P110 REGULATES DNA REPAIR PATHWAYS

5.1

§ 5.1.1 Suppose a differentiable atlas is given on a topological manifold S, as well as an extra chart not belonging to it. Take the intersections of the coordinate-neighbourhood of this chart with all the coordinate-neighbour-hoods of the atlas. If in these intersections all the coordinate transformations from the atlas LSC’s to the extra chart are C^∞, the chart is admissible to the atlas. If we add to a differentiable atlas all its admissible charts, we get a complete atlas, or maximal atlas, or C^∞-structure . The important point is that, given a differentiable atlas, its extension obtained in this way is unique.

A topological manifold with a complete differentiable atlas is a differentiable manifold.

One might think that on a given topological manifold only one complete atlas can be defined — in other words, that it can “become” only one differen-tiable manifold. This is wrong: a fixed topological manifold can in principle accept many distinct C^∞-structures, each complete atlas with charts not ad-missible by the other atlases. This had been established for the first time in 1957, when Milnor showed that the sphere S⁷ accepts 28 distinct complete atlases. The intuitive idea of identifying a differentiable manifold with its topological manifold, not to say with its point-set (when we say “a differen-tiable function on the sphere”, “the space-time”, etc), is actually dangerous (although correct for most of the usual cases, as the spheres Sⁿ with n ≤ 6) and, ultimately, false. That is why the mathematicians, who are scrupulous and careful people, denote a differentiable manifold by a pair (S, D), where D specifies which C^∞-structure they are using. Punctiliousness which pays

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well: it has been found recently, to general surprise, that E⁴ has infinite distinct differentiable structures! Another point illustrating the pitfalls of intuition: not every topological manifold admits of a differentiable structure.

In 1960, Kervaire had already found a 10-dimensional topological manifold which accepts no complete differentiable atlas at all. In the eighties, a whole family of “non-smoothable” compact simply-connected 4-dimensional man-ifolds was found. And, on the other hand, it has been found that every non-compact manifold accepts at least one smooth structure.¹

§ 5.1.2 The above definitions concerning C^∞-atlases and manifolds can be extended to C^k-atlases and C^k-manifolds in an obvious way. It is also pos-sible to relax the Hausdorff conditions, in which case, as we have already said, the unicity of the solutions of differential equations holds only locally.

Broadly speaking, one could define differentiable manifolds without imposing the Hausdorff condition, second-countability and the existence of a maximal atlas. These properties are nevertheless necessary to obtain some very pow-erful results, in particular the Whitney theorems concerning the imbedding of a manifold in other manifolds of higher dimension. We shall speak on these theorems later on, after the notion of imbedding has been made precise.

§ 5.1.3 A very important theorem by Whitney says that a complete C^k-atlas contains a C^k+1-sub-atlas for k ≥ 1. Thus, a C¹-structure contains a C^∞ -structure. But there is much more: it really contains an analytic sub-atlas.

The meaning here is the following: in the definition of a differentiable atlas, replace the C^∞-condition by the requirement that the coordinate transforma-tions be analytic. This will define an analytic atlas, and a manifold with an analytic complete atlas is an analytic manifold . The most important exam-ples of such are the Lie groups. Of course not all C^∞ functions are analytic, as the formal series formed with their derivatives as coefficients may diverge.

§ 5.1.4 General references An excellent introduction is Boothby l975. A short reference, full of illuminating comments, is the 5-th chapter of Arnold l973. Nomizu 1956 is a very good introduction to the very special geometrical properties of Lie groups. The existence of many distinct differentiable struc-tures on E⁴ was found in l983. It is an intricate subject, the proof requiring the whole volume of Freed & Uhlenbeck l984. Before proceeding to the main onslaught, Donaldson & Kronheimer 1991 summarize the main results in a nearly readable way. According to recent rumors (as of November 1994), a suggestion of Witten has led people to obtain all the main results in a much simpler way.

