Homotopy and Framed Cobordisms
4.4 The Lefschetz Number of a Smooth Map
4.4 The Lefschetz Number of a Smooth Map
In this section we introduce the Lefschetz number of a smooth mapf from a closed manifold M to itself as the algebraic count of the fixed point indices.
If the manifold is oriented, the Lefschetz number can also be defined as the intersection number of the graph of f with the diagonal. However, orientability is not required and the Lefschetz number is always a homotopy invariant. The Lefschetz–Hopf theorem asserts that the Lefschetz number is the sum of the fixed point indices whenever the fixed points are all isolated.
The Lefschetz–Hopf Theorem
Assume throughout thatM is a compact smoothm-manifold with boundary, not necessarily orientable, and let f :M →M be a smooth map.
Definition 4.4.1 (Fixed Point Index). An element p∈M is called a fixed point of f if f(p) =p. The set of all fixed points of f is denoted by
Fix(f) :=
p∈M
f(p) =p .
A fixed point p0 ∈Fix(f) is calledisolatedif there exists an open neighbor-hood U ⊂M of p0 such that
f(p)6=p for all p∈U \ {p0}.
Let p0 ∈M \∂M be an isolated fixed point and let U ⊂M \∂M be an open neighborhood of p0 with U∩Fix(f) ={p0} such that there exists a diffeo-morphism φ:U →Rm. Given such a coordinate chart φ:U →Rm, define the open set Ω⊂Rm and the smooth mapη : Ω→Rm by
Ω :=φ(U∩f−1(U))⊂Rm, η:=φ◦f ◦φ−1 : Ω→Rm. Let x0 :=φ(p0) and choose ε >0 such that Bε(x0)⊂Ω. Then the integer
ι(p0, f) := deg
Sm−1 →Sm−1:x7→ x0+εx−η(x0+εx)
|x0+εx−η(x0+εx)|
(4.4.1) is called the fixed point index off at p0. A fixed pointp0 ∈Fix(f)\∂M is called nondegenerate if the linear map 1l−df(p0) :Tp0M →Tp0M is a vector space isomorphism. The map f is called aLefschetz mapif its fixed points are all nondegenerate and Fix(f)∩∂M =∅.
Theorem 4.4.2 (Lefschetz–Hopf ). Let M be a compact manifold with boundary and let f :M →M be a smooth map such that
Fix(f)∩∂M =∅. (4.4.2)
Then the following holds.
(i)If p0 ∈Fix(f) is an isolated fixed point off, then its fixed point index is independent of the choice of the coordinate chartφ used to define it.
(ii)If p0 ∈Fix(f) is a nondegenerate fixed point off, then p0 is an isolated fixed point of f and its fixed point index is given by
ι(p0, f) = sign det(1l−df(p0))
. (4.4.3)
(iii) If f has only isolated fixed points, then X
p∈Fix(f)
ι(p, f) =
m
X
k=0
(−1)ktrace f∗ :Hk(M)→Hk(M)
. (4.4.4) Here H∗(M) denotes the de Rham cohomology ofM. In particular, the left hand side of equation (4.4.4) is a homotopy invariant of f. If is called the Lefschetz number of f and is denoted byL(f).
Proof. See page 78.
In this section we will only prove that the sum of the fixed point indices of a smooth map with with only isolated fixed points and no fixed point on the boundary is a homotopy invariant. The formula (4.4.4) will be established in Theorem 6.4.8.
The strategy for the proof is to show that every smooth map with only isolated fixed points and no fixed points on the boundary is homotopic to a Lefschetz map with the same sum of the fixed point indices (Lemma 4.4.7) and then to show that the sum of the fixed point indices is a homotopy invariant for Lefschetz maps (Lemma 4.4.9). To prove that the Lefschetz number is well defined, we must also show that every smooth map is ho-motopic to a Lefschetz map (Lemma 4.4.8). The proof that the fixed point index at an isolated fixed point is well defined, requires local versions of these results which are of interest in their own rights. In particular, Lemma 4.4.7 asserts the existence of a local perturbation of a map f near an isolated fixed point p0 such that the perturbed map has only nondegenerate fixed points nearp0, the sum of whose indices is the fixed point index of f atp0. This is analogous to Lemma 2.3.3 for isolated zeros of vector fields and
4.4. THE LEFSCHETZ NUMBER OF A SMOOTH MAP 71 Lemma 4.2.15 for isolated intersections. A first preparatory result relates the nondegenerate fixed points of f to the transverse intersections of the graph off and the diagonal inM×M (Lemma 4.4.6).
