Lecture Fish 3 Joel W. Fish July 4, 2015 Contents 1 LECTURE 2 1.1 Recap:................................. 2 1.2 M-polyfolds with boundary..................... 2 1.3 An Implicit Function Theorem................... 5 1.4 Strong bundles............................ 6 1
1 LECTURE 1.1 Recap: 1. Two days ago sc-calculus (a) Chain rule holds (b) Action via reparametrization is sc-smooth 2. Yesterday charts near broken/nodal maps (a) sc-smooth retracts (b) local models for M-polyfolds 3. Today (a) Establish M-polyfolds with boundary and corners (b) Establish an implicit function theorm 1.2 M-polyfolds with boundary Idea: Roughly, our moduli spaces will arise as subsets of a space locally modeled on sc-retracts. Furthermore, most such moduli spaces with have boundary and/or corners, and we will want that boundary/corner structure induced from a boundary/corner structure on the sc-retract itself, which should in turn be induced from a boundary/corner structure in an sc-banach space. As such, we will need to spend some time discussing such boundary/corner structures in sc- Banach spaces (called partial quadrants), a measure of which boundary strata in which a point is contained (called the degeneracy index), and retractions which are well-behaved with respect to these structures (called tame sc-smooth retractions). Definition: A linear sc-isomorphism is a linear sc 0 -map T : E F between sc-banach spaces, which is an isomorphism on every level. Note: linearity guarantees such a map is in fact sc. Definition: A partial quadrant in an sc-banach space E is a closed convex subset C E such that there exists an linear sc-isomorphism T : E R k W satisfying T (C) = [0, ) k W. Observation: The sc-calculus extends to partial quadrants. Definition: An sc-smooth retract is the tuple (O, C, E) where E an sc-banach space, C E is a partial quadrant, and O is the image of an sc-smooth retract r : U U where U C is relatively open. 2
Definition: An M-polyfold (possibly with boundary and corners) is the same as our previous definition, except the charts are of the form (V, φ, (O, C, E)), with (O, C, E) an sc-smooth retract as above. Definition: (degeneracy index) Let C E be a partial quadrant in an sc- Banach space, and choose a linear sc-isomorphism T : E R n W for which T (C) = [0, ) k W. If x C then T (x) = (a 1,..., a k, w) [0, ) k W and we define the degeneracy index d C : C N 0 by d C (x) := #{i {1,..., k} : a i = 0}. Note: For a chart (V, φ, (O, C, E)) for an M-polyfold X, and for x V M, we define the degeneracy index at x associated to this chart by d(x, v, φ, (O, C, E)) := d C (φ(x)). Definition: The degeneracy d X (x) at the point x X is the minimum of all numbers d(x, V, φ, (O, C, E) where (V, φ, (O, C, E)) varies over all smooth charts containing the point x. The degeneracy index of the M-polyfold X is the map d X N 0. Proposition: If X and Y are M-polyfolds and if f : (U, x) (V, y) is a local sc-diffeomorphism around the points x X and f(x) = y Y, then d X (x) = d Y (f(x)). Conclusion: The sc-calculus on retracts (with boundary and corners) detects boundary/corner strata. In other words, M-polyfolds have well-defined boundary/corner structure. Concern: A finite dimensional manifold with boundary and corners should have the property that its boundary is again a manifold with boundary and corners. Similarly, the zero set of a generic smooth section of a bundle over a manifold should be a submanifold with boundary and corner structure induced 3
from the ambient space. In order to gurantee similar results for M-polyfolds, we will need a more restrictive notion of retracts specifically a tame sc-smooth retract. Definition Consider a partial quadrant C E of a sc-banach space. 1. We define the set of boundary points of C to be boundary points of C = {e C : d C (e) = 1} = A 1 A k with each A i a connected component of the set of boundary points of C. 2. Each A i is contained in a unique smallest linear subspace denoted by f i ; we call such f i an extended face. 3. Let F := {f 1,..., f k } denote the set of all extended faces of C. 4. Given f F we let H f denote the closed half sub-space of E which contains C f. 4