1 Quinn 1982.

5.2. SMOOTH FUNCTIONS 135

5.2

§ 5.2.1 In order to avoid constant repetitions when talking about spaces and their dimensions, we shall as a rule use capital letters for manifolds and the corresponding small letters for their dimensions: dim N = n, dim M = m, etc.

§ 5.2.2 A function f : N → M is differentiable (or ∈ C^k, or still smooth) if, for any two charts (U, x) of N and (V, y) of M , the function

y ◦ f ◦ x^<−1> : x(U ) → y(V )

is differentiable (∈ C^k) as a function between euclidean spaces.

Figure 5.1: Coordinate view of functions between manifolds.

§ 5.2.3 Recall that all the analytic notions in euclidean spaces are presup-posed. This function y ◦ f ◦ x^<−1>, taking an open set of Eⁿ into an open set of E^m, is the expression of f in local coordinates. We usually write simply y = f (x), a very concise way of packing together a lot of things. We should keep in mind the complete meaning of this expression (see Figure 5.1): the point of N whose coordinates are x = (x¹, x², . . . , xⁿ) in chart (U, x) is taken by f into the point of M whose coordinates are y = (y¹, y², . . . , y^m) in chart (V, y).

§ 5.2.4 The composition of differentiable functions between euclidean spaces is differentiable. From this, it is not difficult to see that the same is true for functions between differentiable manifolds, because

z ◦ (g ◦ f ) ◦ x^<−1> = z ◦ g ◦ y^<−1>◦ y ◦ f ◦ x^<−1>.

If now a coordinate transformation is made, say (U, x) → (W, w) as in Figure 5.1, the new expression of f in local coordinates is y ◦f ◦w^<−1>. Thus, the function will remain differentiable, as this expression is the composition

y ◦ f ◦ x^<−1>◦ x ◦ w^<−1>

of two differentiable functions: the local definition of differentiability given above is extended in this way to the whole manifold by the complete atlas.

All this is easily extended to the composition of functions involving other manifolds (as g ◦ f in Figure 5.2).

Figure 5.2: A function composition.

§ 5.2.5 Each coordinate xⁱ = uⁱ◦ ψ is a differentiable function xⁱ : U ⊂ N → open set in E¹.

§ 5.2.6 A most important example of differentiable function is a differen-tiable curve on a manifold: it is simply a smooth function from an open set of E¹ into the manifold. A closed differentiable curve is a smooth function from the circle S¹into the manifold.

5.3. DIFFERENTIABLE SUBMANIFOLDS 137

§ 5.2.7 We have seen that two spaces are equivalent from a purely topolog-ical point of view when related by a homeomorphism, a topology-preserving transformation. A similar role is played, for spaces with a differentiable structure, by a diffeomorphism:

A diffeomorphism is a differentiable homeomorphism whose inverse is also smooth.

§ 5.2.8 Two smooth manifolds are diffeomorphic when some diffeomorphism exists between them. In this case, besides being topologically the same, they have equivalent differentiable structures. The famous result by Milnor cited in the previous section can be put in the following terms: on the sphere S⁷one can define 28 distinct smooth structures, building in this way 28 differentiable manifolds. They are all distinct from each other because no diffeomorphism exists between them. The same holds for the infinite differentiable manifolds which can be defined on E⁴.

§ 5.2.9 The equivalence relation defined by diffeomorphisms was the start-ing point of an ambitious program: to find all the equivalence classes of smooth manifolds. For instance, it is possible to show that the only classes of 1-dimensional manifolds are two, represented by E¹ and S¹. The complete classification has also been obtained for two-dimensional manifolds, but not for 3-dimensional ones, although many partial results have been found. The program as a whole was shown not to be realizable by Markov, who found 4-dimensional manifolds whose class could not be told by finite calculations.

5.3

§ 5.3.1 Let N be a differentiable manifold and M a subset of N . Then M will be a (regular) submanifold of N if, for every point p ∈ M , there exists a chart (U, x) of the N atlas, such that p ∈ U , x(p) = 0 ∈ Eⁿ and x(U ∩ M ) = x(U ) ∩ E^m (as in Figure 5.3). In this case M is a differentiable manifold by itself.

