ISSN 1364-0380 (on line) 1465-3060 (printed) 103 Geometry & Topology G T GG T T T Volume 3 (1999) 103–118 G T G T G T G T Published: 28 May 1999 T G T G T G T G T G G G G T T Embeddings from the point of view of immersion theory : Part II Thomas G Goodwillie Michael Weiss Department of Mathematics, Brown University Box 1917, Providence RI 02912–0001, USA and Department of Mathematics, University of Aberdeen Aberdeen AB24 3UE, UK Email: [email protected] and [email protected] Abstract Let M and N be smooth manifolds. For an open V ⊂ M let emb(V, N) be the space of embeddings from V to N . By the results of Goodwillie [4], [5], [6] and Goodwillie–Klein [7], the cofunctor V 7→ emb(V, N) is analytic if dim(N) − dim(M) ≥ 3. We deduce that its Taylor series converges to it. For details about the Taylor series, see Part I (reference [15]). AMS Classification numbers Primary: 57R40 Secondary: 57R42 Keywords: Embedding, immersion, calculus of functors Proposed: Ralph Cohen Received: 10 May 1998 Seconded: Haynes Miller, Gunnar Carlsson Revised: 5 May 1999 c Geometry & Topology Publications 104 Thomas G Goodwillie and Michael Weiss 0 Introduction This is a continuation of [15]. The ideas in this second part are mostly due to Goodwillie; notes and illustrations mostly by Weiss. For introductions and notation generally speaking, see [14] and [15]. We fix smooth manifolds M m and N n , without boundary unless otherwise stated, and we assume m ≤ n. As in [15] we write O for the poset of open subsets of M . Then V 7→ emb(V, N) is a cofunctor from O to Spaces (= fibrant simplicial sets), and from [14] and [15] we have a fairly thorough understanding of its Taylor approximations, the cofunctors V 7→ Tk emb(V, N). Here we show that the Taylor approximations are good approximations. The main result is Corollary 2.5. Sections 4 and 5 contain illustrations. Following are some conventions. Let R be a set with r elements. A functor X from the poset PR of subsets of R to Spaces is an r–dimensional cube of Spaces [3, section 1]. The cube is k–Cartesian if the canonical map X (∅) −→ holim X (S) {S|∅6=S⊂R} is k–connected. A Cartesian cube is one which is k–Cartesian for all k. Since PR as a category is isomorphic to its own opposite, we can use similar termi- nology for cofunctors from PR to Spaces. Such a cofunctor will also be called a cube. It is k–Cartesian if the canonical map X (R) −→ holim X (S) {S|S⊂R,S6=R} is k–connected, and Cartesian if it is k–Cartesian for all k. n+2 n Let R x := [0, ∞)×[0, ∞)×R . An (n+2)–dimensional smooth manifold triad is a paracompact Hausdorff space Q together with a maximal atlas consisting n+2 of open subsets Vi ⊂ Q and open embeddings φi: Vi → R x satisfying the following conditions: • The union of the Vi is Q. −1 • The changes of charts φi(φj ) are smooth where defined, and take points n+2 in R x with vanishing first coordinate (vanishing second coordinate) to points with vanishing first coordinate (vanishing second coordinate). Let ∂0Q be the set of those x ∈ Q which under some chart φi map to a point in n+2 R x with vanishing first coordinate. Also, let ∂1Q be the set of those x ∈ Q n+2 which under some chart φi map to a point in R x with vanishing second coordinate. Then Q is an (n + 2)–dimensional smooth manifold with corners; its boundary is ∂0Q ∪ ∂1Q and its corner set is ∂0Q ∩ ∂1Q. Geometry & Topology, Volume 3 (1999) Embeddings from immersion theory : II 105 The handle index of the manifold triad Q is the smallest integer a such that Q can be built from a collar on ∂0Q by attaching handles of index ≤ a. (It may happen that Q is a collar on ∂0Q. When that is the case we say that the handle index is −∞.) Example Suppose that P is smooth, with boundary, and let f: P → R be smooth. If 0 is a regular value both for f and for f|∂P , then the inverse image of [0, ∞) under f is a manifold triad Q, with ∂0Q = ∂P ∩ Q. (If ∂P = ∅, then of course ∂0Q = ∅.) Any Q ⊂ P which can be obtained in this way will be called a codimension zero subobject of P . 