Chapter 4 - Submersions, Immersions, and Embeddings

Question 1

What is the most essential property of the differential?
For which maps do differentials give good local models?
Key theorem?

Answer 1

Just about the only property independent of basis is rank. Want to use this linear approximation to understand the original smooth map

Constant rank maps, in particular smooth submersions, smooth immersions, and smooth embeddings

Engine powering everything is the rank thm

Question 2

Define: rank of F:M –> N at p, constant rank, full rank, smooth submersion, smooth immersion

Answer 2

The RANK OF F AT p is the rank of the linear map dF_p, ie the rank of the Jacobian in any smooth chart. If F has the same rank everywhere, say F has CONSTANT RANK

rank F is bounded above by the min of {dim M, dim N}. If rank F at p is equal to this upper bound at p, say F has FULL RANK at p. If full rank everywhere, then say F has FULL RANK. These are the most important constant rank maps

SMOOTH SUBMERSION if differential is surjective at each point

SMOOTH IMMERSION if differential is injective at each point

Question 3

Prove: If dFp is surjective, then p has a neighborhood U s.t. F|U is a submersion. If dFp is injective, then p has a neighborhood U s.t. F|U is an immersion.

Answer 3

Choose smooth coordinates about p and F(p). Either hypothesis means the Jacobian of F in coordinates has full rank at p. The set of m x n matrices of full rank is an open subset of M(mxn, R), so by continuity, the Jacobian has full rank in some neighborhood of p.

Points of submersion/immersion cannot be isolated.

Question 4

Examples of submersions and immersions?

Answer 4

1. Projection pi:M1 x … x Mk –> Mi is smooth submersion
2. If gamma:J–>M is a smooth curve, gamma is a smooth immersion <=> gamma’(t) != 0 for all t in J.
3. Projection from tangent bundle TM to M is smooth submersion

Question 5

Define: local diffeomorphism

Answer 5

If M and N are smooth manifolds with or without boundary, a map F:M –> N is called a LOCAL DIFFEOMORPHISM if every point p in M has a neighborhood U s.t. F(U) is open in N and F|U : U –> F(U) is a diffeomorphism

Question 6

What is inverse function theorem for smooth manifolds?

Answer 6

NOTE: We only do this for manifold without boundary!

M, N smooth manifolds, F : M –> N smooth map. If dFp is invertible, then there exist connected neighborhoods U of p and V of F(p) s.t. F|U : U –> V is a diffeomorphism.

i.e. we can show F a local diffeomorphism at a point simply by checking that differential is invertible at that point. Linear isomorphism = local diffeomorphism

The proof of this just applies the ordinary inverse function theorem to coordinate representation of F center at p

Question 7

Properties of Local Diffeomorphisms?

Answer 7

1. Compositions
2. Products
3. Homeomorphism & open map
4. Restriction to open submanifold = local diffeo
5. Every diffeo is a local diffeo
6. Every bijective local diffeo is a diffeo
7. Local diffeo <=> Every point has coord rep that is local diffeo
8. F is local diffeo <=> F submersion & immersion
9. If dim M = dim N and F is either immersion or submersion, F is local diffeo

pg 80

Question 8

What is Rank Theorem? Proof?

Answer 8

Most important theorem about constant rank maps

Thm. Suppose M and N are smooth manifolds of dimensions m and n respectively, and F: M --> N is a smooth map with constant rank r. For each p in M, there exist smooth charts (U, phi) centered at p  and (V, psi) centered at F(p) s.t. F(U) < V in which F has a coordinate representation of the form
F hat(x1, ... , xr, xr+1, ... , xm) = (x1, ... , xr, 0, ... , 0). In particular, if F is smooth submersion = ...
smooth immersion = ...

pg 81-82

Question 9

Corollaries of Rank Theorem?

Proofs?

Answer 9

Cor. Let F: M –> N smooth, M connected. TFAE

1. For each p in M there exists smooth charts containing p and F(p) in which the coordinate rep of F is linear
2. F has constant rank

ie constant rank maps are precisely the ones whose local behavior is the same as that of their differentials

Global Rank Thm. F: M –> N smooth with constant rank.

1. If F is surjective, then it is a smooth submersion
2. If F is injective, then it is a smooth immersion
3. If F is bjiective, then it is a diffeomorphism

Question 10

Define: smooth embedding

Examples? Non-examples?

Answer 10

If M and N are smooth manifolds with or without boundary, a smooth embedding of M into N is a smooth immersion F: M –> N that is also a topological embedding ie a homeomorphism onto its image F(M) < N in the subspace topology.

EXAMPLES

1. Including open submanifold into M
2. Injecting factor Mi into product M1 x … x Mk at any point

NON-EXAMPLES

1. f:R –> R^2, f(t) = t^3, 0) is smooth, topological embedding, not smooth embedding because f’(0) = 0
2. Figure 8 curve: B : (-pi, pi) –> R^2 B(t) = (sin 2t, sin t). Injective smooth immersion, not topological embedding (image is compact in subspace topology, domain is not)
3. Dense Curve on Torus: g: R –> T^2 given by g(t) = (e^2piit, e^2piialphat) where alpha is irrational. Smooth injective immersion, not homeomorphism on to image in subspace topology - g(0) is a limit point of g(Z) but Z has no lp in R. g(R) dense in T^2.

Question 11

Conditions for an injective immersion to be an embedding?

Answer 11

Suppose F : M –> N injective smooth immersion. If any of the following holds, then F is a smooth embedding. (All we need is for F to be a topological embedding)

1. F is an open or closed map
2. F is a proper map
3. M is compact
4. M has empty boundary and dim M = dim N

Question 12

What is Local Embedding Thm? Proof?

