(p.574) Appendix A Smooth maps
(p.574) Appendix A Smooth maps
This appendix clarifies the notion of smooth maps on manifolds with boundary used throughout the book. Section A.1 discusses smooth functions on manifolds with corners. Section A.2 discusses how to extend smooth (homotopies of) embeddings of a closed ball into a manifold M to smooth (homotopies of) embeddings of into M. This is used in Chapter 3, where the extension results are carried over to the symplectic setting. Section A.3 gives an explicit formula for a smooth function that is used in Section 7.1 for the construction of symplectic forms on blow-ups.
A.1 Smooth functions on manifolds with corners
the right upper quadrant in and by
the set of nonnegative integers.
Let be an open set in the relative topology. A function is called smooth if its restriction to is smooth and all its partial derivatives extend to continuous functions on .
In Milnor  and Guillemin–Pollack , a function , defined on an arbitrary subset , is called smooth if, for every , there is an open neighbourhood of x and a smooth function such that the restriction of F to agrees with f. The next theorem shows that, for , their definition of smooth agrees with that in Definition A.1.1.
Let be a compactly supported smooth function and let be an open neighbourhood of Qm. Then there is a compactly supported smooth function such that
The proof is based on the following lemma.
(p.575) Lemma A.1.3
Let and let be sequences such that(A.1.1)
for . Let be a smooth cutoff function such that(A.1.2)
and define by(A.1.3)
Then F is smooth, vanishes for , and satisfies for .
The right hand side in (A.1.3) vanishes for . It converges absolutely and uniformly because for all and hence, for ,
This shows that F is continuous on and . Differentiate the right hand side of equation (A.1.3) term by term to obtain(A.1.4)
Use the inequalities and for all and to show that the right hand side in equation (A.1.4) converges uniformly and absolutely on . Hence F is continuously differentiable and .
To examine the higher derivatives of F, define by
for and . Then, for every there is a constant such that
for all and all . Moreover, and hence(A.1.5)
where is a finite sum, satisfying , and is a polynomial in the functions for . Thus the functions satisfy a uniform bound, independent of k, and hence the series converges absolutely and uniformly. This shows that F is smooth and for all . This proves Lemma A.1.3.
(p.576) Proof of Theorem A.1.2:
We prove the following by induction on n.
Claim. Let be a compactly supported smooth function, let be an open neighbourhood of Qm, and let . Then there exists a compactly supported smooth function such that
for and define the function by and
for and . By Lemma A.1.3 this function is smooth. It is supported in U whenever is chosen sufficiently small. Let and suppose has been constructed. Choose constants
and define by and
We now briefly discuss the notion of a manifold with corners. In the following we will mostly be interested in smooth functions whose domain is either a closed ball B, i.e. a manifold with boundary, or is the product , which is a manifold with corners of codimension 2. For an in-depth discussion of manifolds with corners and their tangent bundles, see Joyce  and the references cited therein.
A manifold with corners (of dimension m) is a second countable Hausdorff topological space M, equipped with an open cover and a collection of homeomorphisms (called coordinate charts) onto open sets (in the relative topology induced by ) such that the transition maps are smooth in the sense of Definition A.1.1. The collection of coordinate charts is called an atlas.
(p.577) Here is the corresponding notion of smooth map.
Let M be an m-manifold with corners and with atlas
A function is called smooth if the composition
is smooth in the sense of Definition A.1.1 for every α. A function is called smooth if its coordinate functions are smooth for . A function with values in a manifold N is called smooth if its composition with an embedding is smooth.
For these to be sensible definitions, we need to know that the corner structure of M, i.e. the stratification of its boundary into pieces of different dimension, is preserved by the coordinate changes. This is not true in the topological category; for example, Q2 is homeomorphic to the half space . However, the next exercise shows that it is true in the smooth category.
Let M be an m-manifold with corners with atlas Let , choose indices such that , and define the coordinate vectors by
Prove that .
