Cobordism Categories

Transcription

1 M Naeem Ahmad Kansas State University KSU Graduate Student Topology Seminar M Naeem Ahmad (KSU) September 10, / 31 We begin with few basic definitions and historical details to put the notion of cobordism theory into mathematical perspective. M Naeem Ahmad (KSU) September 10, / 31

2 Introduction Introduction The subject of cobordism theory forms a significant part of an important branch of mathematics called differential topology. The study of the category of differentiable manifolds and differentiable maps, primarily in relation to the category of topological spaces and continuous map, is called differential topology. Loosely speaking, differential topology is the study of manifolds by topologists using any methods they can find, except imposing structures like Riemannain metrics or connections which tell it apart from the differential geometry. M Naeem Ahmad (KSU) September 10, / 31 Introduction Introduction The following result relates differential topology with group theory. Theorem Let G be a finitely presented group. Then there exists a 4-dimensional manifold M(G) such that π 1 (M(G)) = G. Moreover, two finitely presented groups G and H are isomorphic if and only if the manifolds M(G) and M(H) are homeomorphic. Remark The word problem of figuring out if two finitely presented groups are isomorphic is not solvable. M Naeem Ahmad (KSU) September 10, / 31

3 Classification Problem Classification Problem The primary problem in any subject is the classification of its objects up to isomorphic classes and finding computable invariants to tell the classes apart. Remark The discussion at wrap up of the preceding section tells that the classification problem is not solvable in case of differentiable manifolds. The trouble with the classification problem of differentiable manifolds leads to look for a relation weaker than the isomorphism. This gives rise to the notion of cobordism theory. M Naeem Ahmad (KSU) September 10, / 31 Cobordism Relation Cobordism Relation We begin with a crude but seemingly natural definition of cobordism relation. After noting shortcoming in the definition we will fix it to get a nontrivial and useful theory out of it. Two manifolds without boundary are said to be cobordant if their disjoint union is (diffeomorphic to) the boundary of a manifold. Observation Every manifold M without boundary is the boundary of the manifold M [0, ). This makes any two manifolds without boundary as cobordant to each other in view of the above definition. It is possible to get around this triviality problem with the help of a slight modification in the above definition of cobordism relation: M Naeem Ahmad (KSU) September 10, / 31

4 Cobordism Relation Cobordism Relation Remark It is customary to restrict attention to the class of compact manifolds in the above definition to get a nontrivial theory. In what follows by a manifold we will always mean a smooth compact manifold, unless it is mentioned otherwise. M Naeem Ahmad (KSU) September 10, / 31 Historical Details Historical Details We put together few historical details about few mathematicians who played important roles in the fundamental development and advancement of the cobordism theory. Poincaré In 1885, a French mathematician Henri Poincaré was the first one to implicitly describe the cobordism relation. Actually his idea of homology is basically the same as that of cobordism in use today. M Naeem Ahmad (KSU) September 10, / 31

5 Historical Details Historical Details Pontrjagin In 1947, a Russian mathematician Lev Pontrjagin was the first one to find the classification invariants, namely characteristic numbers, to distinguish the cobordism classes by showing that The characteristic numbers of a closed manifold vanish if it is a boundary. He also tried to study the stable homotopy groups of spheres as the cobordism groups of framed manifolds, and thereby boiled down a homotopoy problem to a cobordism problem. Unfortunately, the lack of knowledge on manifolds prevented it from being of much use in solving the homotopy problem. M Naeem Ahmad (KSU) September 10, / 31 Historical Details Historical Details Rohlin In 1951, a student of Ponrjagin named Vladimir Rohlin solved the cobordism classification problem in dimension 3 by using geometric techniques. The cobordism classification problem in dimensions lower than 3 is considerably elementary as the manifolds in these dimensions have themselves been classified. Thom In 1954, a French mathematician René Thom gave a major breakthrough by reversing the gears in Pontrjagin s key idea. He showed that the cobordism problem is equivalent to a homotopy problem. Luckily, the homotopy problem corresponding to most of the interesting manifold classification questions is solvable. M Naeem Ahmad (KSU) September 10, / 31

6 Now we formally begin with definition of the cobordism category followed by an example and its fundamental properties. M Naeem Ahmad (KSU) September 10, / 31 A cobordism category is a triple (C,, i) such that (i) C is a category having finite sums and an initial object; (ii) : C C is an additive functor and ( (X )) is an initial object for each X ob(c); (iii) i : I is a natural transformation of additive functors, where I is the identity functor; (iv) C has a small subcategory C 0 such that each element of ob(c) is isomorphic to an element of ob(c 0 ). We give an example of the cobordism category. M Naeem Ahmad (KSU) September 10, / 31

