Stratified Morse Theory

Suppose that X is a topological space, f is a real valued function on X, and c is a real number. Then we will denote by X _≤c the subspace of points x in X such that f(x)≤c. The fundamental problem of Morse theory is to study the topological changes in the space X _≤c as the number c varies.

One of the main sets of mathematical results proved in this book is a collection of theorems on the topology of complex analytic varieties. There are generalizations of the Lefschetz hyperplane theorem for complex projective varieties, and generalizations of the theorem that the homotopy dimension of a Stein manifold is bounded by its complex dimension. In this section of the introduction, we give a sketch of the statements of the theorems with motivation and some history. Technically precise statements of the theorems in their most general form are grouped together in Chapter 1 of Part II of the book.

In this chapter we develop the tools from stratification theory which are needed in the proof of the main theorems of Part I (Sects. 3.7, 3.10, 9.5, 10.5, 11.5, 12.5). Sections 1.2 through 1.8 constitute a short course on the main results and techniques of stratification theory, and they summarize the work of many people: [Ab], [AR], [BW], [Ch], [G1], [Ha1], [Ha2], [Hi1], [Hi2], [Hi3], [J], [La], [LT2], [Lo1], [Lo2], [Ma1], [Ma2], [O], [P1], [P2], [T1], [T2], [T3], [T4], [T5], [T6], [Tr1], [Tr2], [V], [Ve1], [Ve2], [W1], [W2], [W3]. Sections 1.8 through 1.10 contain the results on characteristic covectors which are used along with “moving the wall” (Chap. 4) in creating the deformations which are the heart of our main theorems. Section 1.11 is a basic result in stratification theory that is often quoted and whose proof turns out to be surprisingly tricky: the transversal intersection of a manifold M and a Whitney stratified space W admits a “tubular neighborhood” in W, i.e., a neighborhood which is homeomorphic to the total space of a vectorbundle E → W ∩ M.

Classical Morse theory is concerned with the critical points of a class of smooth proper functions f from a manifold Z to the real numbers, called Morse functions. For our generalization, we will let Z be a closed Whitney stratified space in some ambient smooth manifold M. We will need analogues for the notions of smooth function, critical point, and Morse function for this setting. A smooth function on Z will be a function which extends to a smooth function on M. A critical point of a smooth function f will be a critical point of the restriction of f to any stratum S of Z. A proper function f is called Morse if (1) its restriction to each stratum has only nondegenerate critical points, (2) its critical values are distinct, and (3) the differential of f at a critical point p in S does not annihilate any limit of tangent spaces to a stratum other than S. This third condition is a sort of nondegeneracy condition normal to the stratum. If Z is subanalytic (which includes the real and complex analytic cases), then the set of Morse functions forms an open dense subset of the space of smooth functions, and Morse functions are structurally stable, just as in the classical case [P1].

In this chapter the main objects of interest: local Morse data, normal Morse data, and tangential Morse data will be defined. The main theorems are stated in Sects. 3.7, 3.8, and 3.9.

This chapter and the next contain the main technical tools which will be used in Part I. Moving the wall is a rigorous but intuitive technique for verifying the hypotheses and expressing the conclusions of Thom’s first isotopy lemma, which is particularly useful when the isotopy lemma is applied to a complicated geometric situation. The power of this method even in the nonsingular case is illustrated in Sect. 4.5, where we reprove the classical result in Morse theory: crossing a nondegenerate critical point corresponds to attaching a handle.

The definition of local Morse data involves certain choices of allowable parameters ε and δ. The set of all such allowable ε and δ form a region in the (ε,δ) plane of a certain shape, which we call “fringed, of type 0<ε≪δ”. In this chapter we study fringed sets: these are open subsets of the first quadrant in ℝ² whose closure contains a segment of the x-axis ending at the origin, and which are unions of vertical segments. Fringed sets of this type will appear throughout the technical discussions in Part I.

This chapter contains the proofs of Theorems 3.5.3, 3.6.2, 3.9.2, 3.9.3, 3.10, and 3.11.

In this chapter we prove Theorem 3.7: the local Morse data is the Cartesian product of the tangential Morse data with the normal Morse data.

