2.2 Topology
2.2.2 Morse theory
One of the key tools used to study the topology of spaces is Morse theory. Morse theory is the study of the relationship between functions on a space and the shape of the space. Although Morse theory can be applied to spaces of infinite dimension, we are particularly interested in the application of Morse theory to 2-manifolds. Building on variational calculus, Morse theory draws a relationship between the critical points of a smooth function defined on a smooth manifold to the global topology of the manifold [64, 62].
To better understand Morse theory, first let us briefly consider the critical points of functions of one variable, f(x) = y. We can examine the maxima, minima, and inflection points of such a function by examining its critical points. Namely, the critical points of such a function satisfy: f0(x0) = 0. A critical point is said to be non-degenerate if its second derivative is not equal to zero, (f00(x0)6= 0). Likewise for functions of two variables, such asf(x, y) = z, critical points are defined as the points where the gradient is equal to zero (∇f = 0, i.e., the partial derivatives∂f /∂x and∂f /∂y are both equal to zero). A non- degenerate critical point for a function with two variables is one where the determinant of the Hessian of f is not zero. The Hessian is the matrix of second derivatives with the determinant of the Hessian being:
detHf(x, y) =∂2f /∂x2×∂2f /∂y2−(∂2f /∂x∂y)2.
Now, given a smooth manifold, M, a map, f : M → R, which assigns a real number to each point pofM, is a function f onM. Such a function is a Morse function if every critical point of the function f :M →RonM is non-degenerate and isolated. The critical points are isolated if each critical point maps to unique real number.
Consider the case of a function of two variables, Morse theory shows that near a non-degenerate critical point the function can only look fairly simple. In fact, by choosing appropriate local coordinates, a function of two variable will have one of three possible shapes around a non-degenerate critical point:
• f =x2+y2+c
• f =x2−y2+c
• f =−x2−y2+c
wherecis a constant. See Figure 2.3 for an illustration of these functions. By decomposing a smooth manifold into these “shapes”, the global shape and topology of the manifold is revealed. Morse theory also presents methods to classify which type of critical point a given point corresponds to. Specifically, by examining
the number of negative eigenvalues of the Hessian, the critical point can be indexed. That is, a minimum has zero negative eigenvalues, a saddle point has one, and a maximum has two negative eigenvalues. This analysis corresponds to the fact that a minimum has no downhill sides, while an isolated saddle point has two downhill sides, one parallel to the direction of the eigenvector associated with the negative eigenvalue and one anti-parallel. A maximum has downhill sides associated with both directions of both eigenvectors.
A classic result from Morse theory is that given a closed surfaceM and a Morse function,f :M →R. If this function has only two non-degenerate critical points thenM is topologically equivalent to a sphere. For example, a typical Morse function is a height function, and if we consider such a function and a sphere, we see that there are two critical points to the function, corresponding to the maximum and minimum at the north and south pole of the sphere, (see Figure 2.4). More precisely, the number of critical points is equivalent to the Euler characteristic of the surface: χ= #minima−#saddlepoints+ #maxima. Another essential result from Morse theory shows that between the critical points the topology of the manifold is guaranteed not to change (called the deformation lemma).
Figure 2.3: Critical points for a function of two variables
Representations of the possible shapes near critical points for a function of two variables. On the left is f =x2+y2+c, the middle figure representsf =x2−y2+c, and the figure on the right isf =−x2−y2+c.
To further illustrate the relationship of critical points and the global topology of a surface, consider the following geometric interpretation. Given a Morse function,f : M → Rwhich is a height function, that height function defines parallel planes. Now imagine a torus standing on its end, (see Figure 2.4). If we consider each of the tangent planes of the torus, the points with tangent planes that correspond exactly to the height planes will be the critical points of the Morse function. For the example of the torus, as we would expect there will be critical points corresponding to the maximum and minimum (at the north and south poles of the torus) and at the two saddle points of the handle.
There is further geometric interpretation of Morse theory for a 2-manifold correlating the tangent plane of the surface for each point and the planes defined by a height function. Specifically, we consider classifying critical points and trivial points (non-critical points). Similar to the method presented above, where we classify points based on their shape, let us consider analyzing the local shape of the surface, by looking at the relationship between a small circular neighborhood of each point on the surface and the height planes of
Figure 2.4: Illustration of height planes on a sphere and torus
On the left is a sphere and its two critical points, (and their associated tangent planes), at the north and south pole. On the right is a torus with showing height planes matching the tangent planes at critical points for a minimum and a saddle point.
a height function. For a regular (trivial) pointq, the height plane is not the tangent plane. In fact the height plane divides the small circular neighborhood aroundq into two pieces. One piece lays above the height plane and the other below, (see Figure 2.5). Now, consider instead a maximum or minimum point and the small circular neighborhood around each of these points. The horizontal plane of the height function will not intersect either of these points nor their small circle neighborhood. Finally, consider a saddle point – for such a point, again consider the small circular neighborhood around the point and its relationship to the height plane. For a saddle point, its circle will be intersected in four places, dividing the region around the saddle point into four sections, (see Figure 2.5). These observation are important for understanding related methods in the discrete setting, discussed in Chapter 2.
Figure 2.5: Critical points and height planes
On the left is a trivial point and a small circular neighborhood. Note that the height plane divides the circular neighborhood into two regions. On the right is a critical point, a saddle point, with a small circular neighborhood. For isolated saddle point the height plane divides the circular neighborhood into four regions.
Morse theory is a powerful tool for revealing the global topology of smooth 2-manifolds, however, this
thesis is directed to the piece-wise linear discrete setting. We do not start with smooth manifolds, nor is it always easy to construct a smooth Morse function over the surface. There has been recent work on extending Morse theory for piece-wise linear functions on 2-manifolds. Work presented by Ulrike Axen [9] uses a wavefront traversal to generate a Morse function for triangulated meshes and work by Edelsbrunner et al. [26]
extend Morse theory to piece-wise linear functions on 2-manifolds. This work is discussed in more detail in Section 3. The work presented in this thesis builds from Morse theory with some modifications to tune our methods to the discrete setting. We do not find exact critical points on a surface, but instead find critical levels. We discuss these methods in more detail in Section 4.