POINCAR ´ E DUALITY 151 4 Euler Characteristic and Betti Numbers

LetM be a compactm-manifold. TheBetti numbersofM are defined as the dimensions of the de Rham cohomology groups and are denoted by

b_i := dim(Hⁱ(M)), i= 0, . . . , m.

By Corollary 6.2.9 these numbers are finite. Recall that theEuler charac-teristicχ(M) is defined as the sum of the indices of the zeros of a vector field that points out on the boundary (Theorem 2.3.1). The next theorem shows that this invariant is the alternating sum of the Betti numbers. It shows also that the Lefschetz number of a smooth map from M to itself (defined as the sum of the fixed point indices in Theorem 4.4.2) is the alternating of the traces of the induced homomorphism on de Rham cohomology.

Theorem 6.4.8 (Euler Characteristic). Let M be a a compact m-mani-fold with boundary and let f :M →M be a smooth map. Then the Euler characteristic of M is given by

χ(M) =

m

X

i=0

(−1)ⁱdim(Hⁱ(M)) (6.4.9) and the Lefschetz number of f is given by

L(f) =

Proof. The proof has seven steps. The first three steps establish the for-mula (6.4.10) for compact oriented manifolds without boundary.

Step 1. Assume that M is oriented and ∂M =∅. Let τ∆∈Ω^m(M×M) be a closed m-form whose cohomology class is Poincar´e dual to the dia-gonal ∆ :={(p, p)|p∈M}, so (6.4.7) holds with M replaced by M×M and Q:= ∆. Let ωi∈Ω^ki(M) for i= 0,1, . . . , n be closed forms whose cohomology classes [ωi] form a basis of H^∗(M). Then there exist closed forms τ_j ∈Ω^m−kj(M) for j= 0,1, . . . , n such that Their cohomology classes also form a basis of H^∗(M) and

[τ_∆] = and onto the second factor for i= 2.

The existence of theτjsatisfying (6.4.11) and the fact that their cohomology classes form a basis ofH^∗(M) follows directly from Theorem 6.4.1. By the K¨unneth formula in Theorem 6.2.11 the cohomology classes of the differen-tial formsπ₁^∗ωi∧π₂^∗τj form a basis of the de Rham cohomology of M×M. Hence there exist real numberscij ∈Rsuch that

[τ_∆] =X

i,j

c_ij[π^∗₁τ_j∧π^∗₂ω_i]. (6.4.13) We compute the coefficientscij by using equation (6.4.7), which asserts that

Z

6.4. POINCAR ´E DUALITY 153 Since M is a compact oriented manifold without boundary, it follows from Lemma 4.4.6 and Definition 4.4.10 that L(f) = graph(f)·∆. Hence it follows from Theorem 6.4.7 with the triple M, f : P → M, Q replaced by M×M,id×f :M →M×M,∆ that

Step 3. Assume that M is oriented and ∂M =∅. Then (6.4.10)holds.

Let ωi and τj be as in Step 1. Then it follows from (6.4.11) that and so it follows from equation (6.4.14) in Step 2 that

L(f) =X

Step 4. Let M be a compact m-manifold with boundary and letf :M →M be a smooth map such that f(M)∩∂M =∅. Then there exists a com-pact m-manifold N without boundary, a smooth map g:N →N, an open setU ⊂M\∂M, and an embedding ι:M →N such that

g◦ι=ι◦f :U →N, f(M)⊂U, g(N)⊂ι(U), (6.4.15) and the inclusion of U into M is a homotopy equivalence.

Choose a vector fieldX ∈Vect(M) such thatXpoints out on the boundary, letφ: (−∞,0]×M →M be the semi-flow of X, and define

V_ε:={φ(t, p)| −ε≤t≤0, p∈∂M}.

Then Vε is a compact neighborhood of the boundary and φ restricts to a diffeomorphism from [−ε,0]×∂M to V_ε for ε >0 sufficiently small. Fix a constantε >0 so small that this holds and f(M)∩V_ε =∅. Define

N :=M × {±1}/∼, where the equivalence relation is given by

(p,−1)∼(q,+1) ⇐⇒^def

p, q∈V_ε and there exist elements

−ε≤t≤0 andp0∈∂M such that p=φ(t, p0) andq =φ(−ε−t, p0).

