MORSE MOVES IN FLOW CATEGORIES

Transcription

1 ORSE OVES IN FLOW CATEGORIES DAN JONES, ANDREW LOBB, AND DIRK SCHÜTZ Abstract. We ursue the analogy of a framed flow category with the flow data of a orse function. In classical orse theory, orse functions can sometimes be locally altered and simlified by the orse moves. These moves include the Whitney trick which removes two oositely framed flowlines between critical oints of adjacent index and handle cancellation which removes two critical oints connected by a single flowline. A framed flow category is a way of encoding flow data such as that which may arise from the flowlines of a orse function or of a Floer functional. The Cohen-Jones-Segal construction associates a stable homotoy tye to a framed flow category whose cohomology is designed to recover the corresonding orse or Floer cohomology. We obtain analogues of the Whitney trick and of handle cancellation for framed flow categories: in this new setting these are moves that can be erformed to simlify a framed flow category without changing the associated stable homotoy tye. These moves often enable one to comute by hand the stable homotoy tye associated to a framed flow category. We aly this in the setting of the Lishitz-Sarkar stable homotoy tye (corresonding to Khovanov cohomology) and the stable homotoy tye of a matched diagram due to the authors (corresonding to sl n Khovanov-Rozansky cohomology). Contents 1. Introduction lan of the aer 4 2. orse moves in framed flow categories Framed flow categories The Whitney trick in framed flow categories Handle cancellation in framed flow categories The framings of 1-dimensional moduli saces Some simle stable homotoy tyes Framing conventions for 1-dimensional moduli saces Gluing formulae for the Whitney trick Gluing formulae for handle cancellation Examles The (3,4)-torus knot A retzel link 37 AL and DS were both suorted by ESRC grant E/389/1, DJ was suorted by an ESRC graduate studentshi. 1

2 2 DAN JONES, ANDREW LOBB, AND DIRK SCHÜTZ 4.3. The disjoint union of two trefoils 44 References Introduction Given a orse-smale function f : R on a comact Riemannian manifold, it is well-known that there is a handle decomosition of corresonding to f. Suose that this handle decomosition has a handle h i of index i and a handle h i1 of index i 1. If the attaching shere of h i1 intersects the belt shere of h i in exactly one oint then one can obtain a new handle decomosition of in which h i and h i1 are omitted but all other handles remain with suitably adjusted attaching mas. In the orse theory icture, intersections of the attaching shere with the belt shere corresond to flowlines between the critical oints i and i1 which give rise to h i and h i1 resectively. If there is just a single such flowline then the orse function may be modified in a neighbourhood of that flowline so that both critical oints i and i1 are removed. This rocess of modifying the handle decomosition or the orse function is known as handle cancellation. Suose now that the attaching shere of h i1 intersects the belt shere of h i in morethanoneoint, andinarticular intwooints x andx whichhave oosite sign. These corresond to two flowlines between i and i1 which have oosite framing. One would like to cancel x and x against each other and thus reduce the total number of intersection oints by two. In contrast to handle cancellation, there are now toological conditions that need to be satisfied before one can be sure that one can achieve this: in articular we need to be in a situation with a large enough dimension and a large enough degree of connectedness. The rocess by which one can cancel such airs of intersection oints (or such airs of flowlines) is known as the Whitney trick. It is the Whitney trick s failure in general in low dimensions that leads, for examle, to the comlexity of simly-connected smooth 4-manifold toology. In this aer we extend the idea of handle cancellation and the Whitney trick to framed flow categories. A framed flow category can be thought of as a way of encoding the flow data that might arise from a orse function or a Floer functional. Associated to framed flow category C is a stable homotoy tye C by a construction due to Cohen-Jones-Segal [CJS95]. The cohomology of C is designed to recover the orse or Floer cohomology of the inut. Roughly seaking, a flow category C consists of a finite number of Z-graded objects where one thinks of the objects as being critical oints of a Floer functional and the Z-grading as being an absolute aslov index. Then the sace of morhisms from an object of index i to an object of index j is a (i j 1)-dimensional comact manifold-with-corners which one thinks of as being a sace of flowlines between two critical oints. A framed flow category further refines this notion. Examle 1.1. The cu roduct structure on cohomology allows one to distinguish between the saces X 1 = S 2 S 4 and X 2 = C 2. In fact, even u to (based) stable homotoy equivalence, the saces X 1 and X 2 are not the same. The cu roduct is not a stable oeration so it cannot now distinguish them. Rather they can

3 ORSE OVES IN FLOW CATEGORIES 3 Figure 1. art of a surface with three critical oints with resect to the height function. Taking the orse-smale metric to be restriction of the Euclidean metric, we see that there is exactly one flowline between the highest and the middle critical oint, while there are two flowlines of oosite sign between the middle and the lowest critical oint. be distinguished by the observation that the former has a trivial second Steenrod square (a stable cohomology oeration) while for latter it is non-trivial. One could ask what is the simlest framed flow category C i that gives rise to X i = C i for i = 1,2? In both cases, one needs at least one object in each of the cohomological degrees 2 and 4 to generate the cohomology (we are interested in the reduced cohomology and we are working u to based (de-)susension). Let us suose then that C 1 and C 2 each have just two objects which we shall call 2 and 4. Since there are no objects of degree 3 it follows that the sace of morhisms from 4 to 2 is a comact boundaryless 1-manifold (the absence of degree 3 objects should be thought of as a lack of critical oints at which flowlines from 4 to 2 can break). Hence the morhism sace is a disjoint union of circles in both cases. How these cases differ will essentially be in the framings of the circles. Different choices here lead to either X 1 or X 2. What these choices are is discussed in detail in Subsection 3.1. Our main results are the construction of moves analogous to the Whitney trick and to handle cancellation. These aear as Theorems 2.8 and The content of these theorems is in the construction of a framed flow category C W (resectively C H ) from a framed flow category C whose -dimensional morhism saces suggest the ossibility of erforming a Whitney trick (res. handle cancellation). ore secifically: Suose that C is framed flow category with two objects x and y of index differing by 1, such that the morhism sace between them contains two morhisms of differing sign. Then we construct a framed flow category C W with object set Ob(C W ) = Ob(C) and such that the morhism sace between x and y has the same signed count but contains two fewer morhisms. We show that we have (Whitney trick) C W C. On the other hand, suose that C is a framed flow category with two objects x and y of index differing by 1, with exactly one morhism between them. Then we construct a framed flow category C H with Ob(C H ) = Ob(C)\{x,y}, such that (Handle cancellation) C H C.

