The relation between the critical set and the topology of the space will now become more precise. In the smooth setting, the main result of Morse theory states that a smooth manifold with a smooth Morse functionf has the same homotopy type as a finite cell complexK such that each cell of dimension pofK is on one–to–one correspondence to a critical point of indexpoff. This result is still valid in the discrete setting, usingsimple homotopy [Cohen, 1973]
instead of homotopy. In this context, smooth manifolds can be decomposed into discrete structures, and tools of the smooth setting (such as the homotopy
37 II.5. Topological properties
type) and of the discrete setting (such as homology, Euler characteristic) both apply. Therefore, the two parts of this section are actually valid in both settings.
(a) Homotopy and Handle decomposition
The Morse inequalities can be easily deduced from thehandle decomposi-tionof a manifold through a smooth Morse function. A handleHpof dimension n and index p is the Cartesian product of two balls: Hp BpBnp. Attach-ing a handle to a manifold by identifyAttach-ing Sp1 Bnp Hp to a part of the manifold changes the topology of the manifold, and most of the topological changes can be interpreted as a handle attachment. This operation actually changes the homotopy type, defined as follows [Hatcher, 2002].
Homotopy type. Two topological spacesXandY are homotopy equivalent if they can be continuously deformed one into the other. Formally, X and Y have the same homotopy type if there exists four continuous functions f :X ÑY,g :Y ÑX,hX :Xr0,1s ÑX andhY :Y r0,1s ÑY such that hXpx,0q fgpxq,hXpx,1q xandhY py,0q gfpyq,hY py,1q y. For example eachsymbol on figure II.19 corresponds to a homotopy equivalence, while there is no homotopy equivalence on the Ñsymbols.
Figure II.19: Handlebody decomposition of a torus.
Handle decomposition. The Morse theorem states that there is no change in homotopy between cuts of a manifold below and above a level without critical point, and that the change in homotopy type between cuts below and above a level with only one critical point of index p corresponds to attaching
a handle of index p [Fomenko, 1987] (figure II.20). A cell complex HM can then be constructed from a Morse function by successively attaching a handle for each critical point: a handle H0 of index 0 for the absolute minimum f0 of f, then a handle is attached to H0 for the critical point with the smallest value f1 ¡ f0 and so forth. . . A smooth manifold is thus homotopy equivalent to a finite cell complex such that to each critical point of index p corresponds one handle–cell of dimension p: this is the handle decomposition.
This decomposition can also be described from the Smale decomposition introduced at the end of this chapter. The following tools of the discrete setting can then be applied to Kvia HK, in particular the Morse inequalities.
II.20(a): Minimum:
0–handle creating a connected component.
II.20(b): Saddle:
1–handle pinching a boundary.
II.20(c): Saddle:
1–handle joining two boundaries.
II.20(d): Maximum:
2–handle closing a void.
Figure II.20: Each one of the four critical points of a torus corresponds to a handle.
(b) Simple homotopy and Morse inequalities
The usual tools to characterize objects rely on invariants. For example, the homotopy type of a smooth manifold is a topological invariant, i.e. if two manifolds have different homotopy types, then they cannot be equal (home-omorphic). Whereas the handle decomposition HM have the same homotopy type asM, it is not in itself an invariant, since the same manifold with two dif-ferent Morse functions will have two difdif-ferent handle decompositions. However, the handle decomposition of a manifold is a cell complex, on which invariants of algebraic topology apply, at least for characterizing the PL–topology of a man-ifold [Rourke & Sanderson, 1972]. In particular, topological invariants such as singular homology and theEuler–Poincar´e characteristic can be related to the number of critical cells of any Morse function through the Morse inequalities, using the handle decomposition in the smooth setting, or simple homotopy in the discrete one.
39 II.5. Topological properties
Boundary operator. The objects considered by the homology are formal sums of cells having the same dimension called chains cp °
σpPKcσσp, cσ begin coefficients of a ring K. The collection Cp of chains of dimen-sion p is then a free module generated by the cells of dimension p. The central object of homology is the boundary operator Bp : Cp Ñ Cp1, which is simply defined from the orientation r : s of the complex: Bppσpq
°
τp1PKrσp :τp1sτp1 [Cooke & Finney, 1967]. For example, figure II.21 shows the boundary operator on a square made of 4 triangles. We can check for example that the boundary of the whole square is made of the four external edges:B2pf0 f1 f2 f3q e0 e3 e5e7. The definition of r : simplies directly that Bp Bp 1 0, i.e. ImBp 1 kerBp.