ThenN is a compact manifold without boundary, the map M →N :p7→ι(p) := [p,−1]

is an embedding, the set

U :=M\V_ε

is open, and the inclusion of U into M is a homotopy equivalence with a homotopy inverse given by M →U :p7→φ(2ε, p). Choose a smooth func-tion β: [−ε,0]→[−ε,0] such that β(t) =β(−ε−t) =t for t close to −ε, and define the mapg:N →N by

g([p,−1]) :=

[f(p),−1], ifp∈M\Vε, [f(φ(β(t), p₀)),−1], ifp=φ(t, p₀)∈V_ε, g([p,+1]) :=

[f(p),−1], ifp∈M\Vε, [f(φ(β(−ε−t), p0)),−1], ifp=φ(t, p0)∈Vε. This map is smooth and satisfies the requirements of Step 4.

6.4. POINCAR ´E DUALITY 155 Step 5. Assume M is oriented. Then (6.4.10) holds.

In the case ∂M =∅ this was proved in Step 3. Thus assume ∂M 6=∅. By Exercise 4.4.24 and Lemma 4.4.8 we may assume thatf(M)∩∂M =∅andf has only nondegenerate fixed points. Choose the open set U ⊂M and the maps ι:M →N and g:N →N as in Step 4. Then Fix(g) = ι(Fix(f)) and det(1l −dg(ι(p))) = det(1l−df(p)) for each p ∈ Fix(f) by (6.4.15).

Hence, by definition of the Lefschetz number as the sum of the fixed point indices, we haveL(f) =L(g) and thus, by Step 3,

Here the last two equalities follow from Corollary 6.2.10. This proves Step 5.

Step 6. We prove (6.4.10).

Assume first that M is not orientable and ∂M =∅. Assume also, with-out loss of generality, that M is a submanifold of Rⁿ and that f has only nondegenerate fixed points. Then, for ε >0 sufficiently small, the set

N :=

is a smooth manifold with boundary. Moreover, the map r:N →M de-fined by r(p+v) :=pforp∈M and v∈TpM^⊥ with|v|< εis a homotopy equivalence, and the inclusionι:M →N is a homotopy inverse ofr. Define

g:=ι◦f◦r:N →N.

Then Fix(g) = Fix(f) and, for p ∈ Fix(f), we have dg(p)|_TpM = df(p) and dg(p)|_TpM⊥ = 0, and therefore det(1l−dg(p)) = det(1l−df(p)). This implies L(f) =L(g). Since the inclusion ι:M →N is a homotopy equiva-lence with homotopy inverse r, we also have

trace g^∗ :Hⁱ(N)→Hⁱ(N)

= trace f^∗ :Hⁱ(M)→Hⁱ(M)

for each i. Thus, for nonorientable manifolds M without boundary, equa-tion (6.4.10) follows from Step 5. The case of nonempty boundary reduces to the case of empty boundary by the exact same argument that was used in the proof of Step 5 and this proves Step 6.

Step 7. We prove (6.4.9).

By Theorem 4.4.3 the Euler characteristic ofM is the Lefschetz number of the identity map onM and hence (6.4.9) follows directly from (6.4.10). This proves Theorem 6.4.8.

Remark 6.4.9. The zeta function of a smooth map f :M →M on a compact orientedm-manifoldMwithout boundary (thought of as a discrete-time dynamical system) is defined by

ζ_f(t) := exp definition of the Lefschetz numbers (in terms of an algebraic count of the fixed points) the zeta-function of f can be expressed in terms a count of the periodic points off, provided that they are all isolated. If the periodic points off are all nondegenerate then the zeta-function off can be written in the form whereP_n(f) denotes the set of periodic points with minimal periodnand

ι(p, fⁿ) := sign det(1l−dfⁿ(p)),

In particular, the zeta function is rational.

Exercise 6.4.10. Prove that the right hand side of (6.4.16) converges for t sufficiently small. Prove (6.4.17) and (6.4.18). Hint: Use the identities

det(1l−tA)⁻¹= exp trace for a square matrix A and t ∈ Rsufficiently small, and for a fixed point p off that is nondegenerate for all iterates of f.

6.4. POINCAR ´E DUALITY 157