Topological Necessary Conditions for Control Dynamics

The easiest algebraic object we can attach to a space is the graded abelian group $H_{*}(X)$ called the singular homology group of $X$ (with integer coefficients.) One can get quite far with only a vague understanding of what the singular homology measures and the reason is that powerful and effective methods for the computation of $H_{*}(X)$ exist. We outline the Mayer-Vietoris sequence and explain the concept of a long exact sequence of a pair and its relation to excision.

A graded abelian group $G=\oplus G_{k}$ is a direct sum of groups $G_{k}$, $k\in{\mathbb{Z}}_{+}$, such that the group addition is ‘component-wise’, in other words we add elements belonging to the same graded component together. The notation

$$g=\ldots+g^{0}+g^{1}+\ldots$$

for an element of $G$, where $g^{k}\in G_{k}$ is therefore unambiguous.

For a topological space, the $k$th homology group $H_{k}(X;{\mathbb{Z}})$ measures in some sense the ‘holes’ of $X$ that are like $k$ spheres $S^{k}$ (think of a boundary-less space, like a sphere, that does not actually bound anything itself in $X$.) The $0$th group $H_{0}(X)$ is equal to ${\mathbb{Z}}$ if $X$ is path connected. (Recall that a $0$-sphere is the boundary of an interval, i.e. the union of two points.) There is a way of defining reduced homology groups $\tilde{H}_{k}(X)$ so that $\tilde{H}_{0}(X)=0$ for a connected space and so that all higher dimensional groups coincide with the non-reduced ones.

Let us give some examples (we omit the zeroth homology group.) The singular homology of the circle $S^{1}$ is $H_{1}(S^{1})\simeq{\mathbb{Z}}$ and zero for $k>1$. Since $\pi_{1}(S^{1})={\mathbb{Z}}$ also, the homology group contains the same information as the fundamental group of the circle. Note the difference in interpretation, though: In the former case (for $\pi_{1}$), we are thinking of maps from the circle to itself, classifed by the number of net encirclements In the latter, we are thinking of a fixed circle —coinciding in this case with the whole space $S^{1}$— as the generator of a free abelian group; in this sense, we can write

$$H_{1}(S^{1})={\mathbb{Z}}[S^{1}]\simeq{\mathbb{Z}}.$$

For the sphere $S^{m}$ of dimension $m>1$, $H_{m}(S^{m})\simeq{\mathbb{Z}}$ is the only nonzero homology group in positive dimension. Since we also have that the $m$th homotopy group of an $m$ sphere is ${\mathbb{Z}}$, we have not yet obtained anything new, compared with homotopy theory. This is a little misleading: homotopy is both subtler than homology and far more difficult to compute: we do not, even today, have a complete list of the homotopy groups of spheres. Moreover, $\pi_{m+k}(S^{m})$ may very well be nonzero for some $k>0$, while $H_{m+k}(S^{m})=0$ always.

The ‘coincidence’ is really due to a nontrivial theorem, the Hurewicz isomorphism that states that homotopy and homology groups are isomorphic at the first level when one, and hence both, are nontrivial (the abelianization of the possibly nonabelian fundamental group is to be considered, if this happens at the first level.)

A quick check that homology theory does indeed give something new is to compute the homology of the torus $T^{2}$. We have that $H_{2}(T^{2})\simeq{\mathbb{Z}}$ even though $\pi_{2}(T^{2})=0$! (it may be profitable to spend a minute or two pondering the difference.)

Supose a space $X$ is the union of two opne subsets, $X=A\cup B$, with $A,B$ opne and $A\cap B\neq\emptyset$. Then there is a long exact sequence involving the homology groups of the three spaces

$$\ldots\to H_{k}(A\cap B)\to H_{k}(A)\oplus H_{k}(B)\to H_{k}(X)\to H_{k-1}(A\cap B)\to\ldots$$

(1)

This gives a surprisingly powerful tool for the computation of homology. Even without knowing what the maps at each stage are (for which we refer the reader to standard accounts such as [2]) the exactness allows the computation in concrete cases, such as that of the spheres. For this, decompose an $m$-sphere into two slightly overlapping hemispheres $A$ and $B$ so that their intersection $A\cap B$ is deformable to a sphere of dimension $m-1$. We can start an induction with dimension $m=1$ and use the Mayer-Vietoris sequence to obtain

$$\ldots\to H_{k}(S^{m-1})\to H_{k}(A)\oplus H_{k}(B)\to H_{k}(S^{m})\to H_{k-1}(S^{m-1})\to\ldots$$

(2)

yielding, for $k=m$, and since disks have no homology

$$\ldots\to 0\to 0\oplus 0\to H_{m}(S^{m})\to{\mathbb{Z}}\to 0\to\ldots.$$

(3)

We conclude that $H_{m}(S^{m})\simeq{\mathbb{Z}}$, since any exact sequence of the form

$$0\to C\to D\to 0$$

implies that the middle map is one-to-one and onto, i.e. an isomorphism.

