WS 13/14
Marinescu/Herrmann
Homework 4
1. Problem
Consider a triangulation of a surface X with F faces, E edges and V vertices. The
Euler characteristic of X is χ = χ(X) = V − E + F. Show that:
a) 2E = 3F and E = 3(V − χ),
√
b) V ≥ 7+ 49−24χ
,
2
c) A triangulation of the torus has at least 7 vertices. What is the minimal V for
the sphere S 2 and RP2 ?
2. Problem
a) Show that by pasting two Möbius bands along their boundaries we obtain a Klein
bottle.
b) Show that the Klein bottle is the connected sum of two projective planes RP2 #RP2 .
c) Show that T 2 #RP2 = RP2 #RP2 #RP2 .
3. Problem
a) Show that the word α1 α2 α3 α2−1 α1−1 α3−1 represents a torus T 2 .
b) The surface X is represented by the word AabBabC for some words A, B, C.
Show that X is also represented by the word AuBuC.
4. Problem
Consider the closed triangle ∆ in C with vertices −1, 1, i. We identify the points 1
and -1, i.e we consider the equivalence relation ∼ on ∆:
x ∼ y ⇔ x = y or {x, y} = {1, −1}.
a) Calculate the combinatorial Euler characteristic χ(∆/ ∼).
b) Show that ∆/ ∼ is homotopic equivalent to S 1 seen as the interval [−1, 1] with
boundary points identified. Use the invariance of the topological Euler characteristic
to deduce that χtop (∆/ ∼) = 0.
c) We use the same equivalence relation as above on C. What is χtop (C/ ∼)?
5. Problem
Robert likes probabilities. He faces the following Problem:
Let
P M = {a1 , a2 , a3 } be a set with three elements and P (M ) = {f : M → [0, 1] |
a∈M f (a) = 1} the set of probability functions on M . Consider a metric d on
P (M ) given by
sX
(f (a) − g(a))2 .
d(f, g) :=
a∈M
Thus, P (M ) is a metric space. The order of the elements in M should not matter, i.e.
Robert cannot distinguish two probability functions if one of them can be converted
to the other by a permutation of the elements in M . He achieves this by using an
S3 action on P (M )
S3 × P (M ) → P (M )
(σ, f ) 7→ (aj 7→ f (aσ−1 (j) ))
and considering the quotient P (M )/ ∼S3 regarding this action. Robert wonders
a) what is the (topological) Euler characteristic of P (M )/ ∼S3 ?
b)Restricting the action to the subgroup A3 CS3 one gets some quotient P (M )/ ∼A3 .
Calculate χtop (P (M )/ ∼A3 ).
Can you help Robert?