Notation: For an sc-retract (O, C, E), and a smooth point x O, we write F(x) := {f F : x f}. Definition: The minimal linear subspace associated with x C is defined by E x := f. f F(x) Geometrically the minimal linear subspace E x for x C is the set of tangent vectors at a point which are tangent to the stratum {y E : d C (x) = d C (y)}. Definition: The reduced tangent space T R x O is defined as the following subset of the tangent space at x, T R x O = T x O E x. Definition Let r : U U be an sc-smooth retraction defined on a relatively open subset U of a partial quadrant C in the sc-banach space E. The sc-smooth retraction r is called a tame sc-retraction, if the following two conditions are satisfied. 1. d C (r(x)) = d C (x) for all x U. 2. At every smooth point x O := r(u), there exists an sc-subspace A E, such that E = T x O A and A E x. Definition A sc-smooth retract (O, C, E) is called a tame sc-smooth retract if O is the image of an sc-smooth tame retraction. Definition A tame M-polyfold X is an M-polyfold which possesses an equivalent sc-smooth atlas whose charts are all modeled on tame sc-smooth retracts. 1.3 An Implicit Function Theorem Theorem. Let P : Y X be a tame strong bundle, and f an sc-fredholm section having the property that at every point x in the solution set {y X : f(y) = 0}, the linearization f (x) : T x X Y x is surjective and the kernel ker(f (x)) is in good position to the partial cone C x X T x X. Then S is a sub-m-polyfold of X and the induced M-polyfold structure on S is equivalent to the structure of a smooth manifold with boundary and corners. Terms to define: 1. tame strong bundle postpone to the next section 2. sc-fredholm section tomorrow s lecture 5
3. partial cone the partial cone is defined as C x = f F(x) 4. good position Let C E be a partial quadrant in the sc-banach space E, and let N E be a finite dimensional sc-subspace of E. The subspace N is in good position to the partial quadrant C if the interior of N C in N is non-empty, and if N possesses a sc-compliment P, so that E = N P, which has the following property. There exists ɛ > 0 such that for pairs (n, p) N P satisfying p 0 ɛ n 0 the statements n C and n + p C are equivalent. H f 5. sub-m-polyfold Let X be an M-polyfold and let A be a subset of X. The subset A is called a sub-m-polyfold of X, if every a A possesses an open neighborhood V and a sc-smooth retraction r : V V such that 1.4 Strong bundles Motivation: r(v ) = A V. 1. A Fredholm operator is a linear map L : E F between Banach spaces, with closed range, and finite dimensional kernal and co-kernal. 6
2. Fredholm operators are stable under compact perturbations. In other words, for K : E F a compact linear operator, L + K : E F is Fredholm and Ind(L) = Ind(L + F ). 3. A smooth non-linear map f : E F is Fredholm, provided its linearization at each point is Fredholm. 4. The linearized Cauchy-Riemann operator is a Fredholm map from sections of regularity H k+1 to those of regularity H k. And since for these spaces H k+1 compactly embeds into H k, the natural choice of compact perturbation of the Cauchy-Riemann operator is by adding a lower order term, the linearization of which is a section from H k+1 to H k+1. 5. Since in applications the Cauchy-Riemann operator is a section of a bundle, in the sc-calculus the above observation gives rise to a pair of scstructures on local models for sc-bundles one of the form (H k+1 H k ) k Nk 2 of which the Fredholm operator will be a section, and (H k+1 H k+1 ) k Nk 2 of which our compact perturbations (i.e. lower order terms) will be a section. 6. This pair of sc-structures on the bundle is the essence of the notion of a strong bundle, made precise below. Definition: Let U C E where E is a sc-banach space, C is a partial quadrant, and U is a relatively open subset. If F is another sc-banach space, we then define U F to be the set U F equipped with the double filtration given by (U F ) m,k = U m F k for 0 k m + 1. We will view U F U as a trivial bundle. For i {0, 1}, we define the sc-banach spaces (U F )[i] by their filtrations: ( (U F )[i] ) m := U m F m+i for m 0. Question. What is going on here? Definition. A strong bundle map Φ : U F U F is a map which preserves the double filtration and is of the form Φ(u, h) = (φ(u), Γ(u, h)) where the map Γ is linear in h. In addition we demand that the maps are sc-smooth. Φ : (U F )[i] (U F )[i] 7