7 Example Let D be the category having objects as the compact differentiable manifolds and maps as the differentiable maps between such manifolds which take boundary into boundary. We define sum as the disjoint union of manifolds and as the boundary map. The empty manifold is an initial object. For each M ob(d), define i(m) Map( (M), I (M)) = Map( (M), M) as the inclusion map i(m) : (M) M. This gives a natural transformation i : I. By Whiteny embedding theorem, D has a small subcategory D 0 (submanifolds of R ) such that each element of D is isomorphic to an element of D 0. Thus (D,, i) is a cobordism category. M Naeem Ahmad (KSU) September 10, / 31 Let (C,, i) be a cobordism category and X, Y ob(c). We say that X and Y are cobordant if there exist U.V ob(c) such that X + (U) = Y + (V ). We will write it as X Y. Proposition 1 (i) is an equivalence relation on (the set of isomorphism classes of ) ob(c). (ii) X Y implies (X ) = (Y ). (iii) (X ) for each X ob(c), where is an initial object. (iv) X 1 Y 1 and X 2 Y 2 implies X 1 + Y 2 Y 1 + Y 2. M Naeem Ahmad (KSU) September 10, / 31

8 Proof. (i) X + ( ) = X + ( ) implies the reflexivity. The symmetry of follows by the symmetry of =. If X + (U 1 ) = Y + (V 1 ) and Y + (U 2 ) = Z + (V 2 ). Then X + (U 1 + U 2 ) = X + (U 1 ) + (U 2 ) = Y + (V 1 ) + (U 2 ) = Z + (V 2 ) + (V 1 ) = Z + (V 1 + V 2 ). This gives the transitivity. (ii) X + (U) = Y + (V ) implies (X + (U)) = (Y + (V )) and hence (X ) + ( (U)) = (Y ) + ( (V )) and so (X ) + = (Y ) + and thus (X ) = (Y ). (iii) (X ) + ( (X )) = (X ) + = + (X ). (iv) X 1 + (U 1 ) = Y 1 + (V 1 ) and X 2 + (U 2 ) = Y 2 + (V 2 ) implies X 1 + X 2 + (U 1 + U 2 ) = X 1 + X 2 + (U 1 ) + (U 2 ) = Y 1 + Y 2 + (V 1 ) + (V 2 ) = Y 1 + Y 2 + (V 1 + V 2 ). M Naeem Ahmad (KSU) September 10, / 31 An element X of ob(c) is said to be closed if (X ) is (isomorophic to) an initial object. An element X of ob(c) is said to bound if X, where is an initial object. Proposition 2 (i) Y X and X closed implies Y is closed. (ii) X, Y closed implies X + Y is closed. (iii) X bounds implies X is closed. (iv) X, Y bound implies X + Y bounds. (v) Y X and X bounds implies Y bounds. M Naeem Ahmad (KSU) September 10, / 31

9 Proof. (i) Invoke Proposition 1 (ii). (ii) (X + Y ) = (X ) + (Y ) = + =. (iii) Proposition 1 (ii) gives (X ) = ( ) =. (iv) Proposition 1 (iv) gives X + Y + =. (v) Invoke the transitivity of. M Naeem Ahmad (KSU) September 10, / 31 Proposition The collection of equivalence classes of closed objects of C (under ) is a set which has an operation induced by the sum in C. Moreover, this operation is associative, commutative, and has a unit (the class of any object which bounds). Proof. Since two isomorphic objects of C are also equivalent (under ), therefore, the existence of C 0 implies that the collection of equivalence classes of objects (and so of closed objects ) of C is a set. By Proposition 1 (iv) the sum of equivalence classes of object C is well defined, and by Proposition 2 (ii) the sum of two closed objects of C is closed. It follows that the sum in C induces an operation on the of equivalence classes of closed objects of C. M Naeem Ahmad (KSU) September 10, / 31

10 The cobordism semigroup of the cobordism category (C,, i), denoted by Ω(C,, i), is the set of equivalence classes of closed objects in C with the operation induced by the sum in C. Remarks (i) The cobordism semigroup (Ω,, i) may also be viewed as the semigroup of isomorphism classes of closed objects modulo the subsemigroup of isomorphism classes of objects which bound. Indeed, let denote the latter equivalence, and X Y implies X + U = Y + V. By Proposition 1 (iii), U and V bound, which implies X X + U and Y Y + V, hence X X + U = Y + V Y, and so X Y. Conversely, X Y implies X = Y + Z, where Z, that is Z + U = + V. So X + U = Y + Z + U = Y + + V = Y + V. Thus X Y. M Naeem Ahmad (KSU) September 10, / 31 (ii) Ω(D,, i) can be identified with Thom s cobordism group N of unoriented cobordism classes of closed manifolds. In fact, if X is equivalent to Y in N, then there exists V such that V = X Y which implies X V = X X Y. But X X = (X I ), hence X V = Y (X I ), and so X Y. Conversely, X U = Y V implies X Y = T is a geometric argument which involves looking at components and gluing together manifolds with boundary by means of tubular neighborhoods of boundary components. M Naeem Ahmad (KSU) September 10, / 31

