Let us recall two sources of inspiration. Milnor and Wood [Mil58] [Woo71] considered circle bundles S 1 M 3 Σ 2 B = over a closed surface and studied the relationship between the Euler class χ(B) and the existence of a flat connection for different choices of structure group G. When G = U(1) i.e. B is a principle bundle it is well known that there is an integral formula for χ(B) in terms of the curvature form Ω On the other hand if G = PSL(2, R) and represents on S 1 via its action on the circle at infinity of H 2 , the hyperbolic plane, then B has a flat connection iff
(2) |χ(B)| ≤ |χ(τ(Σ))| = |χ(Σ)|
In the boundary case, B = τ(Σ), the tangent bundle, the local trivialization induced by the flat connection identifies tangent spaces at nearby points x, y ∈ H 2 by using geodesic flow to match the unit spheres at both x and y with the ideal circle at infinity.
Wood then showed that (2) also holds even if the structure group is relaxed all the way to Homeo + (S 1 ), the group of orientation preserving homeomorphisms of the circle.
Similarly but in a vastly more general context the Mather-Thurston Theorem [Thu74a] states that for any smooth manifold X (compact, noncompact, bounded, no boundary, of any finite dimension) the natural map
induced by id : Homeo δ (X ) → Homeo(X ) from discrete to compact open topology is an isomorphism on homology H * (; Z). By the Atiyah-Hirzebruch spectral sequence (3) also induces an isomorphism on bordism so the geometrically minded statement, in lowest regularity C 0 , of the MT theorem is as follows. Assume all manifolds have smooth structure. 1 Let V p be a p-dimensional manifold, possibly with boundary, and X q a q-dimensional manifold (also may have boundary) and
V a fiber bundle with transition functions, i.e. structure group = Homeo(X ). The low differentiability form of MT we consider is:
Theorem 1 (Mather-Thurston [Thu74a] ). Suppose B possesses a C 0 -transverse foliation F 0 over a neighborhood N 1 ∂V . Then there is a cobordism (W ;V ;V * ) from V to V * , constant near ∂V and covered by cobordism of bundles (B; B, B * ), so that B * possesses a C 0 -transverse foliation F agreeing with F 0 on a smaller N 0 (∂V ) ⊂ N 1 (∂V ). If B has structure group Homeo 0 (X ) we may arrange that this is also the structure group of B.
Definition 1. We treat as synonymous "C 0 -transverse foliation," "topological transverse foliation," and "topoloigcally flat connection." But we use these phrases both for the C 0 and bilipschitz contexts. Which one is applicable will be clear from the theorem statement; whether they are in the continuous or bilipschitz categories. In the discussion immediately following Theorem 2 of [Thu74b] , Thurston refers to work of his, apparently never published, that bilipschitz foliations of any codimension q on a smooth manifold may be (topologically) isotoped so that each compact portion of each leaf is actually a C ∞ -submanifold. Thus it is only the separation between leaves, as seen in holonomies, that might not be differentiable, and not the leaves themselves. Unfortunately the argument may be lost. We remind the reader of this claim because, if true, it means that we should not think of bilipschitz foliations as particularly wild, and this, in fact, is the natural category for the constructions in the present paper, despite our choice to make theorem statements in the more familiar C 0 category. It is only the holonomies which are not necessarily smooth. So topological foliations should be thought of as "wild" objects.
The original proofs hold if "Homeo" and "C 0 " are both replaced by "bilipschitz." Subsequently, Tsuboi [Tsu85] showed that the MT theorem also holds replacing Homeo(X ) with Diff 1 (X ). Possibly, we could also work in that context, but do not know. Actually we work in the intermediate bilipschitz category, the most natural one for the constructions. We will not change notation but from here forward "Homeo" can always be assumed to be bilipschitz. All statements involving "C 0 " or "Homeo" can be read literally or with Homeo and C 0 everywhere replaced by bilipschitz.
We will explore, in several contexts, adding control to the classical Mather-Thurston theorem, on cobordisms of a general bundle X B
V to a flat bundle
The first refinement is that the cobordism (W ;V,V * ), over the 3-dimensional base V , can always be taken to be a semi-scobordism. In higher dimensions the same statement is true after a stabilization by any product of spheres ∏ J j=1 S i j , i j ≥ 1, with sufficiently many factors J. These results depend on the model C 1 with its π 1 -representation in the bilipschitz category, so they only apply up to that degree of differentiability. Possibly, following [Tsu85] this might be pushed to class C 1 , but this is presently unclear. Because it is more familiar we will write expressions like Homeo 0 (X ), literally meaning the identity component of the group of homeomoprhisms of a manifold X . All statements are correct if read in that way but the reader may mentally replace all occurences of "Homeo" with "bilipschitz" and also have correct statements. Similarly class C 0 can be taken to mean bilipschitz. But not the other way around; if we state a structure is bilipschitz it is because we need that assumption.
One simplification in working with differentiability ≤ class C 1 is that the classifying space for bundles and bundles with Haefliger structures 2 (on a bundle, that is on the bundle of vertical tangents to the original bundle) are homotopy equivalent:
and thus the homotopy fiber is contractible. This expression is often written for the fiber space X = R d (and Γ 0 q defined as the homotopy fiber) but remains true inserting any manifold fiber X [Thu74a] . The homotopy equivalence means that although we draw heavily on the theory of foliation, chiefly [Mei20] , we only encounter Haeflinger structures within certain proofs and not in theorem statements.
From the low differentiability statement of Mather-Thurston (MT) we prove two refinements. Later we consider the case of a Lie group principle bundle, and adapt the flattening over V * to that initial condition.
Before stating our theorems we need to define a highly refined, directional, type of cobordism called a semi-s-cobordism (ssc).
Definition/Discussion. An absolute semi-s-cobordism (ssc) is a manifold triple (W d+1 ;V α ,V * α ) with ∂W = V ⊥ ⊥ −V * , so that there is a simple deformation retraction r : W → V (so that r • inc V s ≃ 2 In this context, a Haeflinger structure is a C 0 -foliation on Vert, the bundle over B of "vertical" tangents, i.e. tangents to the fibers of X, which is transverse to the fibers of Vert. id V ). In the relative case we allow a common boundary ∂V = ∂V * , near which W is assumed to be a product.