The Lefschetz Number
Before carrying out the details, we formulate another theorem that summa-rizes various properties of the Lefschetz number. These properties charac-terize the Lefschetz number axiomatically and hence can also be used to define it. For a smooth manifoldM denote by Map(M, M) the space of all smooth maps f :M →M.
Theorem 4.4.3. Let M be a compact manifold with boundary. Then there exists a function
Map(M, M)→Z:f 7→L(f), (4.4.5) called the Lefschetz number, that satisfies the following axioms for all smooth maps f, g:M →M.
(Homotopy) If f is smoothly homotopic to g, then L(f) =L(g).
(Lefschetz) If f is a Lefschetz map, then L(f) = X
p∈Fix(f)
sign det(1l−df(p)) . (Fixed Point) If L(f)6= 0 thenFix(f)6=∅.
(Hopf ) If Fix(f)∩∂M =∅ and f has only isolated fixed points, then L(f) = X
p∈Fix(f)
ι(p, f).
(Conjugacy) If φ:M →M is a diffeomorphism then L(φ◦f◦φ−1) =L(f).
(Euler) If f is homotopic to the identity, then L(f) =χ(M) is the Euler characteristic of M.
(Graph) If M is oriented and ∂M =∅, then L(f) = graph(f)·∆.
Moreover, every smooth map f :M →M is smoothly homotopic to a Lef-schetz map. Hence the map (4.4.5) is uniquely determined by the (Homo-topy) and (Lefschetz) axioms.
Proof. See page 79.
The Lefschetz Fixed Point Theorem
We remark that the (Fixed Point) axiom in Theorem 4.4.3 is known as the Lefschetz Fixed Point Theorem. We also remark that every continuous mapf :M →M is continuously homotopic to a smooth map and that any two smooth mapsf0, f1 :M →M that are continuously homotopic are also smoothly homotopic and hence have the same Lefschetz number by the (Homotopy) axiom in Theorem 4.4.3. Thus the definition of the Lefschetz number and the Lefschetz Fixed Point Theorem carry over to continuous maps. In this form the Lefschetz Fixed Point Theorem can be viewed as a generalization of the Brouwer Fixed Point Theorem. The Lefschetz Fixed Point Theorem is particularly useful in combination with the formula
L(f) =
m
X
k=0
(−1)ktrace f∗:Hk(M;R)→Hk(M;R)
. (4.4.6)
This formula is proved in Theorem 6.4.8 for smooth maps.
Corollary 4.4.4 (Lefschetz Fixed Point Theorem). Let M be a com-pact manifold with boundary and letf :M →M be a continuous map such thatL(f)6= 0. Then f has a fixed point.
Proof. If f is smooth and has no fixed points then f is trivially a Lef-schetz map and so L(f) = 0 by Theorem 4.4.2. Iff is continuous and has no fixed point, then there exists a smooth map g:M →M without fixed points that is continuously homotopic tof and hence has the same Lefschetz numberL(f) =L(g) = 0. This proves Corollary 4.4.4.
Exercise 4.4.5. LetM ⊂Rkbe a compact submanifold with boundary and letf :M →M be a continuous map (without fixed points). Prove that there exists a smooth mapg:M →M (without fixed points) that is continuously homotopic to f. If f, g : M → M are smooth maps which are continu-ously homotopic, prove that they are smoothly homotopic. Deduce that the Lefschetz number is well defined for continuous maps. Hint: Forε >0 sufficiently small denote theε-tubular neighborhood ofM\∂M by
Uε:=n p+v
p∈M\∂M, v∈Rk, v⊥TpM,|v|< εo
and define the (smooth) mapr:Uε→M\∂M by r(p+v) :=p for p∈M and v∈TpM⊥ with |v|< ε. Assume f(M)⊂M\∂M and use the Weier-straß Approximation Theorem to find a smooth map h : M → Uε such that supp∈M|h(p)−f(p)|< ε. Define ft(p) :=r((1−t)f(p) +th(p)). If f has no fixed points, chooseε <infp∈M|p−f(p)|.
4.4. THE LEFSCHETZ NUMBER OF A SMOOTH MAP 73