The reader who is interested only in Morse theory for singular spaces or for nonproper Morse function may skip this chapter. We will consider the Morse theory of a composition which will eventually be used (in Part II, Sects. 5.1 and 5.1*, with Z = ℂℙ ⁿ ) to prove a conjecture of Deligne [Dl] concerning Lefschetz hyperplane theorems for a variety X and an algebraic map π: X → ℂℙ ⁿ . We will approximate the function f by a Morse function, although the composition fπ: X → ℝ is not Morse (or even Morse-Bott) in any reasonable sense. All attempts to prove Deligne’s conjecture by approximating fπ by a Morse function seem to end in failure because one loses curvature estimates on the Morse index of fπ. Instead, we are forced to “relativize” the Morse theory of f. Our main result is stated in Sect. 9.8.

It is often necessary to consider a “Morse function” f: X → ℝ which is not proper, but which can be extended to a proper function where Z contains X as a dense open subset. For example, Z may be a compactification of some noncompact algebraic variety X ⊂ ℂℙ ⁿ , and f may be a smooth function defined on the ambient ℂℙ ⁿ . We shall assume that it is possible to find a stratification of Z so that X ⊂ Z is a union of strata. Thus, X is obtained from Z by removing certain strata. The main theorems (Sects. 3.7, 3.10, 3.11) continue to apply to X, because we simply remove the same strata from both sides of the homeomorphisms. Since these homeomorphisms were originally proven to be decomposition preserving, it is a triviality that they induce homeomorphisms on unions of pieces in the decomposition. However, Proposition 3.2 is no longer strictly true in this context unless we also consider the effect on X _≤v of critical values v which correspond to critical points p∈Z which do not lie in X. Our main theorems also apply to these “critical points at infinity”. For all the applications which we consider, X will be an open dense subset of Z. However, the results of this chapter apply to any union of strata X ⊂ Z, and so we will not even assume that X is locally closed in Z.

This section is the common generalization of Chapters 9 and 10.

In this chapter we analyze the normal Morse data at a critical point p∈Z of a function f₁: Z → ℝ under the assumption that there exists a second function f₂: Z → ℝ such that the map (f₁,f₂): Z → ℝ² has a nondegenerate critical point at p (see below).

The main technical results of complex Morse theory are the following: Let W be a closed subvariety of a Whitney stratified complex analytic variety Z, and suppose that W is a union of strata in Z. Let f: Z → ℝ be a proper Morse function, which has a nondegenerate critical point p∈Z which lies in some stratum S of complex codimension c in Z. Let λ denote the Morse index of f | S at the point p, and let v=f (p) be the associated critical value. We will consider the Morse theory of f | X, where X = Z − W. Suppose the interval [a, b] contains no critical values of f | Z other than v, and that v∈(a, b).

The proofs of the results stated here will appear in Sect. 5. The four main theorems (Sects. 1.1, 1.1*, 1.2, 1.2*) are followed by four analogous “local” theorems (1.3.1, 1.3.2, 1.3*.1, 1.3*.2) which may be considered as generalizations of the global theorems. These results are further generalized in Sects. 7.2 and 7.3.

In this chapter we describe the local topological structure of a complex analytic variety and a generic complex analytic function on that variety. Most of the material described in this section is fairly well known, see for example [Mi2], [Du], [H1], [H2], [HL3], [LK], [Kp4], [Lê3], [Lê4], [LT1]. However the proofs we give here are rigorous and are easy, given the technique of “moving the wall” which was developed in Part I.