Because r is a (simple) homotopy equivalence all the relative groups H * (W,V ; Z[π 1 (V )]) ∼ = 0, so by Lefschetz duality H * (W,V * ; Z[π 1 (V )]) ∼ = 0 so the map r • inc V * : V * → V is "almost" itself a homotopy equivalence-it would be by Whitehead's Theorem if (r • inc V * ) # : π 1 (V * ) → π 1 (V ) were an isomorphism. As it is we have an exact sequence
In this context Quillen's plus construction [Quil71] (which attaches an equal number of 2-and 3-cells to kill a perfect group while leaving homology unchanged), when applied to the degree one map r • inc V * : V * → V yields V * + homotopy equivalent to V . In fact, V * + s ≃ V since the vanishing of Wh(W,V ) implies the vanishing of the Whitehead obstruction Wh(W,V * ) by duality [Mil71] .
A comment which will soon be useful is that any concantination of ssc
is itself a ssc. The simplest example of an ssc is obtained by taking a compact contractible manifold K n and deleting a ball from its interior;
Warning. It is tempting to denote the Quillen construction by V * − s ≃ V , rather than V * + s ≃ V , because the left boundary V of W is the simpler space, whereas the right boundary V * has been enhanced by a perfect extension. This notation would help keep straight the simple and more elaborate boundaries of W . Also in the context of manifolds of dimension ≥ 5 Quillen's construction can be acomplished by an equal number of 1-and 2-surgeries, in that context cells are not being added, removing that justification for the + notation. But such a change seems on par with changing the sign of the electron (clearly too late); so we stick with the + sign.
The innovation in Theorem 2 below is that the assumed cobordism of the base is "directional." There is the problem end V , which a physicist would think of as the IR manifold, and a solution end V * , the UV end, in which the fundamental group has been elaborated by a perfect extension over which the solution becomes possible. We hope 3 and expect that this will turn out to be a hidden feature of most c-principle theorem where homotopy (h-principle) by itself is inadequate, but the obstructing singularities can be removed by surgery, i.e. by cobordism.
Our new idea on the c-principle is that while holonomy around the boundary of a 2-disk D 2 has traditionally been dealt with by replacing D 2 with S g , a surface of high genus and a single boundary component, a radical change of topology, when the base dimension is ≥ 3, it is often possible to hide the additional genus inside a semi-s-cobordant manifold, requiring only a subtle change in topology.
This paper came together during the isolation of COVID-19. But I always felt surrounded by good friends. I acknowledge with pleasure the mathematicians and physicists who generously shared their thoughts and insights into this project: Parsa Bonderson, Adam Brown, Danny Calegari, Bob Edwards, Yasha Eliashberg, Slava Krushkul, Roman Lutchyn, Chaitanya Murthy, Gael Meigniez, Sam Nariman, Cliff Taubes, and Shmuel Weinberg.
We begin with statements that add control of the bordism W beyond the usual MT Theorem. Theorem 2 is actually a special case of theorem 3 but we state it separately since its proof uses the proof of MT only as a black box. Theorem 3 is more general but requires looking inside Meigniez's proof of MT at a high level of detail. Theorem 4, also with a black box proof, provides in a sense one half of the desired (ssc) information on W ; the retraction r but not the deformation.
The next theorems, 5 and 6, assume that the data on ∂V is of a highly refined nature such as a flat Lie-connection on G-principal bundle or in other contexts an associated bundle. The goal is to find a ssc to a flat topological connection over V * , keeping a basis of transition functions for the flat structure over V * close (in the norm topology) to G inside Homeo 0 (X ), the identity component of the group of homeomorphisms of the fiber. (In the principal bundle case, X = G.)
Unfortunately Theorems 5 and 6 apply only to three dimensional bases V . Extenting them to higher dimensions is our most important unfinished business and presently only a conjecture.
Theorem 2. Suppose dim(V ) = 3 and that the fiber bundle B = X E V has structure group Homeo 0 (X ) and a C 0 -transverse foliation F 0 over N 1 (∂V ), a neighborhood of boundary V . Then there is a ssc (W ;V,V * ) from V to V * , constant near ∂V , such that V * possesses a C 0 -transverse foliation F agreeing with F 0 on some smaller neighborhood N 0 (∂V ) ⊂ N 1 (V ).
It is helpful to think of Theorem 2, and the others, in the language of problem solving. The initial bundle B over V with foliation F 0 near ∂V is the "problem" prob and the "solution" is the bundle at the opposite end V * of the cobordism, or semi-s-cobordism (ssc), with F 0 extending from a germ of ∂V to F over all of V * . We think of the poblem as flattening an initial bundle while changing the base as little as possible and the boundary conditions F 0 not at all. Given a problem prob and a manifold without boundary Q we can pose a new stabilized problem, prob×Q which is the bundle
Note that the original foliation and its pull back are both codimension = q = dim(X ). When we speak of a solution to a stabilized problem there is no requirement that the solution has any product structure.
Theorem 3. Suppose B has structure group lying within bilipschitz(X ), but no assumption of lying within the identity component, and that there is a transverse foliation F 0 of class at least bilipschitz over a neighborhood ∂V . This is our problem, prob. If dim(V ) = p = 3, then prob is ssc to a solution of bilipschitz class. If dim(V ) = p ≥ 4 then for Q = ∏ J j=1 S i j , i j ≥ 1, prob × Q is ssc to a solution of bilipschitz class. The number of factors J is a function of the problem data and can be enormous, but the choice of the factor sphere dimensions i j ≥ 1 is completely arbitrary. (There is no improvement to MT if dim(V ) = 2.) Note. When Theorem 3 is restricted p = dim(V ) = 3 there is a subtle difference with Theorem 2: the structure group has changed from Homeo 0 (X ) to bilipschitz.
One may hope that the stabilization in Theorem 3 for p ≥ 4 is unnecesary, as we know no counterexamples. The next theorem solves the initial problem without stabilization but with less control on the cobordism W .
Theorem 4. Suppose dim(V ) ≥ 3 and B has structure group Homeo(X ) (or bilipschitz(X )) and that there is a C 0 -transverse foliation near ∂V . Then there is a solution, a cobordism (W ;V,V * ) covered by bundles to a solution X B * V * with a topologically flat connection, and with W admitting a retraction r :
Note. A consequence of the retraction r is the existence of the degree 1 map r • inc V * : V * → V . Degree one maps induce a well studied partial order on manifolds, e.g. the cohomology of the target always injects into the source, and π 1 (V * ) → π 1 (V ) is surjective. So as with a ssc, V is the "simple" and V * the "complex" end. Also as with ssc, finite composition of retracting cobordisms are also retracting.
Next we state theorems which add control over the structure group of the solution.
