Definition 0.1.1. Let X be a finite CW complex of dimension n and denote by ci the number of i-cells of X. The Euler characteristic of X is defined as: n X i χ(X) = (−1) · ci. (0.1.1) i=0 It is natural to question whether or not the Euler characteristic depends on the cell structure chosen for the space X. As we will see below, this is not the case. For this, it suffices to show that the Euler characteristic depends only on the cellular homology of the space X. Indeed, cellular homology is isomorphic to singular homology, and the latter is independent of the cell structure on X. Recall that if G is a finitely generated abelian group, then G decomposes into a free part and a torsion part, i.e., r G ' Z × Zn1 × · · · Znk . The integer r := rk(G) is the rank of G. The rank is additive in short exact sequences of finitely generated abelian groups. Theorem 0.1.2. The Euler characteristic can be computed as: n X i χ(X) = (−1) · bi(X) (0.1.2) i=0 with bi(X) := rk Hi(X) the i-th Betti number of X. In particular, χ(X) is independent of the chosen cell structure on X.
Proof. We use the following notation: Bi = Image(di+1), Zi = ker(di), and Hi = Zi/Bi. Consider a (finite) chain complex of finitely generated abelian groups and the short exact sequences defining homology:
dn+1 dn d2 d1 d0 0 / Cn / ... / C1 / C0 / 0
ι di 0 / Zi / Ci / / Bi−1 / 0
di+1 q 0 / Bi / Zi / Hi / 0 The additivity of rank yields that
ci := rk(Ci) = rk(Zi) + rk(Bi−1) and rk(Zi) = rk(Bi) + rk(Hi). Substitute the second equality into the first, multiply the resulting equality by (−1)i, and Pn i sum over i to get that χ(X) = i=0(−1) · rk(Hi). Finally, we apply this result to the cellular chain complex Ci = Hi(Xi,Xi−1), and use the identification between cellular and singular homology.
1 Example 0.1.3. If Mg and Ng denote the orientable and resp. nonorientable closed surfaces of genus g, then χ(Mg) = 1 − 2g + 1 = 2(1 − g) and χ(Ng) = 1 − g + 1 = 2 − g. So all the orientable and resp. non-orientable surfaces are distinguished from each other by their Euler characteristic, and there are only the relations χ(Mg) = χ(N2g).
0.2 Lefschetz Fixed Point Theorem
Let G be a finitely generated abelian group. Given an endomorphism ϕ : G → G, define its trace by Tr(ϕ) = Tr (ϕ ¯ : G/Torsion(G) → G/Torsion(G)) (0.2.1) where the latter trace is the linear algebraic trace of the map ϕ¯ : Zr → Zr. Definition 0.2.1. If X has the homotopy type of a finite cellular (or simplicial) complex and f : X → X is a continuous map, then the Lefschetz number of f is defined to be
dim(X) X i τ(f) = (−1) · Tr(f∗ : Hi(X) → Hi(X)). (0.2.2) i=0 Remark 0.2.2. Notice that homotopic maps have the same Lefschetz number since they induce the same maps on homology.
Example 0.2.3. If f ' idX , then τ(f) = χ(X). This follows from the fact the map induced in homology by the identity map is the identity homomorphism, and the trace of the latter is the corresponding Betti number of X.
Theorem 0.2.4. (Lefschetz) If X is a retract of a finite cellular (or simplicial) complex and if the continuous map f : X → X satisfies τ(f) 6= 0, then f has a fixed point.
Before proving this theorem, let us consider a few examples.
Example 0.2.5. Suppose that X has the homology of a point (up to torsion). Then τ(f) = Tr f∗ : H0(X) → H0(X) = 1.
This follows from the fact that all the other homology groups are zero and that the map induced on H0 is the identity. This example leads immediately to two nontrivial results, the first of which is the Brower fixed point theorem.
Example 0.2.6. (Brower) If f : Dn → Dn is continuous then f has a fixed point.
2n Example 0.2.7. If X = RP , then modulo torsion X has the homology of a point. There- 2n 2n fore any continuos map f : RP → RP has a fixed point.
2 Finally we are led to an example which does not follow from the computation for a point.
Example 0.2.8. If f : Sn → Sn is a continuos map and deg(f) 6= (−1)n+1, then f has a fixed point. To verify this, we compute