1 Excision Estimates Let Y be a smooth n–manifold with boundary. Let Q0,Q1,...,Qr be smooth compact n–manifold triads with handle index qi where n − qi ≥ 3. Suppose that smooth embeddings ei: ∂0Qi → ∂Y have been selected for 0 ≤ i ≤ r, and that their images are pairwise disjoint. For S ⊂ [r] = {0,...,r} let QS be the disjoint union of the Qi with i ∈ S . By emb(QS,Y ) we mean the Space of smooth embeddings f: QS → Y which satisfy f | ∂0Qi = ei and −1 f (∂Y )= ∂0QS . 1.1 Theorem ([4], [5], [6], [7]) The (r + 1)–cube taking a subset S of [r] to r emb(QS,Y ) is (3 − n +Σi=0(n−qi −2))–Cartesian, provided r ≥ 1. Comment Assuming that emb(Q[r],Y ) is nonempty, fix a base point f in emb(Q[r],Y ). Let the image of f in emb(QS,Y ) serve as base point for emb(QS,Y ). Goodwillie shows in [6] that the cube S 7→ Ω emb(QS,Y ) is r (2 − n +Σi=0(n−qi−2))–Cartesian, provided r ≥ 1. The delooped statement, Theorem 1.1 just above, has been proved by Goodwillie and Klein and will appear in [7]. We will need a slight generalization of 1.1 where the Qi are allowed to have dimension m ≤ n. For this and other purposes we need a lemma. 1.2 Lemma Let u: X →Y be a map of (r + 1)–cubes. That is, X and Y are functors from the poset of subsets of [r] to Spaces, and u is a natural transformation. Suppose that Y is k–Cartesian and, for every y ∈ Y(∅), the (r + 1)–cube defined by S 7→ hofiber[X (S) −→Yu (S)] is k–Cartesian. (The homotopy fiber is taken over the image of y in Y(S).) Then X is k–Cartesian. Geometry & Topology, Volume 3 (1999) 106 Thomas G Goodwillie and Michael Weiss Proof of 1.2 Combine [3, 1.18] with [3, 1.6]. Now let Y and Qi be as in 1.1, assuming however dim(Qi) = m ≤ n. As before, embeddings ei: ∂0Qi → ∂Y are specified and n − qi ≥ 3, where qi is the handle index of Qi . We want to show that the conclusion of 1.1 still holds. Without loss of generality, Y is a smooth submanifold of some Rt . Then we can define maps emb(QS,Y ) −→ map(QS, Gn−m) ; f 7→ νf where Gn−m is the direct limit of the Grassmannians of (n − m)–dimensional u linear subspaces of R , for u ≥ 0, and where νf takes x ∈ QS to the intersection of the tangent space of Y at f(x) with the normal space of f(QS) at f(x). In other words, νf classifies the normal bundle of the embedding f . We have therefore a map of (r + 1)–cubes: (•) {S 7→ emb(QS,Y ) } −→ {S 7→ map(QS, Gn−m) } . Since the codomain cube in (•) is Cartesian, the contravariant analog of 1.2 tells us that the domain cube is k–Cartesian provided that, for every ξ in map(Q[r], Gn−m), the cube of homotopy fibers (••) S 7→ hofiberξ [ emb(QS,Y )} −→ map(QS, Gn−m) ] is k–Cartesian. Now it is easy to construct a homotopy equivalence from the ′ ′ homotopy fiber of emb(QS,Y ) → map(QS, Gn−m) to emb(QS,Y ) where QS is the total space of the disk bundle on QS determined by ξ. This construction ′ is natural in S , so (••) is k–Cartesian if the cube S 7→ emb(QS,Y ) is k– r Cartesian, which by 1.1 is the case if k = (3 − n +Σi=0(n−qi−2)). Therefore: 1.3 Observation Theorem 1.1 generalizes to the situation where the Qi have dimension m ≤ n. The corollary below is a technical statement about the cofunctor on O given by V 7→ emb(V, N). Suppose that P is a smooth compact codimension zero subobject of M , and that Q0,...,Qr are pairwise disjoint compact codimension zero subobjects of M r int(P ). For S ⊂ {0,...,r} = [r] let VS be the interior of P ∪ QS where QS is the union of the Qi for i ∈ S . We write V instead of V∅ . Suppose that Qi has handle index qi ≤ n − 3. 1.4 Corollary Assume r ≥ 1. The (r + 1)–cube taking a subset S of [r] to r emb(VS, N) is (3 − n +Σi=0(n−qi−2))–Cartesian. Geometry & Topology, Volume 3 (1999) Embeddings from immersion theory : II 107 Proof Let V¯S be the closure of VS in M . Let emb(V¯S, N) be the inverse limit of the Spaces (simplicial sets) emb(U, N) where U ranges over the neigh- borhoods of VS in M . The restriction from emb(V¯S, N) to emb(VS, N) is a homotopy equivalence. Using this fact and 1.2, we see that it is enough to show that for every embedding f: V¯ → N , the (r + 1)–cube res S 7→ hofiber [ emb(V¯S, N) −−→ emb(V¯ , N) ] r is (3 − n +Σi=0(n−qi −2))–Cartesian.