Consider the example of a principal bundle P = G E V . If G has a bi-invariant metric, 4 then E has a Riemannian metric under which all gauge holonomies are isometries of the fiber G. We are interested in the problem case where F 0 , near ∂V , is a flat Lie(G)-connection. We would like the solution over V * to extend F 0 , if not to a precisely Lie(G)-connection, which is typically impossible, but to a topologically flat connection over V * where all bundle holonomies across a fundamental domain for V * lie in a small neighborhood N(G),
the neighborhood being taken in the norm topology if G is noncompact. Said another way, the solution (V * , ρ), ρ : π 1 (V * ) → bilipschitz(G) will be constructed so that π 1 (V * ) comes equipped with an inversion-symmetric 5 generating set S so that
Theorem 5. Let G be a connected Lie group with a fixed bi-invariant metric, and Lie algebra g. Let (V, Σ) be a compact 3-manifold with boundary Σ and G-principal bundle with a fixed flat 4 This exists for any compact G, some non-compact groups such as Euclidean and hyperbolic transformations, but not SL(2, R), for example. 5 All π 1 -generating sets S are henceforth assumed to be symmetric.
g-connection A i.e. a highly structured F 0 . Then for any norm topology 6 neighborhood N(G) ⊂ bilipschitz 0 (G), the inclusion given by left multiplication, one can find a ssc (W,V,V * ) identity near boundary, and an extension of G B
V to bilipschitz 0 (G) bundle over ∂W = (∂V ) ×I ∪V ∪V * so that restricted to V * , this bundle G B * V has a topologically flat connection in bilipschitz 0 (G) whose associated representation ρ : π 1 (V * ) → bilipschitz 0 (G) satisfies ρ(S) ⊂ N(G) for some generating set S. Note. A defect in this theorem is that the construction does not naturally produce a bundle cobordism B over W . However, in many cases this can be remedied after the fact: For example if G ∼ = SU(2), the group bilipschitz 0 (SU(2)) = bilipschitz 0 (S 3 ) ≃ Diff + (S 3 ) ≃ SO(4), by canonical smoothing of 3D homeomoprhisms [Ham76] and Hatcher [Hat83] so the only obstruction to filling in B is a relative second Siefel Whitney class H 2 (W, ∂W ;
In surgery theory it is not uncommon to guess the answer and construct the normal cobordism after the fact: "first solve, then show your work."
There is a sharper conclusion if we assume that the boundary data over Σ bounds some flat g-connection, i.e. a connection in the visual sense on a G-principle bundle.
Theorem 6. Let G be a connected semi-simple Lie group with Lie algebra g. Let (V, Σ) be a compact 3-manifold with boundary Σ, and B a principal G-bundle with a fixed flat g-connection A which is trivial in H 2 (G δ , Z) i.e. bounds a flat principal G-bundle over some V ′ , ∂V ′ = Σ. Then there is a ssc (W,V,V * ) and an extension B of B over ∂W with B possessing a flat g-connection extending A on Σ.
Note. Again we do not conclude that B is the restriction of a bundle B (of any flavor) over W .
Extensions: Theorems 5 and 6 are special cases of what we actually will prove. In both cases it is not necessary that the bundle B be a G-principal bundle with flat g-connection near the boundary. Instead all that is required is that the fiber X be a manifold with a point set (not necessarily Riemannian) metric and the monodromy of the topologically flat connection lie in the identity component of the group of isometries I 0 (X ) ⊂ bilipschitz 0 (X ). By classical theorems of Montgomery and Zippin, I 0 (X ) is a Lie group and that forms the bridge. Call these extensions Theorems 5 ′ and 6 ′ .
The last theorem (7) quantifies the Milnor-Wood inequality for circle bundles over surfaces, by considering the trade-off between adding lots of genus to the base and keeping the transition functions close to U(1), in the sense of Theorem 5, and keeping the genus lower and allowing transition function (like large boosts from special relativity) which are quite far from rotations. Stronger than in Theorem 5, we can keep the transition function within the Lie group SL(2, R) ⊂ Homeo 0 (S 1 ). The embedding via the usual representation on the hyperbolic disk model. Consider the problem of imposing a sl(2, R) flat connection on a circle bundle B with Euler class χ(B) over a base surface Σ g of genus g. Let g χ(B) (ε) be the smallest genus so that there is a generating set S for π 1 (Σ g ) so that B admits a topologically flat connection with representation ρ : π 1 (Σ g ) → SL(2, R) so that ρ(S) ⊂ N ε (U(1)) ⊂ SL(2, R). Define g χ(B) (ε) similarly by replacing SL(2, R) with Homeo 0 (S 1 ). 7 Clearly g χ(B) (ε) ≤ g χ(B) (ε). It seems reasonable to guess that they are actually equal.
Theorem 7. The function g χ(B) (ε), for ε > 0, is defined into the natural numbers union 0, N + . For ε sufficiently small it obeys the upper bound:
where the error term satisfies
One might ask for a rather strong converse of the form:
, for some c > 0, perhaps with c = 2π. But we have only been able to prove such a statement where the denominator is ε (not ε 2 ), and S is restricted to the standard generators for π 1 (Σ g ). So this question is quite open.
Conjecture 2, below, proposes in a sense a vast generalization of Theorem 7 to higher dimensions and with the additional feature that when dim(base) ≥ 3, it may be possible as in Theorems 2 and 3 to hide the requisitve genus inside a perfect group.
Before stating a broad (and optimistic) conjecture which would cure all defects we see in the theorem statements, and more, we give the simplest example of what we would like to know, but don't.
Example: Recall that the 't Hooft instanton, also known as a generalized Hopf fibration S 3 S 7 S 4 with structure group SU(2). We hope that for every neighborhood N(SU(2)) ⊂ bilipschitz 0 (S 3 ). There is a homology 4-sphere H so that the pull back bundle under the degree one map H → S 4 ,
H admits a topologically flat connection given by ρ : π 1 (H) → Homeo 0 (S 3 ) so that for some generating set S, ρ(S) ⊂ N(SU(2)). This seems to require a new idea.
V be a bundle with structure group Homeo(X ), where X is any manifold with a (point set) metric, and the bundle possessing near ∂V a topologically flat connection F 0 with holonomy lying in I(X ), the group of isometries of the fiber, and let N be any norm-topology neighborhood of I(X ) in Homeo(X ). Then there exists a ssc (W ;V,V * ) covered by a bundle B with structure group Homeo 0 (X ) to a bundle B * V * possessing a topologically flat connection inducing a representation ρ : π 1 (V * ) → Homeo 0 (X ), with the property that ρ(S) ⊂ N for some symmetric generating set S for π 1 (V * ).