4 4 DAN JONES, ANDREW LOBB, AND DIRK SCHÜTZ Remark 1.2. In the roof of the h-cobordism theorem [il65] in high dimensions, three orse moves are used. The third orse move corresonds to the oeration of handlesliding. This move also has an analogue in the setting of framed flow categories, but we intend rather to discuss it in a later aer when we have a new alication for it. For now we note here that using the moves on a framed flow category C one can treat them as oerations simlifying the CW cochain comlex of C. Indeed, the Whitney trick ensures that the absolute count of the -dimensional moduli saces matches the relevant comonent of the differential, handle cancellation acts as Gauss elimination, and handle sliding acts as base change. In this way one could find a framed flow category reresentative S of any finite free cochain comlex C with H (C) = H ( C ) (and furthermore in which the -dimensional moduli saces S are determined by the differential of C) such that S C. Recently, Lishitz-Sarkar [LS14a] have constructed a framed flow category L Kh (D) associated to an oriented link diagram D. The associated stable homotoy tye X Kh (D) := L Kh (D) is invariant under the Reidemeister moves and its bigraded cohomology (graded cohomologically and with resect to a slitting of X Kh (D) as a wedge sum along a second quantum grading) is exactly Khovanov cohomology [Kho]. The authors [JLS15] have associated a framed flow category L n (D) (and associated stable homotoy tye X n (D) = L n (D) ) to an oriented link diagram D (with a choice of decomosition into elementary tangles) and an integer n 2. In the case n = 2 we showed that X 2 (D) = X Kh (D) (u to a choice of bigrading normalization). For n > 2 and D a matched diagram, the bigraded cohomology of X n (D) is sl n Khovanov-Rozansky cohomology [KR8]. Comutations of these stable homotoy tyes has been erformed so far essentially by comutation of cohomology oerations (in articular the first and second Steenrod squares). With the two moves on framed categories corresonding to the Whitney trick and to handle cancellation, we are able to work by hand at the level of the framed flow category, reducing the number of objects and the comlexity of the morhism saces. We use these two moves in examles at the end of the aer, each time reducing the comlexity of a framed flow category until it is essentially as simle as ossible and the associated stable homotoy tye can be seen directly without, for examle, direct comutation of stable cohomology oerations lan of the aer. We start by giving a brief overview of framed flow categories in Subsection 2.1. Then in Subsection 2.2 (resectively 2.3) we discuss how to define the framed flow category C W (res. C H ) arising from erforming the Whitney trick (res. handle cancellation) on a framed flow category C. We show that the Cohen-Jones-Segal construction gives saces for which there is a stable homotoy equivalence C W C (res. C H C ). Then in Section 3 we determine how the Whitney trick and handle cancellation affect the framings on the 1-dimensional moduli saces. These framings may give rise to non-trivial toology in the associated stable homotoy tye (exhibited for examle in a non-trivial second Steenrod square). In rincile this could be done for moduli saces of even higher dimension which may rovide a way to detect unusual stable homotoy tyes, such as those that are invisible to stable cohomology oerations.

5 ORSE OVES IN FLOW CATEGORIES 5 We aly this in Section 4 to the comutation by hand of three stable homotoy tyes. In articular, in Subsection 4.2 we consider the framed flow category L 3 () where is the retzel link (2,2,2). By successive alication of the two moves we reduce the flow category in quantum degree 6 to two objects as in Examle 1.1. The framings on the circle moduli saces then imly that there is a C 2 in X 3 (). In Subsection 4.1, we do something similar to quantum degree 11 of L Kh (T 3,4 ) where T 3,4 is the (3,4) torus knot in the form of the retzel knot diagram (2,3,3). In this case we reduce to three objects and considerations of framings then shows that there is an R 5 /R 2 in X Kh (T 3,4 ). Finally in Subsection 4.3 we consider the Lishitz-Sarkar stable homotoy tye of the disjoint union of twotrefoils, andinthiscasewefindanr 2 R 2 asawedgesummandasredicted by [LLS15, Thm.1]. 2. orse moves in framed flow categories 2.1. Framed flow categories. To define flow categories, we need a sharening of the concet of smooth manifolds with corners. We will give a somewhat shortened resentation here, for more details see [Jän68], [Lau], [LS14a], or [JLS15]. Definition 2.1. Let n be a non-negative integer and let d = (d,...,d n ) be an (n 1)-tule of non-negative integers. Define E d = R d [, ) R d1 [, ) [, ) R dn. Furthermore, if a < b n 1, we denote E d [a : b] = E (da,...,d b1) and set d a:b = d a d b1. Also, let for i {1,...,n}. i E d = R d R di1 {} R di R dn If J {1,...,n} is a non-emty subset, let J E d = j J i E d and J : E d [, ) J be the rojection such that J JEd is constant. Definition 2.2. Let n be a non-negative integer and let d = (d,...,d n ) be an (n 1)-tule of non-negative integers. A smooth n -manifold m is a smooth manifold with corners together with an immersion ı: E d such that (1) corner oints of codimension l in are sent to corner oints of codimension l in E d for all l n; (2) if x has a chart neighborhood [, ) l R ml with x corresonding to [, ) l R ml, there is J {1,...,m} with J = l, ı(x) J E d and the embedding is orthogonal to J E d at x. For i = 1,...,n define i = ı 1 ( i E d ). The immersions can be imroved to embeddings by stabilizing d, and immersions (res. embeddings) are referred to as neat if they satisfy the conditions of Definition 2.2.

6 6 DAN JONES, ANDREW LOBB, AND DIRK SCHÜTZ Definition 2.3. A framed flow category consists of a category C with finitely many objects Ob = Ob(C), a function : Ob Z, called the grading, an (n1)- tule of non-negative integers d = (d k,...,d nk ) and a collection ϕ of immersions satisfying the following: (1) k = min{ x : x Ob(C)} and n = max{ x : x Ob(C)}k. (2) Hom(x,x) = {id} for all x Ob, and for x y Ob, Hom(x,y) is a smooth, comact ( x y 1)-dimensional x y 1 -manifold which wedenoteby(x,y), andwhoseimmersionsarefunctionsı x,y : (x,y) E d [ y : x ]. (3) For x,y,z Ob with z y = m, the comosition ma : (z,y) (x,z) (x,y) is an embedding into m (x,y). Furthermore, { 1 i (z,y) (x,z) for i < m ( i (x,y)) = (z,y) im (x,z) for i > m and i x,y ( q) = (i z,y (),,i x,z (q)). (4) For x y Ob, induces a diffeomorhism i (x,y) = (z,y) (x,z). z, z = y i (5) The immersions ı x,y for x,y Ob(C) extend to immersions which satisfy ϕ(x,y)( q,t 1,...,t d y, x ) = ϕ x,y : (x,y) [ε,ε] d y : x E d [ y : x ] (ϕ z,y (,t 1,...,t d y : z ),,ϕ x,z (q,t d y : z 1,...,t d y : x )) for all (z,y), q (x,z) where z Ob(C). The manifold (x,y) is called the moduli sace from x to y, and we also set (x,x) =. A flow category is basically obtained by droing the immersions. Note that the ϕ x,y are codimension immersions, and we therefore think of them as framings. Again we can obtain embeddings by stabilization. In [CJS95] a stable homotoy tye C is associated to a framed flow category C. We quickly recall the construction in the form given by [LS14a]. Definition 2.4. Let C be a framed flow category embedded into E d for some d = (d k,...,d kn ). For an arbitrary object a in Ob(C) of degree i, recall that for each object b in Ob(C) of degree j < i, we have the embedding ϕ a,b : (a,b) [ε,ε] dj:i [R,R] dj [,R] [,R] [R,R] di1 where R is chosen to be large enough that all moduli saces (a,b) can be embedded in this way. The CW comlex C consists of one -cell (the baseoint) and one (d k d nk1 k i)-cell C(a) for every object a with a = i defined as [,R] [R,R] d k [R,R] di1 {} [ε,ε] di {} {} [ε,ε] d nk1.