A second very useful method for the computation of homology comes from considering pairs $(X,A)$, where $A\subset X$ is a subspace. One gets the long exact sequence for the pair

$$\ldots\to H_{k}(A)\to H_{k}(X)\to H_{k}(X,A)\to H_{k-1}(A)\to\ldots$$

(4)

where the groups $H_{k}(X,A)$ are the relative homology groups. Without giving the exact definition, found in the standard texts, let us mention that in many important cases, these relative groups are isomorphic to the homology groups of the quotient space $X/A$ (see chapter 3 for the definition.) The long exact sequence for a pair is thus extremely useful for the computation of the homological Conley index.

As an example, let us show that the quotient $D^{n}/S^{n-1}$ of a closed ball by its bounding sphere has the homology of the $n$-sphere $S^{n}$. The long exact sequence of the pair $(D^{n},S^{n-1})$ is

$$\ldots\to H_{k}(D^{n})\to H_{k}(D^{n},S^{n-1})\to H_{k-1}(S^{n-1})\to H_{k-1}(D^{n})\to\ldots$$

(5)

and so, at $k=n$, we get

$$\ldots\to 0\to H_{k}(D^{n},S^{n-1})\to{\mathbb{Z}}\to 0\to\ldots$$

(6)

hence $H_{k}(D^{n},S^{n-1})\simeq H_{k}(D^{n}/S^{n-1})\simeq{\mathbb{Z}}$. Similarly, one finds that, for $k\neq n$ (and nonzero), $H_{k}(D^{n}/S^{n-1})=0$.

Given a continuous map $f:X\to Y$, there is an induced map in homology, which we shall denote by $H_{*}(f)$ or $f_{*}$

$$H_{*}(f):H_{*}(X)\to H_{*}(Y)$$

and one checks that homology is a covariant functor from the category Top $=(\text{ Top},C^{0})$ of topological spaces and continuous maps to the category Ab$=(\text{Ab},\text{Hom})$ of abelian groups and homomorphisms between them. Since $H_{*}(X)$ is graded, the above homomorphism is understood to mean that it consists of homomorphisms at each level of homology:

$$H_{k}(f):H_{k}(X)\to H_{k}(Y)$$

for all $k$.

In fact, it would be more precise to say that the functor goes from the category hTop of homotopy equivalence classes of spaces and homotopic maps to the category Ab, since

and

When the space $X$ is a finite-dimensional manifold, the homology groups $H_{k}(X;{\mathbb{Z}})$ are finitely generated; thus, in this case, the basic structure theorem for finitely generated abelian groups is applicable.

In fact, one can describe the torsion group $\tau$ is more detail (see, for example, [4].)

The dimension of the free part of the homology group $H_{k}$ is called the $k$th-Betti number, $b_{k}=\dim H_{k}(X;{\mathbb{Z}})$. The Euler characteristic $\chi(X)$ is the alternating sum of the Betti numbers

$$\chi(X)=\sum_{k}{(-1)}^{k}b_{k}.$$

Since the spaces involved are of the same dimension and the target space is a sphere, we have, by the Hopf theory, that these homotopy equivalence classes are classified by degree.

In the case of a limit cycle $\gamma$, we saw that the Gauss map always has degree zero. Additional necessary conditions are obtained by examining the Gauss map in more detail.

Theorem 9 says roughly that, even though the image of the Gauss map of a limt cycle is ‘thin,’ still it must curve sufficiently in the target sphere so as to intersect all possible hyperplanes.

As for the Lyapunov level sets near a limit cycle, we have

(The proof is a basic topolgical facts about tori and is omitted.) Thus, even though the Gauss map of the gradient vector field of Lyapunov functions is of degree zero on any level (as it should be by the Poincaré-Hopf theorem 4), it covers the unit sphere at least twice.

We see, therefore, that members of the same homotopy equivalence class of maps can have widely different Gauss images. The trick, as far as control is concerned, is to find a representative arising from a control section (see [3].)