We will need a somewhat more controlled version of the Fisher-Epstein Theorem [Fis60, Eps70] .
Fisher actually worked under a stability assumption which became redundant in 1969 when Kirby [Kir69] proved all homeomorphisms are stable as part of his work on the Annulus Conjecture. 8 While adding some control we rely heavily on Fisher's identities. Again all constructions may be done in the bilipschitz category.
In this section ε > 0 will mean a constant, not a function, even if the fiber X is non-compact. In both compact and non-compact case we give X a (point set) metric, and Homeo(X ) the L ∞ -norm topology. A homeomorphism is near id if it moves all points less than a constant ε > 0. If X is compact, of course the norm topology agrees with the compact-open topology but is stronger in the non-compact case. It is immediate that the L ∞ -norm is a bi-invariant metric when restricted to the subgroup of isometries I(X ).
The next proposition is stated in terms of ε ′ and δ , ε ′ >> δ , because it will soon be deployed at a large scale ε >> ε ′ . The proof requires a fine handle decomposition H , all handles have diameter < ε ′ 2 and are equipped with a base point * at (0, 0) ⊂ D q−i × D i . Denote the union of base points by * and fix some homeomorphisms h, supp(h) ⊂ N δ ′ ( * ) which displaces each base point δ ′ a sufficiently small amount δ ′ << δ , so that the 2δ ′ ball around each * lies within its own handle.
, the δ -neighborhood of the identity, then f may be written as a composition of 4n + 4 conjugates of the form: h γ i and (h −1 ) γ i where all the γ i lie in N ε ′ (id) and h is the homeomorphism specified above. n = dim(X ).
Definition 2. For us, a ε-net is merely a discrete collection of points N ⊂ M so that for all m ∈ M, dist(m, N) < ε.
Proof of Proposition 2. Let f ∈ N δ be as in Proposition 2. Using sequential handle straightening ( [EK71] and [Quin82] for the four dim case) we may write f as a composition f = f n • · · · • f 0 of homeo's f i supported near the i-handles, where each successive f i in the composition is somewhat less well-controlled. For example, we can arrange each f i ∈ N 2 i δ (but not f i ∈ N δ ). We will need ε ′ >> 2 n+1 δ . Edwards and Kirby achieve this by successively using the torus trick to construct small topological isotopies bringing f to the identity on (thinned) version of the union of the handles of index
. 0, 1, 2-handles and their "thin" versions. Note that the "thinned" handles are slightly longer in core directions and shorter in co-core directions, giving rise to the name "thinned". The thinned 2-handle (center) is not drawn but would be a larger square, because there is no co-core direction in highest index handles.
Let the homeomorphism h be as above. It is easy to construct n + 1 conjugates h α i , 0 < i < n, of h with the property that We now propose to write f i , supported on H + i , as a product of four conjugates of h:
where r and φ are defined with the help of Figure 2 (see [Fis60] or [Mat71] for specific formuli). The question mark reminds us that we still must check that f i has a representation of this form. In Figure 2 , we draw α i (H + i ) as a single ball although it is actually a discrete collection balls. First we create a geometric series of balls beginning with α i (H + i ) and converging rapidly to a point; call the series α i (T i ) Let r i be a homeo of X which shifts the series one step right with the limit point fixed. φ is defined from α i and the shift map. It installs a conjugate of f i in each term of the series
We may rewrite (7) as a commutator of r i and ((h
have disjoint supports and therefore commute (see Fig. 3 ). Thus shifting r α i to the left in line (8) and canceling yields: (7). Thus f is a composition of at most 4(n + 1) conjugates, as claimed.
Note. We used ε ′ in the statement of Proposition 2 because in applications there will be three scales ε >> ε ′ >> δ .
They will be C 1 , C 2 , C 1 , C 2 , and their compositions C 1 , C 2 , C 1 , and C 2 . Both C 1 and C 2 are homology solid tori together with C 1 and C 2 fixed cobordisms, rel. boundary, to the standard solid torus S 1 × D 2 , and each will come with a homomorphism ρ i : π 1 (C 1 ) → (a relevant structure group), i = 1, 2. C i is the closed complement of certain knot k i in a homology 3-sphere Σ i , and the corresponding homology cobordisms C i will be the closed slice complements for k i in a corresponding homology 4-ball, B i .
Let P 1 and P 2 be the Siefert fibered manifolds over S 2 with (2, 3, 7) and (2, 3, 5) multiplicities, respectively, describing the (three) exceptional fibers. P 1 is an SL(2; R)-manifold whereas P 2 is spherical (actually it is the Poincare homology sphere). Let P − i , i = 1, 2, denote the punctured manifold. Let (γ 1 , ∂ γ 1 ) be any essential embedded arc in (P − 1 , ∂ ) and let (γ 2 , ∂ γ 2 ) be any embedded arc in (P − 2 , ∂ ) representing an element of the binary icosahedral group BI ∼ = π 1 (P 2 ) of order > 2. Note that π 1 (P 1 ) is torsion free, being an infinite cyclic extension of a hyperbolic triangle group.
For i = 1, 2, let Σ i = P i # − P i be the connected sum of the homology sphere and its mirror image,
We may view C as a relative H * -cobordism from C i to S 1 × D 2 . C will be useful in building cobordisms W 0 .
There are natural maps:
order > 2 and α[m] trivial, m the meridian of k and l the longitude. π 1 (P i ) are central extensions of the (2, 3, 5) and (2, 3, 7) triangle groups respectively. For i = 1 π 1 (P i ) is torsion free so the α[l] is merely required to be nontrivial. For i = 2 the aim is to exclude the central element of π 1 (P 2 ) ∼ = BI, the binary icosahedral group. This is easily done by avoiding that element when choosing γ 2 .
Given a homology sphere P there is a "spun" homology sphere Q in one higher dimension with π 1 (P) ∼ = π 1 (Q). Q is an open book with fiber P − and identity monodromy:
Consequently the preceeding set of examples can be extended to all dimensions: Let P k i , k ≥ 3, be P i spun to dim k, i = 1, 2, so P 3 i := P i . Σ k i , B k i are the corresponding doubled homology spheres and homology ball and
i is a homology S 1 × D k which may be viewed as a Z-homology cobordism, constant over the boundary, to the standard S 1 ×D k−1 . Line 10 continues to hold except that when k > 3, there is no meridinal class in π 1 (C k i ) since the meridinal boundary factor is S k−2 , which is simply connected. This explains the capital models. Next we discuss their composition, denoted by the corresponding script letters.