We are now in a position to identify the homotopy Morse data (Part I, Sect. 3.3) for Morse functions on complex analytic varieties. Recall that homotopy Morse data is a pair (A, B) which is homotopy equivalent to some choice of Morse data. The importance of homotopy Morse data is the following: Suppose the pair (A, B) is homotopy Morse data for the Morse function f: X → ℝ at the critical point p, with critical value v= f(p). Suppose v ∈(a, b) and that the closed interval [a, b] contains no other critical values of f. Then there exists a continuous map h: B → X_≤a such that X_≤b is homotopy equivalent to the adjunction space X_≤a⋃ _B A (see Part I, Sect. 3.3). The identification of homotopy Morse data uses the deepest results of Part I and of Part II, Sect. 2: Theorem 3.5.4 of Part I says that local Morse data is Morse data and Theorem 3.7 of Part I says that local Morse data is the product of tangential Morse data with normal Morse data. So, homotopy Morse data is the product of the homotopy type of the tangential Morse data with the homotopy type of the normal Morse data. The homotopy type of tangential Morse data was identified classically by Morse, Thom, and Bott. It is the pair (D ^λ ,∂D ^λ ), where λ denotes the Morse index of the restriction of f to the stratum which contains the critical point p. So, the problem of identifying the homotopy Morse data is reduced to the problem of identifying the homotopy type of the normal Morse data. This was carried out in Sect. 2. In this short chapter we summarize those results.

This chapter contains the estimates on the connectivity of the Morse data which are necessary for the proofs of the main applications. Since the Morse data is the product of the tangential Morse data with the normal Morse data, this requires estimates on both. The estimates on tangential Morse data are made in terms of the remarkable properties of the Levi form of the Morse function (see the appendix, Sect. 4.A). The normal Morse data is analyzed (inductively) by applying the entire apparatus of Morse theory to the complex link, which is a complex analytic space of smaller dimension.

Many of the results in this chapter appeared first in [GM3].

The four results in this chapter are generalizations of Theorems 1.1, 1.1*, 1.2, and 1.2*. We will replace the pair (ℂℙ ^N , L) = (projective space, linear subspace) by the pair (Y, A) = (complex manifold, compact complex submanifold) in the statement of these four theorems. We define the notion of a “q-defective pair”, which implies that the Lefschetz theorem holds for (Y, A) in a range of dimensions which is smaller (by q) than for the pair (ℂℙ ^N , ℂℙ ^N−1 ).

In this section we consider the topology of the complement of a finite collection A = {A₁, A₂,…, A _m } of linear affine subspaces of Euclidean space ℝ ⁿ . (The subspaces can be of arbitrary dimension, and do not necessarily contain the origin.) We give a combinatorial formula for the homology of this complement, which depends only on the partially ordered set P whose elements v are the intersections of the subspaces (ordered by inclusion), and on the dimensions of these “flats” (see Sect. 1.1, “statement of results”). Complements of hyperplanes have received considerable attention during the last ten years ([Ar], [Br], [Ca], [D2], [OS1], [OS2], [Zas]) and a nice survey article on the subject is [Ca]. In the case that the collection A consists of real hyperplanes, our formula reduces to Zaslavsky’s formula [Zas] for the number of connected components of M. If A is the underlying real arrangement of an arrangement of complex hyperplanes in ℂ^n/2 then our formula reduces to the Orlik-Solomon formula ([OS1]). It is interesting that in this case, the complex structure is irrelevant except that it guarantees that each flat is even-dimensional. In particular, our approach clarifies the connection (as observed in [OS1] and [OS4]) between real and complex arrangements in the following corollary.

(Unless otherwise noted, homology with integer coefficients will be used throughout.) Let A_l, A₂, A_m⊂ℝ ⁿ be a finite set of affine subspaces (of possibly various dimensions) of Euclidean space and let be the complement of the union of these subspaces. Associated to this collection of subspaces A there is a partially ordered set P whose elements v correspond to the “flats”, partially ordered by inclusion, with one maximal element T corresponding to the ambient space ℝ ⁿ . We shall use the notation v<w if v and w are distinct elements of P such that the flat |v| is contained in the flat |w|.

SupposeI is a partially ordered set with a unique maximal element T.

Throughout this chapter we will fix an arrangement A = {A₁A_2,…,A_m} of affine subspaces of ℝ ⁿ , let P denote the corresponding partially ordered set of flats which are the intersections of the affine spaces, let T denote the unique maximal element of P corresponding to ℝ ⁿ , and let K(P)denote the order complex of P. We denote by M = ℝ ⁿ − ∪ A the space of interest, i.e., the complement of the affine subspaces in A.

Recall that a partially ordered set L is a geometric lattice if

Consider the central arrangement A in ℝ³ which consists of the coordinate hyperplanes and a skew line ℓ through the origin, as illustrated in the following picture.