Definition. A strong bundle isomorphism is an invertible strong bundle map whose inverse is also a strong bundle map. Definition. A strong bundle retraction is a strong bundle map R : U F U F which satisfies R R = R. Note the map has the form R(u, h) = (r(u), Γ(u, h)) where r : U U is a sc-smooth retraction. We say R is a tame strong bundle retraction provided r is tame. Definition A (tame) local strong bundle retract is denoted by ( K, C F, E F ) where K C F is the image of a (tame) strong bundle retraction R : U F U F ; that is K = R(U F ). Note: The local strong bundle retract (K, C F, E F ) will often be abreviated by p : K O where O = r(u), R(u, h) = (r(u), Γ(u, h)), and p is the projection induced from the canonical projection U F U. Observation: K inherits the double filtration K m,k (m 0 and 0 k m+1), and the filtrations K[i] for i {0, 1} from U F. Furthermore the projection maps p : K[i] O are sc-smooth. Definition. A strong bundle chart for the bundle P : Y X is a tuple (Φ, P 1 (V ), K, U F ) where U C E is an open subset of the partial quadrant C in the sc-banach space E and F is a sc-banach space. Furthermore K = R(U F ) is the image of a strong bundle retraction R : U F U F of the form R(u, h) = (r(u), Γ(r, h)) where r : U U is an sc-smooth retraction onto the retract O = r(u) of (O, C, E) and Γ is linear in h. In addition, V X is an open subset of X, homeomorphic to the retract O by a homeomorphism φ : V O. In addition Φ : P 1 (V ) K is a homeomorphism from P 1 (V ) Y onto the retract K, covering the homeomorphism φ : V O, so the diagram P 1 (V ) V P φ Φ K O p (1) 8
commutes. The map Φ has the property that, in the fibers over x V, the map Φ : P 1 p 1 (φ(x)) is a bounded linear operator between Banach spaces. Definition Two strong bundle charts Φ : P 1 (V ) K and Φ : P 1 (V ) K satisfying V V are compatible, if the transition maps Φ Φ 1 [i] : Φ(P 1 (V V ))[i] Φ (P 1 (V V ))[i] are sc-diffeomorphisms for i = {0, 1}. Definition. Let X be an M-polyfold, let Y be a paracompact Hausdorff space, and let P : Y X be a continuous surjection. We say Y is a strong bundle over X provided it is equipped with an equivalence class of strong bundle atlases. 1. Two atlases are equivalent if their union is an atlas; 2. a strong bundle atlas consists of a collection of compatible charts Definition Given an sc-banach space E and a linear sc-subspace N E (for which N k = N E k ), we say P E is an sc-complement of N provided E k = N k P k for all k N; or more succinctly, E = N P as sc-banach spaces. Definition A sc-fredholm germ (f, x) of a tame strong bundle Y X satisfying f(x) = 0 is in good position if 1. f (x) : T x X Y x is surjective 2. if d X (x) 1 then ker(f (x)) T x X is in good position to the partial cone C x X in the tangent space T x X. Theorem. (Implicit Function Theorem) Let P : Y X be a tame strong bundle, and f an sc-fredholm section having the property that at every point x in the solution set {y X : f(y) = 0}, the linearization f (x) : T x X Y x is surjective and the kernel ker(f (x)) is in good position to the partial cone C x X T x X. Then S is a sub-m-polyfold of X and the induced M-polyfold structure on S is equivalent to the structure of a smooth manifold with boundary and corners. Theorem. Perturbation and Transversality: general position Theorem 5.18 or good position (Thm 5.19); also possibly Thm 5.27 Definition A section of the local strong bundle retract p : K O is a map f : O K satisfying p f = 1 O. The section is called sc-smooth provided f is a section of the bundle p : K[0] O, and called sc + -smooth if it is an sc-smooth section of the bundle p : K[1] O. Such sections have the form f(x) = (x, f(x)) O F, and we call the map f : O F the principal part of the section. 9
References [1] H. Hofer, K. Wysocki, and E. Zehnder, Polyfold and Fredholm Theory I: Basic Theory in M-Polyfolds, arxiv:1407.3185. [2], Sc-Smoothness, Retractions and New Models for Smooth Spaces, Dynamical Systems, vol. 28, October 2010. [3] Dusa McDuff and Dietmar Salamon, J-holomorphic curves and symplectic topology, 2nd ed., American Mathematical Society Colloquium Publications, vol. 52, American Mathematical Society, Providence, RI, 2012. 10