We will use two basic operations to build C from C and C from C. The first is (iterated) longitudinal boundary connected sum, which we call longitudinal sum. For the homology solid tori (in any dimension C := (S k−2 × D 2 ) H ) with boundary ∼ = S k−2 × S 1 . (When the dimension k = 3 a normal framing specifies the "longitude" and fixes the identification of ∂ (S 1 × D 2 ) H with S 1 × S 1 .) In general the sphere S k−2 -factor is the longitude and if I ⊂ S 1 is a fixed interval the longitudinal sum is a gluing
θ the reflection on I. Similarly:
where the last interval factor is normal to C in C. This is illustrated in Fig. 6 for k = 3 in the case of 8-fold and 2-fold iterations. The second way in which we compose homology solid tori is "Bing doubling," implanting a pair of C's in a standard S k−2 × D 2 along a Bing double of its core, or in dimension k > 3 spun Bing doubles (see [Kru18] and our Figures 3 and 5) . The Bing doubling operation relates easily to group commutators, whereas the first operation relates to group multiplication. We actually require ramified Bing doublings [FQ14] , a simple extension corresponding to products of commutators, explained below. When bars are present, C → C , Bing doubling is not replacement, but the attachment of two homology cobordisms (C,C) along Bing pairs. On the boundary of this attachment agrees with the former operation of replacement.
We change capital letters to script to indicate that our models have been composed according to longitudinal sums and Bing doubling.
For us the use of these models, particularly the un-barred C i will be to fill in what Thurston [Thu76] calls holes in foliations, which later morphed into the theory of clefts [Mei20] .
For the proof of Theorem 2, the MT Theorem 1 may be treated as a black box; we just need the statement that W exists and then use surgery techniques and the models C 1 and C 1 to improve W to a ssc while retaining the flat connection over its right end.
In contrast, the proof of Theorem 3 requires following in some detail Meigniez's proof of MT [Mei20] and intervening at the correct moment with models C l 1 and C l 1 , for various dimension l. Proof of Theorem 2. Recall the output of the MT Theorem is a cobordism of X -bundles:
Give W 4 a smooth handle decomposition relative to V and cancel any 0-handles and 4-handles (without changing W ). Since V (and V * ) may be presumed connected, any 1-handle (and any 3handle) may be traded for a 2-handle at the expense of modifying W by a 1-surgery. It is easy to modify interior W in this way but we need carry the bundle cobordism B along. This is one of two places the proof will use that the structure group of B and hence B is Homeo 0 (X ). This guarantees that B| γ is trivial for any simple closed curve γ (scc) in W , allowing the 1-surgery to be covered by a relative cobordism of bundles (B; B; B ′ ) over a (5D) cobordism of W . Three handles are dealt with as 1-handles by turning W upside down. At this point we have reduced to the case where W has handles of index 2 only. The attaching regions of these 2-handles determine a framed link L ⊂ V and dually the 2-handle co-cores determine a framed link L * ⊂ V * . Framed surgery along L produces V * from V , and dually framed surgery along L * produces V from V * . If we could actually do surgery on L * and propogate the topologically flat connection across the surgery we would have solved the flattening problem without changing V by a cobordism. This cannot be done, generally, because surgery bounds each framed longitude of L * by a disk D which can only be covered by a flat bundle if the holonomy around the longitude, ∂ D, is trivial. Traditionally, this problem is solved by replacing D with a genus g surface with circle boundary S g and exploiting the commutator structure of the boundary. But replacing a disk D with a surface S g adds homology and is a much coarser modification than replacement by a ssc manifold. While we do need the commutator structure of π 1 (S g ), in dimension ≥ 3, we are able to hide it in a ssc manifold. So what we do instead is a homologically more subtle replacement: There is a homological version of surgery on L * employing the models C 1 (and C 1 ) which does propogate the flat connection to V * and changes, in the end, V only slightly, by a ssc. See Fig. 4 to visualize these manipulations, and for a pictorial summary of the proof plan. Next we explain how the models C 1 are constructed to permit an extension of the topologically flat connection (i.e. the representation to Homeo 0 (X )), and why the cobordisms indicated in Figure  3 and the discussion above, is in fact a ssc.
To apply Proposition 2, we need an action of π 1 (C 1 ) on Homeo 0 (M), factoring through a homomorphism ρ : π 1 (P 1 ) → Homeo 0 (M) so that for some element α ∈ π 1 (P 1 ), ρ(α) has the property of h in Proposition 2, that for a fine net * , ρ(α) * ∩ * = ∅. Actually we now construct a ρ so every α = e ∈ π 1 (P 1 ), ρ(α) has this property.
As Thurston [Thu74c] observed, π 1 (P 1 ) is naturally a subgroup of PSL(2, R), the universal cover of PSL(2, R). PSL(2, R) is the group of oriented isometries of the hyperbolic plane H 2 and acts faithfully on the circle at infinity S 1 via Möbious transformations. Thus PSL(2, R) acts smoothly on the real line R = S 1 , and taking the end compactification we obtain a continuous (and bi-Lipschitz) action of PSL(2, R) on the closed interval I := [0, 1]. This action is faithful. It is easy to promote this action to a representation π 1 (P 1 ) → Homeo(D n ; id on ∂ ). To do so implant this action on an interval I ⊂ S 1 , then suspend and delete a fixed disk; one obtains a faithful action on D 2 . Iterating implantation, suspension, and deletion, one obtains a faithful action on the n-cell D n .
Finally, given the "net" * ⊂ X , we produce h as in Proposition 2 by allowing ρ to act on a small ball around each point * ∈ * . Then for any α = e ∈ π 1 (P 1 ) the composition ρ(α) (12) ρ : π 1 (P 1 ) → PSL(2, R) → Homeo 0 (X ), α → ρ(α)
has the property of h in Proposition 2.
We construct the ssc W as V × I ∪W 0 , where W 0 , built from a collection of C 1 (call it W 0 = C 1 ), is itself a ssc on N ∼ = ∏ S 1 ×D 2 , a neighborhood of L * , N ֒→ C 1 ←֓ C 1 r ← − (constant on the boundary), so that V * = (V \ N) ∪ (⊥ ⊥ C 1 ) has a topological flat connection agreeing with the restrictions to the flat connection A over ∂ (V \ N) .