7 ORSE OVES IN FLOW CATEGORIES 7 Each cell C(a) is considered a subset of a different coy of [, ) E d. The embedding ϕ can be used to identify articular subsets (1) (a,b) C(b) = C b (a) n C(a) in the following way: C b (a) =[,R] [R,R] d k [R,R] dj1 {} ϕ a,b ( (a,b) [ε,ε] d j:i ) {} [ε,ε] dm {} [ε,ε] da1 C(a). It will be useful to introduce notation for this identification by letting (2) Γ a,b : (a,b) C(b) j C(a) be the identification (a,b) C(b) = C b (a). Let (3) C = d k d nk1 k. Then the attaching ma for each cell C(a) C (Cj1) is defined via the Thom construction for each embedding into C(a) simultaneously. That is, for each subset (a,b) C(b) = C b (a) C(a), the attaching ma rojects to C(b) (which carries trivialisation information), and sends the rest of the boundary C(a)\ b C b(a) to the baseoint. The indeendence of the stable homotoy tye of C on the various choices is discussed in [LS14a, 3] The Whitney trick in framed flow categories. Let (C,ϕ,ı) be a framed flow category containing objects x and y with x = i and y = i1, and such that (x,y) includes two oints, and, with oosite framings. We shall define a new framed flow category, written C W, such that C W C. Definition 2.5. With C as above, we define the object set of C W by Ob(C W ) = {ā : a Ob(C)}. We now give the moduli saces of C W. (1) ( x,ȳ) = (x,y)\{,}. (2) If a Ob(C) is such that (a,x) φ then we have {,} (a,x) (a,y). Let {,} [,1) (a,x) be a collar neighbourhood of this subset and write (,t,) (,1/2t,) for < t < 1/2 and all (a,x). Now we define (ā,ȳ) = ((a,y)\{,} (a,x))/. (3) Similarly, if b Ob(C) is such that (y,b) φ then we have (y,b) {,} (x,b). Let (y,b) [,1) {,} be a collar neighbourhood of this subset and write (,t,) (,1/2t,) for < t < 1/2 and all (y,b). And we define ( x, b) = ((x,b)\(y,b) {,})/.

8 8 DAN JONES, ANDREW LOBB, AND DIRK SCHÜTZ [,1) [,1)\{(,)} [,1) [,1)\{(,)} = = ([,1) [,1)\{(,)}) ([,1) [,1)\{(,)})/ Figure 2. We show how to glue together two coies of [,1) [, 1) \ {(, )} by the orientation-reversing gluing equivalence relation. The dotted line reresents the oen boundary. (4) If a,b Ob(C) and both (a,x) φ and (y,b) φ then we have (y,b) {,} (a,x) (a,b). Let (y,b) [,1) {,} [,1) (a,x) be a neighbourhood of this subset in (a,b). Choose this neighbourhood such that (y,b) {} {,} [,1) (a,x) is a collar neighbourhood of (y,b) {,} (a,x)in(y,b) (a,y)and(y,b) [,1) {,} {} (a,x)is a collar neighbourhood of (y,b) {,} (a,x) in (x,b) (a,x). Now, using the equivalence relation given in Figure 2, we define the equivalence on (y,b) [,1) {,} [,1) (a,x) by requiring (,r,,s,q) (,t,q,u,q) for r,s,t,u [,1), all (y,b), and all q (a,x) if and only if Then we define (r,s) (t,u). (ā, b) = ((a,b)\(y,b) {,} (a,x))/. Note that the equivalence relation may be chosen to be comatible with (2) and (3). (5) In all other cases we define (ā, b) = (a,b). Clearly this defines a flow category C W. Now suose that C comes with a framed embedding (C,ı,ϕ) into the Euclidean sace E d. After ossibly a stabilization and an isotoy, we may assume that ı x,y takes and to the oints (1,,...,) and (1,,...,) in R di resectively.

9 ORSE OVES IN FLOW CATEGORIES 9 Furthermorewemayassumethatforalla,b Ob(C)suchthat a = iand b = i1 we have ı 1 a,b (R {(,...,)}) {,Q}. Also, thinking of the framings of and as ordered d i -tules of orthonormal vectors, we may assume that the framings of and differ only in the first vector, and these vectors are (1,,...,) and (1,,...,) for and resectively. Now, collar neighbourhoods are embedded transversely to the boundaries of E d. So if a Ob(C) such that (a,x) then we may assume that the embedding ı a,y {,} [,1) (a,x) of the collar neighbourhood of {,} (a,x) described in Definition 2.5 satisfies ı a,y {,} [,1) (a,x) (,t,) = ((1,,...,),t,ı a,x ()) and ı a,y {,} [,1) (a,x) (,t,) = ((1,,...,),t,ı a,x ()) for all t [,1) and all (a,x) where the image lies in E d [i1, a ] = R di [, ) E d [i, a ]. Furthermore, we may assume that the framing of this collar neighbourhood is given bytheroductframingof{,} (a,x)(viatheidentificationofnormalbundles using the Euclidean inner roduct). Similarly, we may assume for b Ob(C) with (y,b) φ that we have ı x,b (y,b) [,1) {,} (,t,) = (ı y,b (),t,(1,,...,)) and ı x,b (y,b) [,1) {,} (,t,q) = (ı y,b (),t,(1,,...,)) for all t [,1) and all (y,b), and that the framing is given by the roduct framing of (y,b) {,}. Finally if a,b Ob(C) and both (a,x) φ and (y,b) φ, then we may assumethattheembeddingı a,b,intheneighbourhoodof(y,b) {,} (a,x) given in Definition 2.5, satisfies where the image lies in ı a,b (y,b) [,1) {,} [,1) (a,x) (,r,,s,q) = (ı y,b (),r,(1,,...,),s,ı a,x (q)), ı a,b (y,b) [,1) {,} [,1) (a,x) (,r,,s,q) = (ı y,b (),r,(1,,...,),s,ı a,x (q)) E d [ b, a ] = E d [ b,i1] [, ) R di [, ) E d [i, a ]. And we can assume that the framing on this neighbourhood is given by the roduct framing on (y,b) {,} (a,x). We have illustrated the embeddings of these collar neighbourhoods in Figure 3, where we have included only the interesting factors of E d. Definition 2.6. We give an embedding ī and framing ϕ of C W in the Euclidean sace E d. Firstly, in case (1) of Definition 2.5, the embedding and framing of ( x,ȳ) is defined by restriction of ı x,y and of ϕ. In case (5) of Definition 2.5, the embeddings and framings of C W agree with those of C.

