaces with no punctures
Open Surfaces:
With punctures (and tunnels, too)
All of these are genus 0, but with different numbers of punctures (rims).
genus 0
genus 3 a b c
This is genus 4:
Make 4 cuts through all handle-loops; Subsequent cuts must not intersect ! Some admissible combinations: a a a a b b b b c c c c
genus 0
2-sided, 2 borders
= cylinder, annulus
genus 1
1-sided, 1 border = Möbius band
Confused? – Use Euler Characteristic …

5. Using the Euler CharacteristicD D
Using the Euler Characteristic
Draw a mesh onto the surface, count and calculate: X = V – E + F = Euler Characteristic
V :
-E :
F :
X :
G :
B = S = 1
B = S = 2
from: Genus = ( 2 – X – B ) / S
Closing the gap eliminates 1 edge and 2 vertices;
X := X 1
b := b +1
Genus unchanged
But we don’t actually have to draw a mesh:  Just calculate X incrementally by adding one closure at a time.
“Disk”
Cancels out

6. Apply what we learned to some 2-manifold sculptures:
Surface Analysis
Apply what we learned to some 2-manifold sculptures:
3 cuts needed
B = 2 S = 1
X = –1 (2 cuts)
Genus = (2 – X – B) = 1
 Möbius band + 1 puncture
Independent cutting lines: 1 blue or green – but not both, they would intersect!
B = 4 S = 2
X = –2
Genus = (2 – X – B)/2 = 0
 Sphere with 4 punctures
Any closed-loop cut will divide this surface!
Charles Perry: “Tetra” Brent Collins: “Möbius Shell”
B = 1 S = 2
X = –1 (2 cuts)
Genus = 1
 Torus + puncture
One closed-loop cut is possible.
B = 1 S = 1
X = –21 (22 cuts)
Genus = 22
 Connected sum of 11 Klein bottles.
Carlo Séquin: “Volution_2” Brent Collins: “Heptoroid”