The connection A is trivial on a meridian to L but in general is nontrivial on the framed longitude l i . Conventional surgery would bound each l i by a disk over which the connection cannot extend flatly (unless the holonomy around l i is trivial). To solve this problem we instead do a form of homology surgery explained below.
Since our model cobordism (C i ;C i , S 1 × D 2 ) is a homology product with Z-coefficients, but not group ring Z[Z] coefficients, we next use a trick, Bing doubling, to isolate each copy of C from the fundamental group, as illustrated in Figure 5 .
Surgery will glue in models schematically as in Fig. 5 but with additional ramifications 9 : For a component l ofL, the holonomy hol(l) along l that must extend over C lies in the identity component Homeo 0 (X ). Using local contractibility of Homeo 0 (X ) [EK71] , by subdividing an isotopy hol(l) can be written as a product of small elements
Thus by line (9) hol(l) is a composition of (n + 1)m commutators of the form:
and r i , since it is in N Homeo(X) ε ′ (id), can, like g i , also be expressed (by Proposition 2) as a product of 4n + 4 conjugates of h (or h −1 ) with the conjugate elements now in a larger neighborhood N Homeo(M) ε (id). This is why we previously used the notation ε ′ . (We now see the three widely 9 Ramification (see [FQ14] ) is a technical term for enhancing one Bing pair, as drawn inside the larger solid torus in Figure 5 , to k such pairs. Each pair is successively closer to the boundary; look ahead to Figure 6 . Note that this differs from taking parallel copies of the components of an unramified Bing double. separated scales of the argument: ε >> ε ′ >> δ ). The upshot is that we may now write g as
where the β j are on the ε-scale and the other conjugators are on the ε ′ scale. This leads us to the ramified version of Fig. 5 (Fig. 6 below) necessary to create the model (C 1 ; C 1 , S 1 × D 2 ) required to extend hol(l i ) = ∏ m i=1 g i on the meridianm i of Figures 5 and 6 over C 1 , the homology solid torus used to complete surgery. We have used script C 1 to denote the longitudinal sums and Bing doubling needed to pass from the base model C 1 to the solution to the extension problem. Similarly, we will write C 1 for the corresponding 4D ssc.
Each Bing pair consists on one side of an 4n + 4-fold longitudinal boundary connected sum of homology solid tori C 1 and on the other side of a 2-fold longitudinal boundary connected sum, corresponding to the multiplicities inside the bracket on line (15), and further indicated in Fig. 7 .
The link complement in Fig. 6 is that of a (n + 1)m-ramified Borromean ring. With the addition of the (true) relationl = e to the fundamental group of the link complement (corresponding to the triviality of the A-holonomy along the meridians to the L) the group becomes freely generated by meridinal loops a j and b j to the 2(n + 1)m components in Fig. 6 . This is because filling the hole in Fig. 6 results in a (2n + 2)m-component unlink of solid tori. Notice the presence of the surface relation:
This is what we earlier referenced as the relationship between Bing doubling and the commutator structure on ∂ S g .
Thus equation 16 can be implemented geometrically by filling the (2n + 2)m deleted solid tori in Fig. 6 with copies of the model C 1 , thus realizingm i on the boundary of a homology solid torus C 1 with a sufficiently elaborate fundamental group that g i = hol A (l i ) = hol A (m i ) extends:
To understand how this works we must explain the role on conjugation and inverse. Inverses of holonomies are implemented by applying the automorphism −1 0 0 −1 to the torus ∂C 1 . Up until now we have been a little careless about base points. Holonomies are really just conjugacy classes until base points are introduced. How does one describe the composition of meridinal holonomies g 1 and g 2 when two solid tori T 1 and T 2 are summed along a common longitude to form a third?
Since in all cases the longitudes represent to the identity, the problem dimensionally reduces to asking what the pinch map p : S 1 → S 1 ∨ S 1 represents given that the two wedge factors (petals) separately represent to g 1 and g 2 respectively. The answer is that we have considerable choice. By dragging the base point of S 1 ∨S 1 around an arbitrary element of τ ∈ Homeo(X ) we can realize any conjugacy class of the form [g 1 g τ 2 ]. This key idea, the "simplicity trick," undergirds our "designer extension" of arbitrary meridinal holonomies.
To complete the proof of Theorem 2 we need to describe W 0 (and W ) and verify they are ssc.
To build the 4D bordism W 0 one must take longitudinal sums of the slice-complement C, extending along an additional interval factor the longitudinal boundary connected sums of copies of C. These continue to be relative H * -cobordisms on (S 1 × D 2 , ∂ ). Schematically the cobordism W is shown back in Figure 3 , where the C 1 are represented as thin hatched strips although, in reality, they are ssc.
This completes the construction of W and the extending flat connection on V * = ∂ + W . To see that W r − → V give it the structure of a semi-s-cobordism, note that Figure 3 describes a product except for the hatched strips. The hatched rectangles are of the form: (C 1 ; C 1 , S 1 × D 2 ). S 1 × D 2 is a terminal object 10 in category of three manifolds with boundary S 1 × S 1 and degree one maps, identity on boundary. Using obstruction theory we construct a relative retraction:
While r is not a deformation retraction because π 1 (C 1 ) ∼ = π 1 (S 1 × D 2 ) ∼ = Z, it is a homology isomorphism with integral coefficients.
Because the blocks C 1 are glued to Bing doubles and therefore map trivially into π 1 (W ), the inclusion V → W induces a Z[π 1 ]-homology isomorphism, π 1 = π 1 (V ) ∼ = π 1 (W ) (but distinct from π 1 (V * ) which is much larger). This shows that V → W is a homotopy equivalence. The point here is that triviality of this map means that π 1 (W ) acts on lifts of C 1 only by permuting blocks. It is actually a simple homotopy equivalence since lifted to the universal cover the inclusion of the nonproduct regions, the inclusions: (S 1 ×D 2 ) lifted copy ֒→ C lifted copy are all Z-homology isomorphisms.
As we go beyond dim(V ) = 3 the strategy which proved Theorem 2 will not work. It is true that we may use our models C 1 to "homologically" remove relative 2-handles from the V * end of a cobordism (W ;V,V * ). But because they are only deleted in a homological sense, the attaching regions of the 3-handles do not reach the new V * and we are stuck.
