Discrete Stratified Morse Theory

Abstract

Inspired by the works of Forman on discrete Morse theory, which is a combinatorial adaptation to cell complexes of classical Morse theory on manifolds, we introduce a discrete analogue of the stratified Morse theory of Goresky and MacPherson. We describe the basics of this theory and prove fundamental theorems relating the topology of a general simplicial complex with the critical simplices of a discrete stratified Morse function on the complex. We also provide an algorithm that constructs a discrete stratified Morse function out of an arbitrary function defined on a finite simplicial complex; this is different from simply constructing a discrete Morse function on such a complex. We then give simple examples to convey the utility of our theory. Finally, we relate our theory with the classical stratified Morse theory in terms of triangulated Whitney stratified spaces.

Appendix A: Preliminaries on Classical and Stratified Morse Theory

For completeness, we include here a review of the basics of (stratified) Morse theory. Given a topological space ${X}$, studying the relation between the critical points of a Morse function (or a stratified Morse function) on ${X}$ and the topology of ${X}$ requires more care in the smooth setting in comparison with the discrete setting. Most of our review originates from the seminal work of Goresky and MacPherson [18].

1.1 A.1 Classical Morse Theory

Let ${X}$ be a compact, differentiable d-manifold and $f:{X}\rightarrow {{\mathbb {R}}}$ a smooth real-valued function on ${X}$. For a given value $a \in {{\mathbb {R}}}$, let ${X}_a = f^{-1}(-\infty ,a] = \{x \in {X}\mid f(x) \le a\}$ denote the sublevel set. Morse theory studies the topological changes in ${X}_a$ as a varies.

Morse functions. A point $x \in {X}$ is critical if the derivative at x equals zero. The value of f at a critical point is a critical value. All other points are regular points and all other values are regular values of f. A critical point x is non-degenerate if the Hessian, the matrix of second partial derivatives at the point, is invertible. The Morse index of the non-degenerate critical point x is the number of negative eigenvalues in the Hessian matrix, denoted as $\lambda (x)$.

Definition A.1

$f:{X}\rightarrow {{\mathbb {R}}}$ is a Morse function if all critical points are non-degenerate and its values at the critical points are distinct.

Results. We now review two fundamental results of classical Morse theory (CMT).

Theorem A.1

(CMT Theorem A; [18, p. 4], [10, p. 129])   Let $f:{X}\rightarrow {{\mathbb {R}}}$ be a differentiable function on a compact smooth manifold ${X}$. Let $a< b$ be real numbers such that $f^{-1}[a,b]$ is compact and contains no critical points of f. Then ${X}_a$ is diffeomorphic to ${X}_b$.

On the other hand, let f be a Morse function on ${X}$. We consider two regular values $a < b$ such that $f^{-1}[a,b]$ is compact but contains one critical point u of f, with index $\lambda$. Then ${X}_b$ has the homotopy type of ${X}_a$ with a $\lambda$-cell (or $\lambda$-handle, the smooth analogue of a $\lambda$-cell) attached along its boundary ([18, p. 5] and [10, p. 129]). We define Morse data for f at a critical point u in ${X}$ to be a pair of topological spaces (A, B) where $B \subset A$ with the property that as a real value c increases from a to b (by crossing the critical value f(u)), the change in ${X}_c$ can be described by gluing in A along B [18, p. 4]. Morse data measures the topological change in ${X}_c$ as c crosses critical value f(u). We have the second fundamental result of Morse theory,

Theorem A.2

(CMT Theorem B; [18, p. 5], [29, p. 77])   Let f be a Morse function on ${X}$. Consider two regular values $a < b$ where $f^{-1}[a,b]$ is compact and contains one critical point u of f, with index $\lambda$. Then ${X}_b$ is diffeomorphic to the space ${X}_a \cup _{B} A$, where $(A,B) = (D^{\lambda } \times D^{d-\lambda }, ({\partial }D^{\lambda }) \times D^{d-\lambda })$ is the Morse data, d is the dimension of ${X}$, $\lambda$ is the Morse index of u, $D^{k}$ denotes the closed k-dimensional disk, and ${\partial }D^{k}$ is its boundary.

1.2 A.2 Stratified Morse Theory

Morse theory can be generalized to certain singular spaces, in particular to Whitney stratified spaces [18, 30].

Stratified Morse function. Let ${X}$ be a compact d-dimensional Whitney stratified space embedded in some smooth manifold ${{\mathbb {M}}}$. A function on ${X}$ is smooth if it is the restriction to ${X}$ of a smooth function on ${{\mathbb {M}}}$. Let ${\bar{f}}:{{\mathbb {M}}}\rightarrow {{\mathbb {R}}}$ be a smooth function. The restriction f of ${\bar{f}}$ to ${X}$ is critical at a point $x \in {X}$ iff it is critical when restricted to the particular manifold piece which contains x [3]. A critical value of f is its value at a critical point.