Exercise A.1.6 shows that there is a well-defined map
which assigns to each point the number of components of the vector that vanish for some, and hence every, index α such that . For define the set
The boundary of M is the set
and the interior of M is the set
Prove that is an -manifold with corners.
(p.578) A.2 Extension
We now show that ball embeddings have smooth extensions. Let denote the closed ball of radius .
Theorem A.2.1 (Smooth embedding extension)
Let M be a smooth manifold without boundary and fix a constant .
(i) Every embedding extends to an embedding whose image has compact closure.
(ii) Let be a smooth map such that is an embedding for every t. Then there is a smooth extension of ψ, whose image has compact closure, such that is an embedding for every t.
(iii) Let be as in (ii) and let be embeddings, whose images have compact closure, such that for . Then the extension Ψ in (ii) can be chosen such that for .
The proof is based on the following lemma.
Lemma A.2.2 (Smooth extension)
Fix a constant .
(i) Every smooth function extends to a compactly supported smooth function .
(ii) Every smooth function extends to a compactly supported smooth function .
(iii) Let be a smooth function and let be compactly supported smooth functions such that for . Then the extension F in (ii) can be chosen such that for .
Assertion (i) is trivial for (take any smooth cutoff function). To prove it for choose a smooth cutoff function that satisfies (A.1.2). For , define by
for . Choose a sequence of real numbers ck such that for all and all . Define by and
for and . Then F is smooth by Lemma A.1.3 and this proves (i). The same formula proves (ii) because of the smooth dependence of F on f. To prove (iii), assume that is any compactly supported smooth extension of f and let be compactly supported smooth functions (p.579) such that for . Let be a smooth cutoff function that satisfies (A.1.2). Then the function
satisfies (iii). This proves Lemma A.2.2.
Proof of Theorem A.2.1:
Choose any embedding of M (or of an open subset of M that contains the image of f and has compact closure) into some Euclidean space . Let be a tubular neighbourhood of the image of the embedding and denote by the projection. Use part (i) of Lemma A.2.2 to construct any smooth extension of ψ to a function with values in and compose it with π to obtain a smooth extension for some . Shrinking , if necessary, we may assume without loss of generality that ϕ is an embedding. Now choose a diffeomorphism
such that for . Then the map , defined by
for and , is an embedding and agrees with ψ on . The same argument, using part (ii) of Lemma A.2.2, proves assertion (ii).
We prove (iii). The same argument as in the proof of (i) and (ii), using part (iii) of Lemma A.2.2, gives rise to a smooth map , defined on an open neighbourhood of , such that , for , the image of ϕ has compact closure, and the map
is a embedding for every t. Now choose a smooth function such that for all t, and tends to infinity for and . Then choose a smooth function
such that is a diffeomorphism satisfying for all t, and . Then the function , defined by
for and , satisfies the requirements of (iii). This proves Theorem A.2.1.
(p.580) A.3 Construction of a smooth function
The following construction is used in Section 7.1.
There exists a smooth function
for all and all
Consider first the case and fix a constant . Then it is a standard problem in analysis to construct a function that satisfies (A.3.1). Moreover, the set of all such functions is a convex open set in the set of all smooth functions that satisfy the constraints for and for . By varying and using this convexity, one can prove the existence of a smooth function that satisfies (A.3.1) for , and then, by varying similarly, for all . Below we construct the functions explicitly.
Complete the details of the above proof.
Proof of Lemma A.3.1
The explicit construction of requires two preparatory steps.
Step 1. There is a smooth function such that(A.3.2)
Let and be smooth functions such that(A.3.3)
The map is well-defined and smooth and satisfies the first two equations in (A.3.2). Moreover,
and hence also satisfies the integral condition in (A.3.2). This proves Step 1.
(p.581) Step 2. There exists a smooth function
for all and all .
Let be as in Step 1 and define(A.3.6)
for and . If then the integral vanishes and so . Moreover, since for every , we have
This shows that for and for . Moreover, it follows from equation (A.3.2) in Step 1 that
Hence for . This proves Step 2.
Let be as in Step 2. Then the function