Proof of Theorem 3. The plan is to follow Meigniez's proof of MT via quasi-complementary foliations up to its final step [Mei20] . In this proof, the problem is now not just a bundle but also a 10 Explicitly to construct the degree one map (C 1 , ∂ ) → (S 1 × D 2 , ∂ ), recall K = ker(H, (∂ C 1 ; Q) → H 1 (C 1 ; Q)) ∼ = Q (a consequence of Lefschitz duality and the sequences for pairs). Lifting K into H 1 (∂ C 1 ; Z) we find a 1-submanifold in ∂ C 1 bounding a 2-submanifold ∆ ⊂ C 1 . Map ∆ degree one to * × D 2 ⊂ S 1 × D 2 and then extend (degree one) (C 1 \ ∆, ∂ ) → (S 1 × D 2 \ * × D 2 , ∂ ). Note that ∂ (C 1 \ ∆ 1 ) is connected. More generally, higher genus handle bodies are also terminal objects in the corresponding sense.
Haefliger structure on that bundle. Meigniez treats a range of smoothness from bilipschitz to C ∞ . The problem is solved (the Haefliger structure made "regular" so that it induces a foliation) up to singularities along "clefts" Σ i in the total space B of X B
V . This cleft Σ is a true obstruction to solving the problem over V . But [Mei20] solves the problem by changing V by cobordism. He sequentially projects connected components of Σ, Σ i , to V , where they are individually still embedded manifolds, and removes them through a type of generalized surgery. These generalized surgeries allow little control over the topology of the cobordism W . Our approach is to replace the project cleft components with the models of type C 1 , from our collection. This allows much more control of W . We realize W as a ssc, but the cost is we must dimensionally stabilize W to match the topology of the successive clefts to that of our models.
When the regularity is less than or equal to class C 1 , as we have noted, the bundle B can be augmented, without making essential choices, with a Haefliger structure γ with normal bundle the vertical tagents, Vert. If V has boundary, γ is required to be regular near ∂V . γ is, by definition, a class C 0 foliation F for Vert, which is transverse to fibers. γ is called regular if it is also transverse to Z(V ), the zero-section of Vert. Using a fine triangulation K of Vert, and many ingenious adaptations of Thurston's constructions [Thu76] from the 1970's, control of F w.r.t. K is gradually obtained. Although it is not possible to homolop γ to full regularity i.e. to make the Haefliger structure everywhere transverse to the fiber, this is achieved in the complement of a dimension p − 2 embedded submanifold Σ ⊂ B, p = dim(V ). (Note in [Mei20] this is called the ([cleft core] ∩M × 1) = ∂ Σ, but we have no need of the part of Σ in the interior of the cobordisms W and only need the "core" so we have simplified the notation slightly, calling the "projected boundary of the cleft" simply the "cleft". Also in [Mei20] our B is written M.) Furthermore, from the simplicial, actually "prismatic," structure K, it is deduced that 1) the components of Σ all have small diameter 2) the topology of each compoenent can be separately controlled to be either S 1 × S p−3 or the (p − 2)-torus (S 1 ) p−2 , and 3) individually each component of Σ projects 1-1 as embedded submanifolds of V , which we now call the clefts, Σ i , l i=1 Σ i = Σ. If the clefts Σ i were pairwise disjoint, the replacement device we describe next would prove a stronger version of Theorem 3 without the stabilizing manifold Q. The replacement models we know how to build require a cleft Σ i to have the form M × S k for some closed manifold M and 1 ≤ k ≤ p − 2. It is possible that with more powerful machinery for building replacement models with more general boundary conditions, Thm. 3 might still be proven without stabilization.
A priori, it may seem that the cleft models are just what is needed as each contains a circle factor. The difficulty comes from the apparently unavoidable crossing of clefts Σ i ∩ Σ j = ∅ ⊂ V , and the structure of the final induction in [Mei20] . Although [Mei20] and this paper heal the cleft Σ 1 (Σ l in [Mei20] ) by replacing its neighborhood with a different model, in our case C 1 , in both cases there is a degree one map (called "a" in the last paragraphs before the end of section 5 [Mei20] ) from the replacement to what is replaced, a generic argument based on Thom transversality for which Meigeniez credits Poenaru allows Σ i , i > 1 to be "pulled" through the replacement C 1 , and become the new (projected) clefts for a cobordant problem over a modified V , called V * . These new clefts, Σ ′ 2 , . . . , Σ ′ l , are still individually embedded in V * , still small when measured by the natural retraction back to V , but their topology may now be quite complicated, with no circle factor present. Since the replacement model in [Mei20] is factor preserving, Σ i × D 2 is replaced with Σ i × S g , where S g is the genus g surface with one boundary component, the loss of control on the topology of Σ ′ i , i ≥ 2, presents no problem for him. Because our replacement models C k 1 require an S k factor, 1 ≤ k ≤ p − 2, we resort to crossing with (any) such factor to augment Σ ′ j ⊂ V * to Σ ′ j × S k ⊂ V * × S k , 2 ≤ j ≤ l, a stabilized problem. Proceeding then in l steps, healing first Σ 1 ⊂ V to get V * , healing Σ ′ 2 × S k ⊂ V * × S k to get Σ ′′ 3 × S k ′ ⊂ V * * × S k ′ we end up building, after stabilizing by a product of l − 1 spheres, a composition of l ssc W . W is a ssc by composing retractions. The result is an ssc from the original problem stabilized by Q to a solution over V * · · · * l-times . The reason stabilization is not necessary in Thm. 3 when p = dim(V ) = 3, is in that case all Σ i ⊂ V are circles and are pairwise disjoint by general position.
Having described the relation with [Mei20] here are some further details of our construction. First, for concreteness assume we choose S 1 × S p−3 for the topology of Σ 1 . The initial replacement of Σ 1 × D 2 with our model C 1 is via the cobordism (which we built as a slice complement) W 1 := C 1 . Σ ′ 2 is some (p − 2)-submanifold of V * but we do not know its topology. Crossing everything now with S k 1 , the new cleft is Σ ′ 2 × S k 1 , its neighborhood is Σ ′ 2 × S k 1 × D 2 with boundary Σ ′ 2 × S k 1 × S 1 . We have in our stockpile a model C k 1 +1 1 , with a ssc C k 1 +1 1 to D k 1 +1 × S 1 . Now cross this model with Σ ′ 2 and glue the result to V * times S k 1 to form the next chamber, W 2 . So far we have built
Sequentially build W :
(20) W = W 1 × S k 1 × · · · × S k l ∪W 2 × S k 2 ×···×k l ∪ · · · ∪W l−1 × S k l ∪W l Each chamber is a ssc by the argument used in Theorem 2, so the composition W is as well.