10 1 DAN JONES, ANDREW LOBB, AND DIRK SCHÜTZ Figure 3. The three diagrams illustrate the framed embeddings of collar neighbourhoods of {,} (a,x), (y,b) {,}, and (y, b) {, } (a, x) resectively. In the first two cases we have rojected to the roduct of the first coordinate of R di = E d [i1,i] and the relevant [, ) factor of E d, and in the third case we have rojected to the roduct of the first coordinate of R di and the two relevant [, ) factors. In the omitted factors the embeddings are given by the mas ı a,x, ı y,b, and (ı y,b,ı a,x ). The arrows give the vector that corresonds to the first coordinate of the framing of {,}. All subsequent vectors in the framing of agree with those of Q. In cases (2), (3), and (4) of Definition 2.5, we define the framing and embedding of C W to differ only from those of C in a small neighbourhood of the gluing regions given in those cases. How the framings and embeddings differ is described in Figure 4. Itremainstoconcludethatthestablehomotoytyeassociatedto(C W,ī, ϕ)agrees with that associated to (C,ı,ϕ). In fact, we can say a little more. roosition 2.7. We write C and C W resectively for the CW-comlexes (and not just the stable homotoy tye of those comlexes) determined by alying the construction due to Cohen-Jones-Segal to the framed flow categories (C, ı, ϕ) and (C W,ī, ϕ). Then we write f a and fā resectively for the attaching mas of the cells C(a) and C(ā) for all a Ob(C).

11 ORSE OVES IN FLOW CATEGORIES 11 Figure 4. The three diagrams illustrate the framed embeddings of oen subsets of (ā,ȳ), ( x, b), and (ā, b) containing the subsets in which moduli saces of C have been glued together. The coordinates rojected to agree with those of Figure 3. In the coordinates that have not been shown, and outside the regions shown, the embeddings and the framings are inherited from (C,ı,ϕ). For each t [,1] there exist mas F a,t : {t} C(a) Y a 1 t where Y i t is defined inductively for increasing i by Yt i = Yt i1 Fa,t {t} C(a) where the union is taken over all cells C(a) of dimension i and Y t = {t}. These F a,t are such that the mas F a : [,1] C(a) Y a 1 : (t,x) F a,t (x) are continuous where Y i is defined inductively for increasing i by and Y = {t}. Y i = Y i1 Fa [,1] C(a) Furthermore, identifying C(a) with {} C(a) and C(ā) with {1} C(a) we have that F a, = f a and F a,1 = fā.

12 12 DAN JONES, ANDREW LOBB, AND DIRK SCHÜTZ Figure 5. We show the sequence of attaching mas F a,t from roosition 2.7 as for t 1/2. It is then easy to see that C = i Y i and C W = i Y i 1 are both subsets of i Y i, and that this larger sace deformation retracts onto both C and onto C W. Hence the following is immediate. Theorem 2.8. The saces C and C W from roosition 2.7 are homotoy equivalent. roof of roosition 2.7. We wish to see that we can continuously deform the attaching mas of the comlex C to arrive at the attaching mas of the comlex C W. The attaching mas of the cells of C are determined by the framed embedding (C, ı, ϕ). Ideally we would like a smooth deformation through framed embeddings to arrive at the framed embedding (C,ī, ϕ), but since C C W this cannot be achieved. Instead we ass from (C,ı,ϕ) to (C W,ī, ϕ) through the deformations in Figure 5. We have illustrated in Figure 5 the attaching ma f a,t f 1 a,t (C(y)) thought of as a ma to C(y) for t 1/2 (the reader should be able to fill in the ictures for 1/2 t 1 herself) where (a,x) φ. The attaching mas are constructed in the usual way: via a framing and then alying the Thom construction, with the excetion illustrated in the third and fourth diagrams of Figure 5. In all diagrams the thick arrows reresent (a vector in) the framing, and in fact are the fibres that are going to wra once over the first coordinate of [ǫ,ǫ] di thought of as a factor of C(y). In the third and fourth diagram there is one thick arrow with a single head and a double tail, while all other arrows are homeomorhic to the interval [ǫ,ǫ]. These singular fibres again ma to the first coordinate of [ǫ,ǫ] di, with the mas