Definition A.2

f is a stratified Morse function if ([3] and [18, p. 13]):

• 1. labelC1 All critical values of f are distinct.

• 2. At each critical point u of f, the restriction of f to the stratum S containing u is non-degenerate.

• 3. The differential of f at a critical point $u \in S$ does not annihilate (destroy) any limit of tangent spaces to any stratum $S'$ other than the stratum S containing u.

Conditions 1 and 2 imply that f is a Morse function when restricted to each stratum in the classical sense. Condition 2 is a non-degeneracy requirement in the tangential directions to S. Condition 3 is a non-degeneracy requirement in the directions normal to S [18, p. 13].

Results. Now we state the two fundamental results of stratified Morse theory.

Theorem A.3

(SMT Theorem A; [18, p. 6])   Let ${X}$ be a Whitney stratified space and $f:{X}\rightarrow {{\mathbb {R}}}$ a stratified Morse function. Suppose the interval [a, b] contains no critical values of f. Then ${X}_a$ is diffeomorphic to ${X}_b$.

Theorem A.4

(SMT Theorem B; [18, p. 8 and p. 64])   Let f be a stratified Morse function on a compact Whitney stratified space ${X}$. Consider two regular values $a < b$ such that $f^{-1}[a,b]$ is compact but contains one critical point u of f. Then ${X}_b$ is diffeomorphic to the space ${X}_a \cup _B A$, where the Morse data (A, B) is the product of the normal Morse data at u and the tangential Morse data at u.

To define tangential and normal Morse data, we have the following setup. Let ${X}$ be a Whitney stratified subset of some smooth manifold ${{\mathbb {M}}}$. Let $f:{X}\rightarrow {{\mathbb {R}}}$ be a stratified Morse function with a critical point u. Let S denote the stratum of ${X}$ which contains the critical point u. Let N be a normal slice at u, that is, $N={X}\cap N' \cap B^{{{\mathbb {M}}}}_{\delta }(u)$, where $N'$ is a sub-manifold of ${{\mathbb {M}}}$ which is traverse to each stratum of ${X}$, intersects the stratum S in a single point u, and satisfies $\mathrm dim\,{S}+\mathrm dim\,{N'}=\mathrm dim\,{{{\mathbb {M}}}}$. $B_\delta ^{{{\mathbb {M}}}}(u)$ is a closed ball of radius $\delta$ in ${{\mathbb {M}}}$ based on a Riemannian metric on ${{\mathbb {M}}}$. By Whitney’s condition, if $\delta$ is sufficiently small then ${\partial }B^{{{\mathbb {M}}}}_{\delta }(u)$ will be transverse to each stratum of ${X}$, and to each stratum in ${X}\cap N'$, fix such a $\delta > 0$ [18, p. 40].

The tangential Morse data for f at u is the pair

$$\begin{aligned} (P, Q) = (D^{\lambda } \times D^{s-\lambda }, ({\partial }D^{\lambda }) \times D^{s - \lambda }), \end{aligned}$$

where $\lambda$ is the (classical) Morse index of f restricted to S, f|S, at u, and s is the dimensional of stratum S [18, p. 65]. The normal Morse data is the pair

$$\begin{aligned} (J, K) = (N \cap f^{-1}[v-\varepsilon , v+\varepsilon ], N \cap f^{-1}(v-\varepsilon )), \end{aligned}$$

where $f(u)=v$ and $\varepsilon >0$ is chosen such that f|N has no critical values other than v in the interval $[v-\varepsilon ,v+\varepsilon ]$ [18, p. 65]. The Morse data is the topological product of the tangential and the normal Morse data, where the product of pairs is defined as $(A, B) = (P, Q) \times (J,K) =(P \times J , P \times K \cup Q \times J)$.

Theorem A.4 corresponds to the Main Theorem of [18, p. 65], which has the following homotopy consequences. Suppose ${X}$ is a Whitney stratified space, $f:{X}\rightarrow {{\mathbb {R}}}$ is a proper stratified Morse function, and [a, b] contains no critical values except for a single isolated critical value $v\in (a,b)$ which corresponds to a critic point p in some stratum ${{\mathbb {S}}}$ of ${X}$. $\lambda$ is the Morse index of $f|_{{{\mathbb {S}}}}$ at the point p.

Theorem A.5

(SMT homotopy consequences; [18, Sect. 3.12, p. 68])   The space ${X}_{b}$ has the homotopy type of a space which is obtained from ${X}_{a}$ by attaching the pair

$$\begin{aligned} (D^{\lambda }, {\partial }D^{\lambda }) \times (\mathrm {cone}(l^{-}), l^{-}). \end{aligned}$$

Here, $l^{-}$ is the lower half link of ${X}$ where $l^{-} = N \cap f^{-1}(v-\varepsilon ) \cap B_{\delta }^{{{\mathbb {M}}}}$.