Answer 12

Thm. F:M–>N is smooth immersion <=> Every point in M has a neighborhood U s.t. F|U is a smooth embedding.

Proof. <= obvious since full rank on neighborhoods.
=> If we ignore minor technical complications caused by boundary points this is simple. Let p in M
1. Rank Thm gives coordinate rep F(x1, … , xm) = (x1, … , xm, 0, … ,0) on some neighborhood U1 of p. So F locally injective.
2. There exists a precompact neighborhood U of p s.t. U closure < U1. Then F|U closure is an injective continuous map with compact domain, so topological embedding by closed map lemma. Restricting to U retains top embedding so F|U is top embedding and smooth immersion => smooth embedding

Question 13

Define: section, local section

Answer 13

If p:M–>N is any continuous map, a SECTION of p is a continuous right inverse for p, ie a continuous map q:N–>M s.t. p o q = Id_N.

A LOCAL SECTION of p is a continuous map q defined on some open subset U < N and satisfying the analogous relation p o q = ID_U.

Question 14

What is Local Section Theorem? Proof?

Answer 14

Thm. p : M –> N is a smooth submersion <=> every point of M is in the image of a smooth local section of p.

Pf. First say p is a smooth submersion. For any x in M, we can choose coordinates centered at x s.t. p(x1, … ,xn) = (x1, … , xm). Define sigma(x1, … , xm) = (x1, … , xm, 0, … , 0).
Next, say each point of M is in the image of a smooth local section. p o sigma = IdU implies dp o dsigma = Id so dp surjective.

pg88-89

Question 15

Discuss the analogy between smooth submersions in smooth manifold theory and quotient maps in topology. Prove important theorems

Answer 15

Prop. If pi:M –> N is a smooth submersion, then. pi is an open map and if pi is surjective, it is a quotient map.

Pf. In coordinates, every point of M has a neighborhood st that pi looks like projection. Projection is open map.
Or do using local section…pg 89

Thm. (Characteristic Property of Surjective Smooth Submersion) Let pi: M –> N be a smooth submersion. For any smooth manifold P with or without boundary, a map F: N –> P is smooth <=> F o pi is smooth.

Pf. Use local section for harder direction pg 90

Thm. (Passing Smoothly to the Quotient) Let pi: M –> N be a surjective smooth submersion. If P is a smooth manifold with or without boundary and F: M –> P is a smooth map that is constant on the fibers of pi, then there exists a unique smooth map F tilda : N –> P s.t F tilda o pi = F.

Thm. (Uniqueness of Smooth Quotients) Suppose that M, N1, N2 are smooth manifolds, and p1:M –> N1, p2:M –> N2 are surjective smooth submersions that are constant on each other’s fibers. Then there exists a unique diffeomorphism F: N1 –> N2 s.t F o p1 = p2.

pg 90-91

Question 16

Define: covering map, smooth covering map, base of the covering, covering manifold, universal covering manifold

Answer 16

A COVERING MAP between topological spaces is a surjective continuous map pi: E –> M between connected, locally path-connected spaces with the property that each point of M has a neighborhood U that is EVENLY COVERED, meaning that each component of pi^-1(U) is mapped homeomorphically onto U by pi.

If E and M are connected smooth manifolds with or without boundary, a map pi: E –> M is called a SMOOTH COVERING MAP if pi is smooth and surjective, and each point in M has a neighborhood U s.t. each component of pi^-1(U) is mapped diffeomorphically onto U by pi.

The space M is called the BASE OF THE COVERING, and E is called a COVERING MANIFOLD OF M. If E is simply connected, it is called the UNIVERSAL COVERING MANIFOLD OF M.

Question 17

Properties of Smooth Coverings? Proofs?

Answer 17

1. Every smooth covering map is a local diffeomorphism, a smooth submersion, an open map, and a quotient map
2. An injective smooth covering map is a diffeomorphism
3. A topological covering map is a smooth covering map <=> it is a local diffeomorphism

pg 91

Question 18

How can the local section theorem be strengthened for smooth covering maps?

Answer 18

If pi: E –> M is a smooth covering map, then pi is a surjective smooth submersion, so the Local Section Thm applies. However, now if p in E, and U is an evenly covered open subset of M containing pi(p) = q, there exists a UNIQUE smooth local section sigma: U –> E such that sigma(q) = p.

pf?

pg 92

Question 19

Can you always find a compatible smooth structure on a topological covering space E of a connected smooth n-manifold M making E into a smooth covering manifold? Proofs?

Answer 19

Yes. If pi: E –> M is a topological covering map, then E is a topological n-manifold, and has a unique smooth structure s.t. pi is a smooth covering map.

Pf. First show E an n-manifold. Use pi local homeomorphism…

Construct smooth structure on E: let p in E and U be an evenly covered subset of M containing pi(p) s.t. (U, phi) is a chart for M (just chose U small enough). Then if U^tilda is component of pi^-1(U) containing p, define phi^tilda = phi o pi|U^tilda. (U^tilda, phi^tilda) is a chart on E containing p.

Finally pi is smooth covering map since coordinate rep in term of pairs of charts (U^tilda, phi^tilda) and (U, phi) is identity map

Question 20

Prove existence of Universal Covering manifold for smooth manifolds

Answer 20

This just comes down to the fact that a topological universal cover exists for any reasonable space. By results of this chapter, any topological covering space of a smooth connected manifold has a unique smooth structure, so in particular the topological universal cover has a unique smooth structure making it into a smooth universal covering manifold.

Question 21

Criterion for determining if map is smooth covering map?

Answer 21

A sufficient criterion: Suppose E and M are nonempty connected smooth manifolds with or without boundary. If pi: E –> M is a proper local diffeomorphism, then pi is a smooth covering map.

pg 94