Fixed Point Indices and Manifolds with Collars

This paper concerns a formula which relates the Lefschetz number L(f) for a map f:M -->M' to the fixed point index I(f) summed with the fixed point index of a derived map on part of the boundary of M. Here M is a compact manifold and M' is M with a collar attached.


INTRODUCTION
This Paper represents the first third of a Ph. D thesis [16] written by the first author under the direction of the second author at Purdue University in 1990. The Thesis is entitled " Fixed Point Indices and Transfers, and Path Fields" and it contains, in addition to the contents of this manuscript, a formula analogous to (1.1), which relates to Dold's fixed point transfers and a study of path fields of differential manifolds in order to relate the formula in this manuscript with an analogous formula involving indices of vector fields.
Let M be a compact differentiable manifold with or without boundary ∂M . Assume V is a vector field on M with only isolated zeros. If M is with boundary ∂M and V points outward at all boundary points, then the index of the vector field V equals Euler characteristic of the manifold M . This is the classical Poincaré-Hopf Index Theorem. (A 2-dimensional version of this theorem was proven by Poincaré in 1885; in full generality the theorem was proven by Hopf [11] in 1926). In particular, the index is a topological invariant of M ; it does not depend on the particular choice of a vector field on M .
Marston Morse [13] extended this result to vector fields under more general boundary conditions, namely, to any vector field without zeros on the boundary ∂M ; he discovered the following formula: where χ(M ) denotes the Euler characteristic of M and ∂ V is defined as follows: Let ∂ M be the open subset of the boundary ∂M containing all the points m for which the vectors V (m) point inward; and let ∂V be the vector field on the boundary ∂M obtained by first restricting V to the boundary and then projecting V | ∂M to its component field tangent to the boundary. Then ∂ V = ∂V | ∂ M . Furthermore, in the same paper, Morse generalized his result to indices of vector fields with nonisolated zeros. This is the formula (1.1).Now (1.1) was rediscovered by D. Gottlieb [7] and C. Pugh [15]. D. Gottlieb further found further interesting applications in [8], [9] and [10]. Throughout this paper we shall call formula (1.1) the Morse formula for indices of vector fields.
We consider maps f : M → M ′ from a compact topological manifold M to M ′ where M ′ is obtained by attaching a collar ∂M × [0, 1] to M . If f has no fixed points on the boundary ∂M , we prove Theorem (2.2.1) which is the fixed point version of the Morse formula: where This paper is organized as follows: In Section 2, §1, we list some properties of fixed point indices; our first main result, Theorem (2.2.1), is proven in Section 2, §2.

FIXED POINT VERSION OF THE MORSE FORMULA
In this section, we use the definition of fixed point index and some well known results on fixed point index given by Dold in [3] or [5,Chap. 7] to obtain an equation for fixed point indices [Theorem 2.2.1] analogous to the Morse equation for vector field indices described in the introduction. §1. Fixed point index and its properties. [3] defined the fixed point index I(f ) and proved the following properties.
Given a map f : V → X and V is a union of open subsets V j , j = 1, 2 . . . , n such that the fixed point sets F j = F (f ) ∩ V j are mutually disjoint. Then for each j, I(f | Vj ) is defined and Let f : V → X be a constant map. Then If f is a map from a compact EN R X to itself, then (2.1.6) COMMUTATIVITY.
If f : U → X ′ and g : U ′ → X are maps where U ⊆ X and U ′ ⊆ X ′ are open subsets, then the two composites gf : For our purposes it is useful to reformulate the properties of additivity (2.1.2) and homotopy invariance (2.1.7) in the form of the following propositions. These reformulations are found in R. F Brown's book [2b], and they form part of an axiom system for the fixed point index. The five axioms are a subset of Dold's properties. They consist of localization, homotopy invariance , addititvity, normalization and commutivity. We will show that the main formula will follow from these axioms. We will give an alternate proof in the next subsection §2.  Proof: Since H = H| V ×I is a homotopy from f 0 to f 1 , it suffices to verify that the set F = {x ∈ V |H(x, t) = x for some t} is compact. Let {x j } be a sequence in F converging to x ∈ V = V ∪ Bd(V ). There exists a subsequence {t j } of those t's in I such that H(x j , t j ) = x j . Since I is compact, a subsequence of {t j } converges to a point t ∈ I. By the continuity of H, we have H(x, t) = x. On the other hand, we know that H(x, t) = x for all x ∈ Bd(V ); thus, x ∈ V and H(x, t) = x. Consequently, x ∈ F . Therefore, F is a closed subset of a compact space, hence F is compact. This proves the proposition. §2. The main formula. For specificity we define the retraction r: Let r : M ′ → M be the retraction from M ′ to M given by the formula: Now we can formulate the main result of the section. Now, assume r ′ is any retraction from M ′ to M such that r ′ maps the collar ∂M × [0, 1] into the boundary ∂M . Then the following Theorem is true: (2.2.1) THEOREM. Proof: First, we prove the formula: Since r ′ f is a self map from M to M , so We have: Let us decompose the map r ′ f | V2 : The commutativity (2.1.6) implies that Combining equations (i), (ii) and (iii), we obtain (iv) This completes the proof of the formula holding for any retraction r ′ . The following two lemmas will show that the terms in equation (iv) are the same no matter which retraction r ′ is chosen.
(2.2.2) LEMMA. The retraction r is homotopic to r ′ Proof: Consider the homotopy H t : M ′ → M , 0 ≤ t ≤ 1, defined as follows: Clearly, H 0 = r and H 1 = r ′ . So, rf and r ′ f are homotopic. Proof: By LEMMA (2.2.2) rf and r ′ f are homotopic and, consequently, since the Lefschetz number L is a homotopy invariant. Equations (iv) and (v) and equation (v) with r replacing r ′ imply that This concludes the proof of THEOREM (2.2.1).  (i) if f (S n−1 ) ⊂ D n , then f has a fixed point.
Proof: The commutativity (2.1.6) implies that It is easy to see that the fixed point set of the map f We now define a homotopy G s , 0 ≤ s ≤ 1, as the composite of the following maps where the map H s is defined as follows: where t is a constant, 0 < t ≤ 1.
since the map H 0 = Identity, we have G 0 (x, t) = H 0 (f r(x, t)) = f r(x, t), and where r • f is a map from ∂ M to ∂M and g : I → I, g(t) = t, is the constant map. Furthermore, the restriction G s | Bd(∂ M×I) has no fixed points for any 0 ≤ s ≤ 1.
To see this, we look at a point x ∈ Bd(∂ M ). We know then f (x) ∈ ∂M and rf (x) = f (x) = x, therefore, The last equality holds because ∂ M × (0, 1] contains the fixed point set of (f r| r −1 (V ) ).

Proof of Theorem (2.2.1):
Consider the composite M