Simplicial Structures in Topology
Simplicial Structures in Topology
Simplicial Structures in Topology
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III.3 Some Applications 121<br />
has no fixed po<strong>in</strong>ts and so Λ( f )=0; but f is homotopic to the identity 1 S 1 through<br />
H(e 2πit ,s)=e 2πi(t+s 1 12 )<br />
and therefore Λ(1 S 1)=0 (because the Lefschetz Number is <strong>in</strong>variant up to homotopy);<br />
however, 1 S 1 has only fixed po<strong>in</strong>ts.<br />
We now consider an important corollary to the Lefschetz Fixed Po<strong>in</strong>t Theorem,<br />
known as the Brouwer Fixed Po<strong>in</strong>t Theorem. Aself-map of a space X is a map of<br />
X <strong>in</strong>to itself.<br />
(III.3.4) Corollary. Every nonconstant self-map of a connected acyclic polyhedron<br />
has a fixed po<strong>in</strong>t.<br />
Proof. Let |K| be such a polyhedron 3 ; by hypothesis, the homology of |K| is all<br />
trivial, except for H0(K;Q) ∼ = Q. Let us suppose that f is simplicially approximated<br />
by g: K (r) → K. Letx0 be a vertex of |K| and let us suppose that {x0} + B0(K,Q)<br />
is a generator of H0(K,Q). On the other hand,<br />
C0(g)ℵ0({x0})=g(x0)<br />
and g(x0) is a vertex of K because g is simplicial. S<strong>in</strong>ce |K| is connected, g(x0)<br />
is homological to {x0} and therefore Λ( f )=1. Lefschetz theorem allows us to<br />
conclude that f must have a fixed po<strong>in</strong>t. �<br />
Before stat<strong>in</strong>g the next corollary, let us look <strong>in</strong>to the def<strong>in</strong>ition of the degree of<br />
a self-map of a sphere. It is easy to see that the n-sphere S n (n ≥ 1) is the geometric<br />
realization of a simplicial complex Σ n .Letf : S n → S n be a given map and<br />
g: (Σ n ) (r) → Σ n be a simplicial approximation of f . Σ n has trivial homology <strong>in</strong> all<br />
dimensions except for 0 and n, when it is isomorphic to the Abelian group Z. We<br />
recall that Hn((Σ n ) (r) ) ∼ = Zn((Σ n ) (r) ) has only two possible generators (differ<strong>in</strong>g by<br />
their orientation); let z be one of them. Then, there exists an <strong>in</strong>teger d such that<br />
Hn(g)(z)=dz. This number, which is obviously <strong>in</strong>dependent from the homotopy<br />
class of the map f ,thecyclez, and the simplicial approximation g of f ,isthedegree<br />
of f (notation: gr ( f )). 4<br />
(III.3.5) Lemma. For every map f : S n → S n ,<br />
Λ( f )=1 +(−1) n gr ( f ) .<br />
Proof. The Lefschetz number is def<strong>in</strong>ed for the homology with rational coefficients.<br />
S<strong>in</strong>ce Σ n is connected, Tr H0( f ,Q)=1; we only need to prove that Tr Hn( f ,Q)=<br />
gr ( f ), to which end it is sufficient not<strong>in</strong>g that<br />
Zn((Σ n ) (r) ,Q) ∼ = Zn((Σ n ) (r) ) ⊗ Q . �<br />
3 See examples of acyclic complexes <strong>in</strong> Sect. II.4.<br />
4 Case n = 1 is particularly <strong>in</strong>terest<strong>in</strong>g: when we move along the cycle z, its image g(z) wraps the<br />
correspond<strong>in</strong>g generat<strong>in</strong>g cycle of Z1(S 1 ) d times around S 1 .