§ 5.3.2 This decomposition in coordinate space is a formalization of the intuitive idea of submanifold we get when considering smooth surfaces in E³. We usually take on these surfaces the same coordinates used in E³, adequately restricted. To be more precise, we implicitly use the inclusion i:

Surface → E³ and suppose it to preserve the smooth structure. Let us make this procedure more general.

§ 5.3.3 A differentiable function f : M → N is an imbedding when (i) f (M ) ⊂ N is a submanifold of N ;

(ii) f : M → f (M ) is a diffeomorphism.

Figure 5.3: M as a submanifold of N .

The above f (M ) is a differentiable imbedded submanifold of N . It corre-sponds precisely to our intuitive idea of submanifold, as it preserves globally all the differentiable structure.

§ 5.3.4 A weaker kind of inclusion is the following. A smooth function f : M → N is an immersion if, given any point p ∈ M , it has a neighbour-hood U , with p ∈ U ⊂ M , such that f restricted to U is an imbedding. An immersion is thus a local imbedding and every imbedding is automatically an immersion. The set f (M ), when f is an immersion, is an immersed submani-fold . Immersions are consequently much less stringent than imbeddings. We shall later (§ 6.4.33 below) give the notion of ntegral submanifold.

§ 5.3.5 These things can be put down in another (equivalent) way. Let us go back to the local expression of the function f : M → N (supposing n ≥ m). It is a mapping between the euclidean spaces E^mand Eⁿ, of the type y ◦ f ◦ x^<−1>, to which corresponds a matrix (∂yⁱ/∂x^j). The rank of this matrix is the maximum order of non-vanishing determinants, or the number of linearly independent rows. It is also (by definition) the rank of y ◦ f ◦ x^<−1>and (once more by definition) the rank of f . Then, f is an immersion iff its rank is m at each point of M . It is an imbedding if it is an immersion and else an homeomorphism into f (M ). It can be shown that these definitions are quite equivalent to those given above.

5.3. DIFFERENTIABLE SUBMANIFOLDS 139

§ 5.3.6 The mapping f : E¹ → E² given by f (x) = (cos 2πx, sin 2πx)

is an immersion with f (E¹) = S¹ ⊂ E². It is clearly not one-to-one and so it is not an imbedding. The circle f (E¹) is an immersed submanifold but not an imbedded submanifold.

§ 5.3.7 The mapping f : E¹ → E³ given by the expression f (x) = (cos 2πx, sin 2πx, x)

is an imbedding. The image space f (E¹), a helix (Figure 5.4), is an imbedded submanifold of E³. It is an inclusion of E¹ in E³.

Figure 5.4: A helix is an imbedded submanifold of E³.

§ 5.3.8 We are used to think vaguely of manifolds as spaces imbedded in some Eⁿ. The question naturally arises of the validity of this purely intuitive way of thinking, so convenient for practical purposes. It was shown by Whit-ney that an n-dimensional differentiable manifold can always be immersed in E²ⁿand imbedded in E²ⁿ⁺¹. The conditions of second-countability, complete-ness of the atlas and Hausdorff character are necessary to the demonstration.

These results are used in connecting the modern treatment with the so-called

“classical” approach to geometry (see Mathematical Topic 10). Notice that eventually a particular manifold N may be imbeddable in some euclidean manifold of lesser dimension. There is, however, no general result up to now

fixing the minimum dimension of the imbedding euclidean space of a dif-ferentiable manifold. It is a theorem that 2-dimensional orientable surfaces are imbeddable in E³: spheres, hyperboloids, toruses are perfect imbedded submanifolds of our ambient space. On the other hand, it can be shown that non-orientable surfaces without boundary are not, which accounts for our in-ability to visualize a Klein bottle. Non-orientable surfaces are, nevertheless, imbeddable in E⁴.

Part II

DIFFERENTIABLE