Simplicial Structures in Topology
Simplicial Structures in Topology
Simplicial Structures in Topology
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124 III Homology of Polyhedra<br />
1-simplexes:<br />
2-simplexes:<br />
{0,y0},...,{0,yn−1}<br />
{y0,∞},...,{yn−1,∞}<br />
{y0,y1},...,{yn−1,y0}<br />
{0,y0,y1},...,{0,y1,y2},...,{0,yn−1,y0}<br />
{∞,y0,y1},...,{∞,y1,y2},...,{∞,yn−1,y0} ,<br />
whose geometric realization is also homeomorphic to S 2 . Note that<br />
f (0)=0 , f (∞)=∞ and f (xs)=ys.<br />
Let z be a generator of H2(L;Z) (the sum of all oriented 2-simplexes of L) andz ′<br />
be the correspond<strong>in</strong>g generator of K (a subdivision of L); we can easily see that<br />
H2( f )(z ′ )=nz, which means that gr ( f )=n.<br />
S<strong>in</strong>ce the degree of a map is <strong>in</strong>variant up to homotopy, we conclude that<br />
gr (P)=n. Assum<strong>in</strong>g that P is not a surjection, there should exist a p ∈ S 2 such<br />
that P(S 2 ) ⊂ S 2 \p ≈ R 2 . We could then <strong>in</strong>terpret P as a map from S 2 to R 2 and,<br />
therefore, homotopic to the constant map from S 2 to 0 ∈ R 2 (if X is a topological<br />
space and h: X → R 2 is a cont<strong>in</strong>uous function, then h is homotopic to the constant<br />
map to 0 through the homotopy ht(x)=tx). But a constant map has degree 0,<br />
contradict<strong>in</strong>g the fact that P has degree n > 0. �<br />
Exercises<br />
1. Let A: S 1 → S 1 be the antipodal function. Prove that gr A = 1.<br />
2. Prove that there is no retraction of the unit disk D n ⊂ R n onto its boundary S n−1 .<br />
3. A polyhedron |K| has the fixed po<strong>in</strong>t property if every map f : |K|→|K| has at<br />
least one fixed po<strong>in</strong>t. Prove that the fixed po<strong>in</strong>t property is <strong>in</strong>variant up to homeomorphism.<br />
4. Let |K| be a polyhedron with the fixed po<strong>in</strong>t property. Prove that if |L| is a retraction<br />
of |K| then also |L| has the fixed po<strong>in</strong>t property.<br />
5. Consider the space<br />
X = {(x0,x1,x2) ∈ R 3 |(∀i = 0,1,2)xi ≥ 0}<br />
with the topology <strong>in</strong>duced by R 3 and let f : X → X be a given cont<strong>in</strong>uous function.<br />
Prove that it is possible to f<strong>in</strong>d a unit vector�v ∈ X and a nonnegative real number λ<br />
such that f (�v)=λ�v.