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==Basic concepts== [[Image:Saddle point.png|thumb|right|A saddle point]] To illustrate, consider a mountainous landscape surface <math>M</math> (more generally, a [[manifold]]). If <math>f</math> is the [[Function (mathematics)|function]] <math>M \to \mathbb{R}</math> giving the elevation of each point, then the [[inverse image]] of a point in <math>\mathbb{R}</math> is a [[contour line]] (more generally, a [[level set]]). Each [[Connected component (topology)|connected component]] of a contour line is either a point, a [[closed curve|simple closed curve]], or a closed curve with [[Singular point of a curve|double point(s)]]. Contour lines may also have points of higher order (triple points, etc.), but these are unstable and may be removed by a slight deformation of the landscape. Double points in contour lines occur at [[saddle points]], or passes, where the surrounding landscape curves up in one direction and down in the other. [[Image:Saddle contours.svg|thumb|left|Contour lines around a saddle point]] Imagine flooding this landscape with water. When the water reaches elevation <math>a</math>, the underwater surface is <math>M^a \,\stackrel{\text{def}}{=}\, f^{-1}(-\infty, a]</math>, the points with elevation <math>a</math> or below. Consider how the topology of this surface changes as the water rises. It appears unchanged except when <math>a</math> passes the height of a [[Critical point (mathematics)|critical point]], where the [[gradient]] of <math>f</math> is <math>0</math> (more generally, the [[Jacobian matrix]] acting as a [[linear map]] between [[Tangent space|tangent spaces]] does not have maximal [[Rank (linear algebra)|rank]]). In other words, the topology of <math>M^a</math> does not change except when the water either (1) starts filling a basin, (2) covers a saddle (a [[mountain pass]]), or (3) submerges a peak. [[Image:3D-Leveltorus.png|thumb|right|The torus]] To these three types of [[Critical point (mathematics)|critical points]]{{Em dash}}basins, passes, and peaks (i.e. minima, saddles, and maxima){{Em dash}}one associates a number called the index, the number of [[Linear independence|independent]] directions in which <math>f</math> decreases from the point. More precisely, the index of a non-degenerate critical point <math>p</math> of <math>f</math> is the [[Dimension (vector space)|dimension]] of the largest subspace of the [[tangent space]] to <math>M</math> at <math>p</math> on which the [[Hessian matrix|Hessian]] of <math>f</math> is negative definite. The indices of basins, passes, and peaks are <math>0, 1,</math> and <math>2,</math> respectively. Considering a more general surface, let <math>M</math> be a [[torus]] oriented as in the picture, with <math>f</math> again taking a point to its height above the plane. One can again analyze how the topology of the underwater surface <math>M^a</math> changes as the water level <math>a</math> rises. [[Image:3D-Cylinder and disk with handle.png|thumb|left|A cylinder (upper right), formed by <math>M^a</math> when <math>f(q)<a<f(r)</math>, is homotopy equivalent to a 1-cell attached to a disk (lower left).]] [[Image:3D-Cylinder with handle and torus with hole.png|thumb|right|A torus with a disk removed (upper right), formed by <math>M^a</math> when <math>f(r)<a<f(s)</math>, is homotopy equivalent to a 1-cell attached to a cylinder (lower left).]] Starting from the bottom of the torus, let <math>p, q, r,</math> and <math>s</math> be the four critical points of index <math>0, 1, 1,</math> and <math>2</math> corresponding to the basin, two saddles, and peak, respectively. When <math>a</math> is less than <math>f(p) = 0,</math> then <math>M^a</math> is the empty set. After <math>a</math> passes the level of <math>p,</math> when <math>0 < a < f(q),</math> then <math>M^a</math> is a [[Disk (mathematics)|disk]], which is [[homotopy equivalent]] to a point (a 0-cell) which has been "attached" to the empty set. Next, when <math>a</math> exceeds the level of <math>q,</math> and <math>f(q) < a < f(r),</math> then <math>M^a</math> is a cylinder, and is homotopy equivalent to a disk with a 1-cell attached (image at left). Once <math>a</math> passes the level of <math>r,</math> and <math>f(r) < a < f(s),</math> then <math>M^a</math> is a torus with a disk removed, which is homotopy equivalent to a [[Cylinder (geometry)|cylinder]] with a 1-cell attached (image at right). Finally, when <math>a</math> is greater than the critical level of <math>s,</math> <math>M^a</math> is a torus, i.e. a torus with a disk (a 2-cell) removed and re-attached. This illustrates the following rule: the topology of <math>M^{a}</math> does not change except when <math>a</math> passes the height of a critical point; at this point, a <math>\gamma</math>-cell is attached to <math>M^{a}</math>, where <math>\gamma</math> is the index of the point. This does not address what happens when two critical points are at the same height, which can be resolved by a slight perturbation of <math>f.</math> In the case of a landscape or a manifold [[Embedding|embedded]] in [[Euclidean space]], this perturbation might simply be tilting slightly, rotating the coordinate system. One must take care to make the critical points non-degenerate. To see what can pose a problem, let <math>M = \R</math> and let <math>f(x) = x^3.</math> Then <math>0</math> is a critical point of <math>f,</math> but the topology of <math>M^{a}</math> does not change when <math>a</math> passes <math>0.</math> The problem is that the second derivative is <math>f''(0) = 0</math>{{Em dash}}that is, the [[Hessian matrix|Hessian]] of <math>f</math> vanishes and the critical point is degenerate. This situation is unstable, since by slightly deforming <math>f</math> to <math>f(x) = x^3 +\epsilon x</math>, the degenerate critical point is either removed (<math>\epsilon>0</math>) or breaks up into two non-degenerate critical points (<math>\epsilon<0</math>).
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