13 ORSE OVES IN FLOW CATEGORIES 13 determined by continuity of f a,t. In the factors not illustrated the framings and embeddings of course do not change with t. The attaching mas f x,t f 1 x,t (C(y)) thought of as mas to C(y) are obtained from Figure 5 by just looking at the horizontal boundaries of the diagram. The attaching mas f x,t f 1 a,t (C(b)) for those b with (y,b) φ are obtained by rotating the diagrams by π/2. The attaching mas f a,t f 1 a,t (C(b)) for a,b with (a,x) φ and (y,b) φ are swet out by rotating the diagrams in 5 by π/2. Finally, for all other a,b the mas f a,t f 1 a,t (C(b)) do not vary with t Handle cancellation in framed flow categories. Throughout this subsection, let C denote a framed flow category (C,ı,ϕ) with two of its objects having a one-oint moduli sace between them, (x,y) =. Let x = i and y = i1. We shall show that the sace arising from the handle-cancelled framed flow category C H is stably homotoy equivalent to the sace arising from the original framed flow category C. This aears as Theorem 2.17 but before we can state it we need to define, embed, and frame C H. We denote the CW comlex associated to a framed flow category S by the Cohen-Jones-Segal construction (as oosed to its stable homotoy class) by S and we term this the realisation of S. Definition 2.9. Denote by C H the flow category whose object set is given by Ob(C H ) = {ā : a (Ob(C)\{x,y})} and whose moduli saces are given by (ā, b) = (a,b) f ( (x,b) (a,y) ) where f identifies the subsets and (x,b) (a,x) (y,b) (a,y) (a,b) (x,b) ((x,y) (a,x)) ((y,b) (x,y)) (a,y) We call C H the cancelled category (relative to x and y) of C. (x,b) (a,y). It follows from the existence of collar neighbourhoods for n -manifolds, see [Lau, Lemma 2.1.6], that (ā, b) is a ( a b 1)-dimensional a b 1 -manifold, and that C H is a flow category, with object grading inherited from C. For the framed flow category C, we must rovide framed neat embeddings of C H, so that we can form C H. Recall that C(x) is a (Ci)-cell (for some C >> ) and a single coy of C(y) = (x,y) C(y) is identified with a subset on the boundary of C(x) (in fact on the (i1)-face) via C y (x) =[,R] [R,R] db [R,R] di2 {} ı x,y ( (x,y) [ε,ε] d i1 ) {} [ε,ε] d i {} [ε,ε] da1 i1 C(x). Note that we can assume ı x,y embeds the oint (x,y) = { } as ı x,y ( ) = in R di1. The framing of that oint gives a homeomorhism between C y (x) and the (C i1)-cell C(y).

14 14 DAN JONES, ANDREW LOBB, AND DIRK SCHÜTZ Choose a homeomorhism f : C(x)\int(C y (x)) C y (x) which is the identity on C y (x) and is smooth on all boundaries of C(x) of codimension 1. Let A f = ( C(x)\int(C y (x))) [,1] C y (x)/ (where is defined by (,s) (,t) for all C y (x) and s,t [,1] and (q,1) f(q) for all q C(x)\int(C y (x))) be a quotient of the maing cylinder of f. Note that the boundary of A f is naturally identified with C(x) and choose a identification, smooth on the interior, A f = C(x) resecting this on the boundary. Definition 2.1. For t [,1], define the ma Ψ t : C(x)\int(C y (x)) C(x) by comosing the inclusion ( C(x)\int(C y (x))) {t} C(x)\int(C y (x)) [,1] C y (x) with the quotient. Recall from Definition 2.9 that (ā, b) is formed by gluing the two saces (a,b) and (x,b) (a,y) along their common boundaries. Thus, in order to embed (ā, b), we shall define an embedding for each of these saces searately, and emhasisehowthegluingworks. Theformerofthetwoisembeddedwithitsoriginal embedding from (C,ı,ϕ), while a framed embedding Γ x,b a,y of the roduct moduli saces (x,b) (a,y) is described in Lemma 2.11 along with a descrition of the gluing. A framed embedding Γā, b of the moduli saces (ā, b) is then described in Lemma Finally, some alteration is needed to ensure that these embeddings are neat embeddings, and this is described in Lemma In the remainder of this section let a,b Ob C \{x,y} with a = m > n = b. We shall write Γ a,b : (a,b) C(b) = C b (a) C(a) for the inclusions. Lemma There is an embedding Γ x,b a,y : (x,b) (a,y) C(b) C(a). oreover, this embedding can be defined to agree with Γ a,b on the boundary subset ( (y,b) (x,y) ) (a,y) (x,b) ( (x,y) (a,x) ). roof. Consider each (x, b) embedded into Euclidean sace as ı x,b : (x,b) [ε,ε] dn di1 [R,R] dn [,R] [,R] [R,R] di1. Varyingtin[,1]rovidesanintervalofframedembeddedsubsacesΨ t Cb (x)(c b (x)) inside C(x), and in articular we now consider the framed embedded subsace Ψ 1 Cb (x)(c b (x))insidec y (x)(seefigure8). Thisrovidesanembeddingof(x,b) C(b) into [,R] [R,R] db [R,R] di2 {} ı x,y ( (x,y) [ε,ε] d i1 ) {} [ε,ε] di {} [ε,ε] da1 = C y (x). given by Ψ 1 Cb (x) Γ x,b : (x,b) C(b) C y (x).

15 ORSE OVES IN FLOW CATEGORIES 15 Now, abusing notation slightly, let Γ 1 x,y : C y (x) (x,y) C(y) be the obvious homeomorhism, so that Γ 1 x,y Ψ 1 Cb (x) Γ x,b : (x,b) C(b) C(y) rovides an embedding of (x,b) C(b) into a one-oint roduct of C(y). Next, to embed the roduct moduli sace (x, b) (a, y) C(b), consider the identification Γ a,y : (a,y) C(y) C y (a) i1 C(a) with (x,b) C(b) embedded into the C(y) comonent via Γ 1 x,y Ψ 1 Cb (x) Γ x,b. Then define the embedding (4) Γ x,b a,y : (x,b) (a,y) C(b) C(a) by Γ x,b a,y (,q,δ) = Γ a,y( q,γ 1 x,y Ψ 1 Cb (x) Γ x,b (,δ) ) (see Figure 9). Notice that the embeddings Γ x,b a,y and Γ a,b agree on ( (y,b) (x,y) ) (a,y) since Ψ 1 is the identity there, but on (x,b) ( (x,y) (a,x) ) they disagree. We shall next fix this by adding a collar (x,b) (a,x) [,1] C(b) which we glue onto (x,b) (a,y) C(b) along (x,b) (a,x) {1} C(b) in the obvious way. Note of course that this results in the same sace and we shall abuse notation by referring to the sace with the collar also as (x,b) (a,y) C(b). Consider the embedding (5) Γ x,b a,y : (x,b) (a,y) C(b) C(a) that is defined as Γ x,b a,y away from the collar, and defined as Γ x,b a,y(,q,t,δ) = ( Γ a,x q,ψt Cb (x)(γ x,b (,δ)) ) on oints (,q,t,δ) within the collar. Varying t from to 1 has the effect of tracing from the embedding Γ a,b (when t = ) to the (when t = 1). We need to check that the ma on the collar embedding Γ x,b a,y (x,b) (a,x) [,1] does not affect the intersection with the collar neighbourhood (y,b) (a,y) [,1], which is (y,b) (a,x) [,1] 2. In fact, since Γ x,b sends the boundary subset (y,b) (x,y) C(b) to C b (x) C y (x) C y (x), Ψ t has no effect on this articular collar. Hence, Γ x,b a,y rovides an embedding satisfying the required roerties. SincetheembeddingsdefinedintheroofofLemma2.11mayseemalittleabstract, let us consider an examle. Examle Let C Ex be a framed flow category with Ob(C Ex ) = {a,x,c,y,b} such that a = 2, x = c = 1, and y = b =. Here is an illustration of C Ex a 1 2 x c q 1 q 2 q 3 y b

