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.
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Acknowledgements
Bei Wang is supported in part by NSF IIS-1513616 and NSF DBI-1661375. We would like to thank Vin de Silva and Davide Lofano for their valuable comments regarding the definition of stratified simplicial complexes. We are also grateful to the anonymous referees for carefully reading this manuscript and providing insightful comments.
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Appendix A: Preliminaries on Classical and Stratified Morse Theory
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.
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2.
At each critical point u of f, the restriction of f to the stratum S containing u is non-degenerate.
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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
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
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
Here, \(l^{-}\) is the lower half link of \({X}\) where \(l^{-} = N \cap f^{-1}(v-\varepsilon ) \cap B_{\delta }^{{{\mathbb {M}}}}\).
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Knudson, K., Wang, B. Discrete Stratified Morse Theory. Discrete Comput Geom 67, 1023–1052 (2022). https://doi.org/10.1007/s00454-022-00372-1
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DOI: https://doi.org/10.1007/s00454-022-00372-1