11 Constructions Constructions Now we discuss a couple of constructions in order to describe the ways to tailor new cobordism categories out of a given one together with another category. Construction 1 Given a cobordism category (C,, i), a category X having finite sums and an initial object, and an additive functor F : C X. Let X ob(x), form a category C/X having objects as pairs (C, f ) with C ob(c) and f Map(F (C), X ), and maps given by letting Map((C, f ), (D, g)) be the set of maps ψ Map(C, D) such that the diagram M Naeem Ahmad (KSU) September 10, / 31 Constructions Constructions F (ψ) F (C) F (D) f g X commutes. If ob(c) is an initial object and φ : F ( ) X the unique map, then (, φ) is an initial object of the category C/X. For (C, f ), (D, g) ob((c, f ), (D, g)), we have F (C + D) = F (C) + F (D) in X and since sum is an initial object in its category, we get a well defined map f + g : F (C + D) X, and (C, f ) + (D, g) = (C + D, f + g) in C/X. M Naeem Ahmad (KSU) September 10, / 31

12 Constructions Constructions Define (C, f ) := ( C, f F (i(c)), and (ψ) := ψ i(c) for ψ : C D to get the functor : C/X C/X, and ĩ(c, f ) := i(c) : C C to get the natural transformation ĩ : I. We have that (C,, ĩ) is a cobordism category. Remark If we take (C,, i) as the category (D,, i), X the category of topological spaces and continuous maps, and F : D X the forgetful functor. Then Ω(D/X,, ĩ) is the unoriented bordism group N (X ). M Naeem Ahmad (KSU) September 10, / 31 Constructions Constructions Construction 2 Let (C,, i) be a cobordism category and E a small category. Define the category Fun(E, C) having objects as functors Ψ : E C and maps the natural transformations. If ob(c) is an initial object, then the constant functor Φ : E C given by A is an initial object of Fun(E, C). For F, G ob(fun(e, C)), let H : E C by letting H(A) be a sum of F (A) and G(A) and let (j F ) A := j F (A) : F (A) H(A) and (j G ) A := j G(A) : G(A) H(A) be the maps exhibiting H(A) as the sum. We have that j F and j G exhibit H as a sum of F and G. Define (F ) := F : λ (F (λ)), and ĩ(f ) Map( (F ), F ) = Map( F, F ) by ĩ(f )(A) = i(f (A)) for all A E. We have that (Fun(E, C),, ĩ) is a cobordism category. M Naeem Ahmad (KSU) September 10, / 31

13 Constructions Constructions Remark Let (C,, i) be the category (D,, i), and E the category of one object A with Map(A, A) a finite group G. A functor F : E D is given by a choice of manifold X = F (A) and homomorphism G Map(X, X ). The finiteness of G implies that the induced map G X X is a differentiable action of the group G on the manifold X. We have that Ω((Fun(E, D),, ĩ)) is the unoriented cobordism group of (unrestricted) G-actions. M Naeem Ahmad (KSU) September 10, / 31 Appendix Appendix For the sake of completeness, we include here few categorical definitions which were teletyped above to keep the flow of main ideas smooth. M Naeem Ahmad (KSU) September 10, / 31

14 Appendix Appendix A category C consists of (i) a class of objects, denoted by ob(c); (ii) a set of maps f : X Y for all X, Y ob(c), denoted by Map(X, Y ); (iii) the composite gf Map(X, Z) for all f Map(X, Y ), g Map(Y, Z) and for all X, Y, Z ob(c) satisfying the axioms: Associativity. h(gf ) = (hg)f for all f Map(X, Y ), g Map(Y, Z), and h(z, W ); Identity. For each Y ob(c) there exists 1 Y Map(Y, Y ) such that 1 Y f = f and g1 Y = g for all f Map(X, Y ) and g Map(Y, Z). M Naeem Ahmad (KSU) September 10, / 31 Appendix Appendix A category is said to be small if the class of its objects is a set. An object of a category C is said to be an initial object if Map(X, Y ) is singleton for every Y ob(c). Let {Y α } α Λ be an indexed collection of objects in a category C. Let S{Y α } denote a category such that an object of it is an indexed collection of maps {f α } α Λ of C having the same range and a map of it with domain {f α : Y α Z} and range {g α : Y α W } is a map h : Z W of C for which hf α = g α holds for all α Λ. An initial object of S{Y α }, if it exists, is called a sum of the collection {Y α }. M Naeem Ahmad (KSU) September 10, / 31

15 Appendix Appendix A functor (covariant) from a category C to D consists of (i) an object function which assigns to every object X of C and object F (X ) of D; (ii) a map function which assigns to every map f : X Y of C a map F (f ) : F (X ) F (Y ) of D satisfying the axioms: (a) F (1 X ) = 1 F (X ) ; (b) F (gf ) = F (g)f (f ). M Naeem Ahmad (KSU) September 10, / 31 Appendix Appendix Let F and G be two functors from a category C to a category D. A natural transformation ψ : F G is a function from the objects of C to maps of D such that the diagram F (X ) F (f ) F (Y ) ψ(x ) ψ(y ) G(X ) G(f ) G(Y ) is commutative, for every map f : X Y of C. M Naeem Ahmad (KSU) September 10, / 31

16 The End The End Thank you!!! M Naeem Ahmad (KSU) September 10, / 31