16 (a,b) = q1 1 q 3 2 (a,y) = 1 q DAN JONES, ANDREW LOBB, AND DIRK SCHÜTZ in which the -dimensional moduli saces are all single oints that are labelled in blue. The 1-dimensional moduli saces need to be given as well, and these can be drawn as Now assume that ı is a neat embedding of C Ex relative d = (d,d 1 ) = (1,1). The cells needed to construct the CW comlex C Ex are: C(a) = [,R] [R,R] [,R] [R,R] C(x) = [,R] [R,R] {} [ε,ε] = C(c) C(y) = {} [ε,ε] {} [ε,ε] = C(b). Each of these cells can be considered as a subset of E = R R R R. oreover, E = ( {} R R R ) ( R R {} R ) is a 3-dimensional 2 -manifold, which can be illustrated by flattening out the corner-lane {} R {} R to give a homeomorhism E = R 3 (c.f. [LS14a, Figure 3.3]). Under this homeomorhism, the boundary C(a) has subsets identified with certain moduli saces that are deicted in Figure 6. Figure 6. Examle: C Ex. Identifications of cells in C(a). In blue is the embedding Γ a,b and in green the embedding Γ a,y of (a,b) C(b) and (a,y) C(y), resectively. The rightmost (white) cube is C c (a) = C(c) and the leftmost (white) cube is C x (a) = C(x). The cell C(x) has arts of its boundary identified with both C y (x) and C b (x), and can be deicted on its own as in Figure 7. The cell C(y) is green and the cell C(b) is blue. Further, the deformation Ψ t on C(x) (which is iecewise smooth on faces) sends C(x) \ C y (x) through C(x) to C y (x).

17 ORSE OVES IN FLOW CATEGORIES 17 Figure 7. Examle: C Ex. The cell C(x). This results in an embedding Γ 1 x,y Ψ 1 Cb (x) Γ x,b of (x,b) C(b) in the cell C y (x) = C(y). This embedding is highlighted in blue in Figure 8, where the images of each face are outlined. Figure 8. Examle: C Ex. The result of collasing C(x) using Ψ 1. Now recall that the embedding Γ x,b a,y is defined in Equation 4 as the embedding Γ a,y of (a,y) C(y) with C b (x) embedded into C(y) as above. This embedding, together with the embedding Γ a,b is deicted in Figure 9. The cells C x (a) and C y (a) are indicated by dashed lines since they corresond to the objects that are being cancelled. The embedding Γ a,b is highlighted blue as before, and the embedding Γ x,b a,y is highlighted urle. Notice that the two framed intervals do not agree on their boundaries corresonding to (x,b) (a,x) and (x,b) ( (x,y) (a,x) ). This is the urose of the alteration of Γ x,b a,y in the roof of Lemma The embedding Γ x,b a,y is defined in Equation 5 by altering the embedding Γ x,y a,y in a collar neighbourhood of (x,b) (a,y) C(b). The alteration uses the deformation Ψ t and takes lace inside C x (a); it is highlighted red in Figure 1. In this figure, the embedding Γ x,b a,y of (x,b) (a,y) C(b) is deicted as the concatenation of both the red and urle framed embedded intervals. Lemma There is an embedding Γā, b : (ā, b) C(b) C(a).

18 18 DAN JONES, ANDREW LOBB, AND DIRK SCHÜTZ Figure 9. Examle: C Ex. The embedding Γ a,b (in blue) and Γ x,b a,y (in urle). Figure 1. Examle: C Ex. The embeddings Γ a,b and Γ x,b a,y. roof. Since (ā, b) = (a,b) f (x,b) (a,y) (Definition 2.9), we may use Γ a,b and Γ x,b a,y to embed (ā, b) into C(a). The fact that these embeddings agree on the gluing f given in Definition 2.9 was shown in Lemma Examle Consider again the framed flow category C Ex from Examle The embeddings Γ a,b and Γ x,b a,y are deicted in Figure 1. The embedding Γ x,b a,y agrees with Γ a,b on Let ((x,b) (a,x)) C(b) ((a,b) C(b)). denote the rojection ma, and let Π[b : a] : C(a) E d [b : a] Ξ[b : a] : [ε,ε] dn dm1 C(b)

19 ORSE OVES IN FLOW CATEGORIES 19 be the inclusion which takes value on all omitted coordinates of C(b). Observe that we can recover the embeddings as the comosition ı a,b : (a,b) [ε,ε] dn dm1 E d [b : a] ı a,b = Π[b : a] Γ a,b (id (a,b) Ξ[b : a]). One might hoe to be able to define embeddings of the moduli saces (ā, b) in a similar way. However, these embeddings would not obviously be neat embeddings since boundary oints of both (a,b) and (x,b) (a,y) that become interior oints of (ā, b) would need to be identified with art of the interior of n C(a). This is the case in Examles 2.12 and 2.14, where the embedding Γā, b rotrudes into the 1-face of C(a) (the red framed interval in Figure 1). Instead, we alter these embeddings slightly as outlined in the roof of the following lemma. Lemma For each ā, b in Ob(C H ), there are framed neat embeddings īā, b : (ā, b) [ε,ε] dn dm1 E d [b : a] roof. We shall give a homotoy of homeomorhisms h t : C(a) C(a) for t 1, which are smooth away from the corners of C(a). The urose of this homotoy is to move from h = id C(a) to the ma h 1 which redefines the cornered structure of C(a) in a useful way. In articular, if one ulls back the face structure of C(a) through h t one will obtain a cornered manifold C(a) t isomorhic to C(a) = C(a). The homotoy h t will be chosen so that the embedding h 1 (Γ a,b Γ x,b a,y ) will be neat. We shall define the embedding īā, b as the embedding Π[b : a] h 1 (Γ a,b Γ x,b a,y ) ((id (a,b) id (x,b) (a,y) ) Ξ[a : b]). Let us begin by defining the face structure of C(a) 1. It will be enough, for each oint of C(a) 1 to know whether or not it lies in the closure of the k-face for each k < m. If im(γ a,x ) (resectively im(γ a,y )) then, since Γ a,x (res. Γ a,y ) is injective, we have that Γ 1 a,x() is a well-defined oint of (a,x) C(x) (res. Γ 1 a,y() is a well-defined oint of (a, y) C(y)). rojecting to the second coordinate gives a oint C(x) (res. C(y)). If there is a oint q in the k-face k C(x) such that Ψ t (q) = for some t 1 (res. such that Ψ 1 (q) = ) then we have that is in the k-face of C(a) 1. For all other im(γ a,x ) we have that is in the k-face of C(a) 1 iff it is in the k-face of C(a). That the embedding Γ a,b Γ x,b a,y is neat with resect to the face structure of C(a) 1 is clear from the definition of Γ a,b Γ x,b a,y. Finally we wish to see that we can realize the face structure of C(a) 1 as the ullback of the face structure of C(a) = C(a) through h 1 for some homotoy h s. We shall give the corresonding face structures C(a) s, from which it will be clear that such an h s exists. Let f : C(x) [,1] be an injective height function smooth where it makes sense on the boundaries of C(x), such that f 1 (1) is a oint, f 1 () = C y (x), f 1 (s) is homeomorhic to C y (x) for < s < 1, and the intersection of f 1 [,s] with any

