4. Exercises

definition of smooth maps to di erential spaces, we reformulate the definition of smooth maps between manifolds.

If M is an m-dimensional smooth manifold and U is an open subset of Rk then a map f : M −→ U is a smooth map if and only if all components fi : M −→ R are in C∞(M) for 1 ≤ i ≤ k. If we don’t want to use components we can equivalently say that f is smooth if and only if for all ρ C∞(U) we have ρf C∞(M). This is the logic behind the following definition. Let (X, C) be a di erential space and (X , C ) another di erential space. Then we define a morphism f from (X, C) to (X , C ) as a continuous map f : X −→ X such that for all ρ C we have ρf C. We denote the set of morphisms by C(X, X ). The following properties are obvious from the definition:

(1)id : (X, C) −→ (X, C) is a morphism,

(2)if f : (X, C) −→ (X , C ) and g : (X , C ) −→ (X , C ) are morphisms, then gf : (X, C) −→ (X , C ) is a morphism,

(3)all elements of C are morphisms from X to R,

(4)the isomorphisms (as defined above) are the morphisms

f : (X, C) −→ (X , C )

such that there is a morphism g : (X , C ) −→ (X, C) with gf = idX and fg = idX .

We define the di erential of a morphism as follows.

Definition: Let f : (X, C) → (X , C ) be a morphism. Then for each x X the di erential

dfx : TxX → Tf(x)X

is the map which sends a derivation α to α where α assigns to [g]f(x) Cx the value α([gf]x).

4.Exercises

(1)Let U Rn be an open subset. Show that (U, C∞(U)) is equal to (U, C(U)) where the latter is the induced di erential space structure which was described in this chapter.

(2) Give an example of a di erential space (X, C(X)) and a subspace Y X such that the restriction of all functions in C(X) to Y doesn’t give a di erential space structure.

(3)Let (X, C(X)) be a di erential space and Z Y X be two subspaces. We can give Z two We can give Z two di erential structures:

First by inducing the structure from (X, C(X)) and the other one by first inducing the structure from (X, C(X)) to Y and then to Z. Show that both structures agree.

(4)We have associated to each smooth manifold with a maximal atlas a di erential space which we called a smooth manifold and vice versa. Show that these associations are well defined and are inverse to each other.

(5)a) Let X be a topological space such that X = X1 X2, a union of two open sets. Let (X1, C(X1)) and (X2, C(X2)) be two di erential spaces which induce the same di erential structure on U = X1 ∩X2. Give a di erential structure on X which induces the di erential structures on X1 and on X2.

b)Show that if both (X1, C(X1)) and (X2, C(X2)) are smooth manifolds and X is Hausdor then X with this di erential structure is a smooth manifold as well. Do we need to assume that X is Hausdor or it is enough to assume that for both X1 and X2?

(6)Let (M, C(M)) be a smooth manifold and U M an open subset. Prove that (U, C(U)) is a smooth manifold.

(7)Prove or give a counterexample: Let (X, C(X)) be a di erential space such that for every point x X the dimension of the tangent space is equal to n, then it is a smooth manifold of dimension n.

(8)Show that the following di erential spaces give the standard structure of a manifold on the following spaces:

a)Sn with the restriction of all smooth maps f : Rn+1 → R.

b)RPn with all maps f : RPn → R such that their composition with the quotient map π : Sn → RPn is smooth.

c)More generally, let M be a smooth manifold and let G be a finite group. Assume we have a smooth free action of G on M. Give a di erential structure on the quotient space M/G which is a smooth manifold and such that the quotient map is a local di eomorphism.

(9)Let (M, C(M)) be a smooth manifold and N be a closed submanifold of M. Show that the natural structure on N is given by the restrictions of the smooth maps f : M → R.

(10)Consider (Sn, C) where Sn is the n-sphere and C is the set of smooth functions which are locally constant near (1, 0, 0, . . . , 0). Show that (Sn, C) is a di erential space but not a smooth manifold.

(11) Let (X, C(X)) and (Y, C(Y )) be two di erential spaces and f : X → Y a morphism. For a point x X:

a) Show that composition induces a well-defined linear map Cf(x) → Cx between the germs.

b)What can you say about the di erential map if the above map is injective, surjective or an isomorphism?

(12)Show that the vector space of all germs of smooth functions at a point x in Rn is not finite dimensional for n ≥ 1.

(13)Let M, N be two smooth manifolds and f : M → N a map. Show that f is smooth if and only if for every smooth map g : N → R the composition is smooth.

(14)Show that the 2-torus S1 × S1 is homeomorphic to the square I × I with opposite sides identified.

(15)a) Let M1 and M2 be connected n-dimensional manifolds and let

φi : Bn → Mi be two embeddings. Remove φi(12 Bn) and for each x 12 Sin−1 identify in the disjoint union the points φ1(x) and φ2(x). Prove that this is a connected n-dimensional topological manifold.

b)If Mi are smooth manifolds and φi are smooth embeddings show that there is a smooth structure on this manifold which outside Mi \ φi(12 Dn) agrees with the given smooth structures.

One can show that in the smooth case the resulting manifold is unique up to di eomorphism. It is called the connected sum, denoted by M1#M2 and does not depend on the maps φi.

One can also show that every compact orientable surface is diffeomorphic to S2 or a connected sum of tori T 2 = S1 ×S1 and every compact non-orientable surface is di eomorphic to a connected sum of projective planes RP2.