B2pf0q e0 e1e2 B1pe0q v1v0 B2pf1q e3 e4e1 B1pe1q v4v1 B2pf2q e5 e6e4 B1pe2q v4v0 B2pf3q e2e6e7 B1pe3q v2v1 B0pv0q 0 B1pe4q v4v2 B0pv1q 0 B1pe5q v3v2 B0pv2q 0 B1pe6q v4v3 B0pv3q 0 B1pe7q v3v0 Figure II.21: The boundary operator on a small square model.
Homology. This can be written as an exact sequence [Hatcher, 2002]:
t0uoo B0 C0oo B1 C1oo B2 C2oo B3 oo Bn CnooBn 1 t0u
The direction of the arrows in the diagram comes from the fact that the boundary operator decreases the dimension of the chains. Thep–th homology group HppKq is defined by kerBp{ImBp 1. Those groups are topological invariants: homeomorphic spaces have the same homology groups. Their ranks βppKqare called theBetti numbers ofK. For example on figure II.21, we have kerB2 t0u, ImB2 kerB1 K4, ImB1 K4, kerB0 C0 K5. We get H0 K, H1 H2 t0u, i.e. β0 1, β1 β2 0.
Euler–Poincar´e characteristic. Denoting by #p the number of cells of dimension p of K, the Euler–Poincar´e characteristic is the alternate sum of these quantities:χpKq °
pPZp1qp#ppKq. In that sense, it can be defined
on the Handle decomposition of a smooth manifold. The Euler–Poincar´e characteristic is also a topological invariant, since it can be written in terms of the Betti numbers: χpKq °
pPZp1qpβppKq. For example on figure II.21 we had χ# 58 4β 10 01.
Figure II.22: Collapse of a heptagon: collapses do not alter the Euler charac-teristic.
Simple homotopy Homotopy of cell complexes is usually defined over their geometric realization. However, a weaker version of homotopy equivalence, calledsimple homotopy, can be defined combinatorially by successive collapses and extensions [Cohen, 1973]. If τp1 σp are two cells of a cell complex K and τ is not the face of any other cell of K, then K collapses onto Kz tτ, σu (figure II.22). The reverse operation of a collapse is called anextension. If one complex can be obtained from K by a sequence of collapses and extensions, it is said to have the same simple homotopy type of K.
Discrete decomposition. The main theorem of Forman’s discrete Morse theory states that a cell complex with a discrete Morse function f is simple homotopy equivalent to another cell complex having exactly one cell of
II.23(a): Simple torus, with 1 critical vertex, 2 criti-cal edges and 1 criticriti-cal face.
II.23(b): Solid torus with 1 criti-cal vertex and 1 criticriti-cal edge.
Figure II.23: Optimal Morse functions characterize the complex, for example to distinguish between a toric surface and a solid torus.
41 II.5. Topological properties
dimension p for each critical cell of f of dimension p [Forman, 1995]. For example, figure II.5 shows the discrete decomposition of a torus with only four critical points, which differs from the decomposition of a solid torus (figure II.23). This theorem actually puts, to our knowledge, Forman’s Morse theory as the only discrete version of Morse theory that proves this homotopy equivalence.
Figure II.24: A non–optimal discrete Morse function on a torus:m0 1β0, m1 3¥2β1 and m2 2¥1β2.
Morse inequalities. The number of p–cells in the decomposition of K by a Morse functionf is the number of critical elements of index p, which we will denote mppfq. Since the Euler–Poincar´e characteristic is a topological invari-ant, the characteristic of this decomposition is the same as the characteristic of K, and can thus be written: χpMq °
pPZp1qpmppfq °
pPZp1qpβp. This equality is weak, since it can be deduced from a stronger set of inequal-ities, called the Morse inequalities: °
p¤kp1qkpβp ¤ °
p¤kp1qkpmppfq. By summing these inequalities, we get βp ¤ mppfq. There are two obstruc-tions for these inequalities to become equalities: either the Morse function is not optimal (figure II.24), the worst case being all cells critical (fpσpq p, V H and V 0), or the cell complex has some finer topological character-istics that homology does not detect, for example homotopy–only features or collapsibility [Crowleyet al., 2005].