Applied Topology, Fall 2016
Problem Set # 3
Due on Thursday, October 20th 2016.
1. (a) Let S be a finite set of points in an arbitrary metric space. First prove that the containment
VR(S, r) ⊂ Č(S, 2r) ⊂ VR(S, 2r) holds for every r ≥ 0. Then give a counterexample showing
that the complexes are not always equal.
(b) Let S ⊂ Rd be a finite set of points in general position. For each of the two containment claims
below, either prove them or provide a counterexample:
i. For all r > 0, Alpha(S, r) ⊂ Č(S, r) ∩ Del(S),
ii. For all r > 0, Č(S, r) ∩ Del(S) ⊂ Alpha(S, r).
2. The Euler Characteristic of a simplicial complex K equals
χ(K) =
dim
XK
(−1)i ni ,
i=0
where ni is the number of simplices of dimension i.
(a) Find a triangulation of a torus with 7 vertices and compute its Euler characteristic.
(b) For each positive integer n, let ∆n be the simplicial complex consisting of an n-simplex together
with all of its faces. Prove that χ(∆n ) = 1.
(c) Let K be a simplicial complex in R2 with more than one triangle, for which the union of
simplices |K| is a bounded convex set. Let e be an edge, on the boundary of K (ie. contained
in a single triangle t) and K 0 a simplicial complex obtained by removing e and t. Prove that
K 0 is a simplicial complex and that χ(K) = χ(K 0 ). Prove that the Euler characteristic of a
compact, convex set in R2 does not depend on a triangulation and compute it.
3. Related to the notion of the Euler characteristic is Euler’s formula:
Take any graph that has been drawn in R2 as a planar graph (a graph is planar if we can draw it in
the plane so that none of its edges intersect). Then, if V is the number of vertices, E is the number
of edges, and F is the number of faces in this graph, we have the following relation:
V − E + F = 2.
For any planar graph G, we can define a face of G to be a connected region of R whose boundary
is given by the edges of G. For example, the following graph has four faces, as labeled:
If you are interested in the proof, Prof. Eppstein prepared not one, but twenty of them. You can
find them online at: https://www.ics.uci.edu/ eppstein/junkyard/euler/.
You will use this formula to get information about a molecule in chemistry, called fullerene.
Fullerenes have lots of strange applications ranging from cancer treatments to superstrong materials; some people even conjecture that they are the seed behind all organic life on Earth (read
for example http://www.universetoday.com/76732/buckyballs-could-be-plentiful-in-the-universe/).
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For this reason, chemists would like to understand them! In chemistry, carbon molecules want to
do very specific things:
(a) Each carbon wants to be connected to three other carbons.
(b) Those connections do not want to cross.
(c) All the faces in this graph are formed by cycles consisting either of 5 or 6 edges.
Using Euler’s formula prove the following remarkable result:
Any fullerene has precisely 12 cycles consisting of 5 edges.
Not every fullerene corresponds to molecules in real life. In particular, one rule that chemists
have noticed that all fullerenes obey is that they never have two adjacent pentagonal faces: this is
probably because the pentagon is not a shape that carbons are terribly happy in, and the stress
of having any carbon in two such faces probably makes any such molecule unstable. Therefore, it
seems likely that any viable fullerene will have to have all of its pentagonal faces isolated. By the
proposition above, it has precisely 12 such faces. Does such a fullerene exist? The answer is yes
(see the picture below). And this one actually exists in reality! It is called buckminsterfullerene,
after Richard Buckminster Fuller.
4. Given a simplicial complex K and σ a simplex of K, the open star of σ is the set of simplices of K,
denoted by St(σ), defined by
St(σ) := {τ ∈ K | τ ⊃ σ}.
The simplicial complex consisting of St(σ) and all faces is called the closed star St(σ). The link
Lk(σ) is the collection of simplices in St(σ) which are disjoint from σ:
Lk(σ) := {τ ∈ St(σ) | σ ∩ τ = ∅}.
a
b
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(a) What are the open stars, closed stars and links of the simplices [a], [b] and [ab] in the complex
depicted on the next page?
(b) Prove or give a counterexample: Lk(σ) is always a simplicial complex.
(c) Prove or give a counterexample: For every edge [ab] in a simplicial complex,
Lk[ab] = Lk[a] ∩ Lk[b].
5. Let K be the simplicial complex below (the boundary of a tetrahedron).
• Write down the chain complex by specifying vector spaces of chains and writing boundary
maps as matrices.
• Simplify the boundary matrices ∂0 , ∂1 , ∂2 , ∂3 so that the nonzero columns are linearly independent.
• Find the cycles Z0 , Z1 , Z2 , Z3 .
• Find the boundaries B0 , B1 , B2 , B3 .
• Homology groups H0 (K), H1 (K), H2 (K), H3 (K).
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