20 2 DAN JONES, ANDREW LOBB, AND DIRK SCHÜTZ face of C(x) is homeomorhic to a disc. Now we use the same notation as used above when we were giving the face structure of C(a) 1. If there is a oint q in the k-face k C(x) with f(q) s such that Ψ t (q) = for some t 1 (res. such that Ψ 1 (q) = ) then we have that is in the k-face of C(a) s. For all other im(γ a,x ) we have that is in the k-face of C(a) s iff it is in the k-face of C(a). Examle Consider the recurring examle of the framed flow category C Ex. TheembeddingΓā, b, asseeninfigure1, embeds(ā, b) C(b)in C(a). However, the embedding rotrudes into the 1-face of C(a). Redefining the face structure of C(a) as in Lemma 2.15 will ush this interval into the -face only, and the result is illustrated in Figure 11. Figure 11. Examle: C Ex. Theresultofredefiningthefacestructure of C(a). So, if (C,ı,ϕ) is a framed flow category containing two objects x and y with (x,y) =, then there is a framed flow category (C H,ī, ϕ) whose objects and moduli saces are given in Definition 2.9. We are now in a osition to state and rove the main theorem of this subsection. Theorem Let (C,ı,ϕ) be a framed flow category containing two objects x and y with (x,y) =. The realisation C is stably homotoy equivalent to the realisation C H of the cancelled category. roof. Therealisation C H isbuiltuinductivelyfromcellsc(ā)forobjectsāof C H with increasing indices as rescribed by the Cohen-Jones-Segal construction. The way in which these cells are attached corresonds to the framed neat embeddings īā, b defined in Lemma Firstly observe that the air of skeleta C (i2) and C H (i2) are identical. Further attaching all cells C(ā) for objects ā with index equal to i1, we have C (i1) = C H (i1) C(y) C H (i1). For objects ā of C H of index m x = i the cells C(a) are attached to C (m1) inductively via the Thom construction in the usual way, corresonding to the identifications Γ a,b : (a,b) C(b) C(a).

21 ORSE OVES IN FLOW CATEGORIES 21 The cells C(ā) are attached to C H (m1) corresonding to the identifications that come from the embeddings īā, b constructed in this subsection. To show that the CW comlexes roduced as a result of these methods are homotoy equivalent, consider the CW comlex X homotoy equivalent to C, and that is obtained from C by collasing the cell C(x) via Ψ t. X = C /(Ψ s (x) Ψ 1 (x),x ( C(x)\int(C y (x))), s 1). CollasingC(x)in C givesadescritionofx asacwcomlexwithtwofewercells than C, and where the attaching mas of X are given in terms of the attaching mas for C and the ma Ψ 1. Indeed, the cell C(a) attaches to X (m1) by alying the Thom construction to the embeddings Γā, b = Γ a,b Γ x,b a,y defined in Lemma In Lemma 2.15 the neat embeddings īā, b of the framed flow category C H are defined via erturbations of the embeddings Γā, b inside C(a). Finally, the Isotoy Extension Theorem ensures that there is a global isotoy in the Euclidean sace E d [b : a] between the two embeddings which extends the isotoy in C(a). This gives a homotoy equivalence C X C H of CW comlexes. 3. The framings of 1-dimensional moduli saces 3.1. Some simle stable homotoy tyes. In this subsection we consider three non-trivial stable homotoy tyes, the last two of which were demonstrated to be wedge summands of the Lishitz-Sarkar sace X Kh (K) for articular knots K in [LS14b]. These saces have low-dimensional reresentatives as based CW comlexes C 2, R 4 /R 1, and R 5 /R 2. We give framed flow categories which give rise to each of these saces. These framed flow categories are the simlest such ossible in the sense that they have the minimal number of objects required to give the cohomology of the associated saces, and their -dimensional moduli saces contain no cancelling airs. Let m 3 and let e m, e m1, and e m2 be three cells for which the dimension is indicated by the suerscrit, and let b be a baseoint. The grou is isomorhic to Z/2. [ e m2,{b} e m ] = [S m1,s m ] = π m1 (S m ) We can give reresentatives for the two elements of this grou in the following way. Let K = S 1 S m1 be an embedded circle. The normal bundle to the circle is then a trivial D m -bundle over S 1. There are two ways to frame this normal bundle u to homotoy equivalence (corresonding to the fact that π 1 (SO(m)) is a 2-element grou). Each framing then determines a ma S m1 S m by the Thom construction, and exactly one of the framings will induce the non-trivial element of π m1 (S m ). In the Cohen-Jones-Segal construction, a framed flow category C consisting of two objects i and i2 gives rise to the stable homotoy tye of a CW comlex consisting of a baseoint b and one cell of each degree m and m 2 for some m >>. (This CW comlex would then undergo(de)susension sothat its reduced cohomology would be suorted in degrees i and i 2). Given that the reduced

22 22 DAN JONES, ANDREW LOBB, AND DIRK SCHÜTZ cohomology of Σ i2 C 2 is 2-dimensional, this is the framed flow category with the smallest number of objects that might give rise to the stable homotoy tye of Σ i2 C 2. The attaching ma S m1 = e m2 {b} e m = S m is given by alying the Thom construction to a disjoint union of framed embedded circles lying in e m2. These framed embedded circles are exactly the framed embedded moduli sace ( i2, i ). In terms of the Lishitz-Sarkar framing conventions, if ( i2, i ) consists of r circle comonents framed and s circle comonents framed 1, then the induced attaching ma e m2 {b} e m is the wedge sum of r homotoically non-trivial mas and s homotoically trivial mas S m1 S m. If the attaching ma is non-trivial(equivalently if r is odd), then the resulting stable homotoy tye is that of Σ i2 C 2 (see Examle 5.6 in [JLS15]). If the attaching ma is trivial (equivalently if r is even) then the resulting stable homotoy tye is of course that of a wedge of oore saces Σ i2 (S 2 S 4 ). A similar analysis alies to the cases Σ i2 (R 4 /R 1 ) and Σ i3 (R 5 /R 2 ) (see Examle 5.5 in [JLS15] for descritions which are a single Whitney trick away from the following descrition). In the case of Σ i2 (R 4 /R 1 ) we take a framed flow category C consisting of three objects i, i1, and i2 (which again is the smallest number of objects that could give rise to the cohomology of Σ i2 (R 4 /R 1 ). The resulting cell comlex is built from the baseoint b and three cells e m, e m1, e m2. Since we know the differentials in the cochain comlex, we know that the signed count of the number of oints in ( i2, i1 ) must be 2 and the signed count of the number of oints in ( i1,) must be. By the Whitney trick then we may assume that the signed count agrees with the absolute count. This forces ( i2, i ) to be a boundaryless framed 1-manifold, in other words a disjoint union of framed embedded circles lying in e m2. As before, if the number of -framed circles is odd then we obtain Σ i2 (R 4 /R 1 ), but if the number of -framed circles is even then we obtain a wedge of oore saces Σ i (R 2 S ). The case of Σ i3 (R 5 /R 2 ) arises from a flow category of three objects i, i1, and i2 in which ( i2, i1 ) is emty, ( i1, i ) consists of two oints with the same framing, and ( i2, i ) contains an odd number of -framed circles (when the number of -framed circles is even then we obtain a wedge of oore saces Σ i1 (S 3 R 2 )) Framing conventions for 1-dimensional moduli saces. The formulae in [LS14b] and [JLS15] require a way to encode the framing information of the 1-dimensional moduli saces. articularly for intervals one has to be quite secific, and here we give a summary of [JLS15, 5]. Recall that the 1-dimensional moduli saces are framedly embedded into some R di [, ) R di1. Circle comonents stay away from the boundary of this Euclidean half-sace. The framing together with a tangent direction of the circle reresents an element of H 1 (SO(d i d i1 1)), a grou which is Z/2 rovided d i d i1 2, as we shall assume. We then assign this element as the framing information of the circle. Remark 3.1. As the tangential direction is taken into account, we get the following curious side effect: if a circle is trivially embedded into R 3 with trivial framing, the

23 ORSE OVES IN FLOW CATEGORIES 23 resulting element of H 1 (SO(3)) is non-trivial. This is because the tangent direction with one normal direction erforms one rotation in SO(2) with the second normal direction being constant. Similarly, if we take a circle as the fibre of the Hof fibration with framing coming from the ull-back of a framing at a oint, the resulting element of H 1 (SO(3)) is trivial. For interval comonents J we also obtain an element fr(j) Z/2 which deends on a coherent system of aths joining the different framings of endoints. We assume that each -dimensional moduli sace is framed using the standard framing, that is, the ositive framings are framed via (e 1,...,e di ) in R di (that is, using the standard basis), and negative framings are framed via (e 1,e 2,...,e di ) in R di. Interval comonents of 1-dimensional moduli saces are therefore framed so that the boundary, when embedded in R di {} R di1 is framed via (e 1,...,e di,e di1,...,e eie i1 ),(e 1,e 2,...,e di,e di1,...,e eie i1 ), (e 1,...,e di,e di1,e di2...,e eie i1 ),(e 1,e 2,...,e di,e di1,e di2...,e eie i1 ) which we denote by,,,, resectively. Definition 3.2. A coherent system of aths joining,,, is a choice of ath ϕ 1 ϕ 2 in SO(m 1) from ϕ 1 to ϕ 2 for each air of frames ϕ 1,ϕ 2 {,,, } satisfying the following cocycle conditions: (1) For all ϕ {,,, } the loo ϕϕ is null-homotoic; (2) For all ϕ 1,ϕ 2,ϕ 3 {,,, } the ath ϕ 1 ϕ 2 ϕ 2 ϕ 3 is homotoic to ϕ 1,ϕ 3 relative to the endoints. Coherent systems of aths exist, we will use the one described in [LS14b, Lm.3.1]. To describe it, we will refer to the first coordinate of R di as the e 1 -coordinate, to the first coordinate of R di1 as the e 2 coordinate, and to the coordinate of [, ) as the ē-coordinate. For ϕ 1,ϕ 2 {,,, } define ϕ 1 ϕ 2 as follows: (i),,, : Rotate 18 around the e 2 -axis, such that the first vector equals ē halfway through. (ii), : Rotate 18 around the e 1 -axis, such that the second vector equals ē halfway through. (iii), : Rotate 18 around the e 1 -axis, such that the second vector equals ē halfway through. (iv),,, : Rotate 18 around the ē-axis, such that the second vector equals e 1 halfway through. The framing of a 1-dimensional moduli sace (z,x) is now encoded in a function fr: π ((z,x)) Z/2, already given on circle comonents, and which is or 1 on interval comonents deending as the interval is coherently framed or not.

24 24 DAN JONES, ANDREW LOBB, AND DIRK SCHÜTZ e e 1 e2 Figure 12. Here we show how to comute the framings of the glued intervals in (ā,ȳ) in the case that a > 1 x. The to left big cuboid reresents a ositively framed oint in (a, x), while the to right reresents a negatively framed oint in (a,x). Also shown in to diagrams are the ends of the intervals in (a,y) which have endoints (,) and (,). The horizontal lane is, as usual, a corner of Euclidean sace. The bottom left and bottom right diagrams show the framing on art of the new 1-dimensional moduli sace in (ā,ȳ) created by gluing the two ends together Gluing formulae for the Whitney trick. If C is a framed flow category containingtwoobjectsx,y with(x,y) = {,}where isframedositivelyand negatively, then we can form the framed flow category C W. We have seen that C and C W are stably homotoy equivalent. For the uroses of comutation, in this subsection we determine how the framings of the 1-dimensional moduli saces of C W can be determined from those of C. roosition 3.3. Let C and C W be as above, and suose that a Ob(C) is such that a > 1 x. Now, (ā,ȳ) is obtained from (a,y) by gluing n airs of endoints of intervals of (a,y) together where n is the number of oints in (a,x). The framing of the moduli saces of (ā,ȳ) can then be calculated by summing the frames of the contributing moduli saces of (a,y), and adding 1 for each oint of (a,x) at which there is a gluing. roof. The situation is illustrated in Figure 12, where we consider a articular oint of (a, x) (which is either ositively or negatively framed) and the gluing that corresonds to it. We can see (in both the ositively and the negative framed case) thattheframingofthegluingregiongivesaathinso(3)whichisarotationaround the e 2 axis by 18 in which halfway through the first vector is ointing in the e