To complete the proof we must check that the replacements C k 1 can be engineered to solve the representation extension problem. In the stabilization direction, S k , even if we chose k = 1, the holonomy is trivial owing to the product form of stabilization.
A nice feature of [Mei20] is that he allows us to choose arbitrarily any nontrivial fiber X holonomy φ (he calls it φ 1 ) around the norm circle to Σ i . So at first it looks like the technology of bounded simplicity does not need to be invoked. However, there appeared to be a small gap. In any early arXiv version of [Mei20] , "any" meant any nontrivial C 1 -diffeomorphism in the identity component Diff 1 (X ). Unfrotunately, our simplest models C k 1 produce holonomy h (as in Thm. 2 proof), h ∈ bilipschitz 0 (X ).
Fortunately in the current version of [Mei20] the author has verified that his result does hold in the bilipschitz category so we can avoid the more complicated replacement used in the proof of Theorem 2 and build C 1 and C 1 as a straightforward Bing double of a pair ((C 1 ,C ′ 1 ); (C 1 ,C ′ 1 )), and demand from [Mei20] Theorem A ′ of section 3 that the clefts be built with normal holonomy = [α C 1 , α C ′ 1 ], the commutator of two models with slightly different choice of h so as not to commute, see our section 3 on dynamics. However, as an alternative to relying on Meigniez's recent extension to the bilipschitz category, the next paragraph does supply the details needed to pass from a model with normal holonomy in bilipschitz(X ) or even in Homeo(X ) to clefts with holonomy in C 1 .
We employ the same tricks, longitudinal sums and iterated Bing doubles, used to prove Theorem 2. What is necessary is to express a (any) nontrivial element θ = id ∈ Diff 1 0 (X ) as a product of commutators of products-L conjugates of the bilipschitz homeomorphism h ′ : X → X , such as the h in the proof of Thm. 2, which acts on some ball B q ⊂ X via a faithful action of ∆ (2, 3, 7) .
This formula leads to a construction of an composition (C 1 , C 1 ) of the simple model (C 1 ,C 1 ) along the lines summarized in Figures 6 and 7 . The model (C 1 , C 1 ) solves the stabilize clefreplacement problem allowing the cobordisms W i to be built.
Proof of Theorem 4. The proof is entirely formal. It uses the relative Mather-Thurston theorem (see [McD80] ) and an induction over a handle decomposition of X .
Let H be a relative handle decomposition of X . It is trivial to reduce the structure group to Homeo δ 0 (M) over ∂ X ∪ H 0 ∪ H 1 , over the boundary and the 0-and 1-handles. Now consider a 2-handle h 2 . By (relative) MT the bundle over h 2 is bordant, rel the attaching region ∂ − (h 2 ), to a solution over a surface k 2 , and a small thickening of k 2 , where the structure group will be discrete. Observe that (h 2 , ∂ h 2 ) ≃ (D 2 , ∂ D 2 ) × D p−2 enjoys the universal extension property of accepting degree one maps:
Apply this property to a relative surface k 2 . This deals with the 2-handles. Now if h 3 is a 3-handle, its core boundary now receives maps from a union of solutions k 2 (one k 2 for every facet of the attaching map of h 3 to the lower index handles). In fact, the just constructed cobordisms over the (cores of the) 2-handles fit together with core(h 3 ) to define a 3D flattening problem (solvable again by relative MT). Its solution k 3 again has a universal property:
, and so maps degree 1 to h 3 (see the schematic in Figure 8 which has been reduced by one dimension for legibility).
The new problem h 3 ∪(∪ i b i 3 ) is bordent via b 4 to the solution k 3 with reduction to Homeo δ 0 (M). So far we have described how to contract a cobordism, with traction, over the 2 and 3-handles.
Proceed by induction, handle by handle, to build W as a union of the bordism b i d . By construction the retraction to V is built out of a union of degree one maps to the handle-cores of H .
Next we turn to the theorems whose initial data include a more specialized structure over ∂V . Proof of Theorem 5 ′ . The first step is to build some 3-manifold Z, ∂ Z = Σ, with a flat g-connection extending A over Σ. Then we will work on modifing Z to V * , with a ssc (W ;V,V * ) from V to V * . Strangely, the Z we construct is nonorientable (the only occurence of a nonorientable manifold in this paper) but that will present only a momentary difficulty. During the modification process the quality of the connection will be downgraded from Lie type to Homeo 0 (G) but we will retain control of where a generating set S for π 1 (V ) lands.
Lemma 1. Given a flat g-connection A on a closed manifold surface Σ and N ⊂ Homeo 0 (G) as in the theorem statement, there is a 3-manifold Z, ∂ Z = Σ, over which A extends as a flat gconnection with an associated ρ : π 1 (Z) → Homeo 0 (G) with the additional property that π 1 (Z) admits a generating set S so that ρ(S) ⊂ N.
Proof of Lemma 1. First consider as a warm-up the case Σ = T 2 , a 2-torus. Let Y 2k j be the closed complement of the (4k j + 1)-component link, a ramified Bing double, shown in Figure 9 : a core circle linked by 2k j Bing pairs. The factor 2 j is the root 11 needed to bring a problem holonomy around a scc l ⊂ T 2 , hol(l) = ρ(l 0 ) ∈ G within N 0 (id) ⊂ G ⊂ Homeo 0 (G), and k is a multiplicity associated with writing that 2 j th root as a product of commuters. We can set k = 4(dim(G) + 1), according to Proposition 2.
Writing a presentation for π 1 (Y 2k j ) in terms of the peripheral meridians and longitudes we have
The connection A over Σ = T 2 fixes ρ(l 0 ), ρ(m 0 ) ∈ I 0 , our job is to construct an extension first over π 1 (Y 2k j ) as a flat Homeo 0 (G)-connection, with the following symmetry: for 1 ≤ i ≤ k j. Since m 0 , l i , and l ′ i are all free variables, this is easily done by arrange that
Then Z is obtained by gluing up the last 4k j toral ∂ -components of Y 2 jk via the "identity":
preserving Homeo 0 (X )-connection.
. ., γ 2 jk } the γ's introduced by the glueings, ρ(γ i ) = e, 1 ≤ i ≤ 2k j. This completes the proof of the "warmup" case of Lemma 1, where Σ ∼ = T 2 . From here we finish the proof of Theorem 5 and then return to the general case of Lemma 1.
Note. Z is non-orientable since the gluing up of pairs of torus boundary components is via the identity map. To obtain an orientable glue up Z one would need to glue by orientation reversing maps, which are not compatible with the representation we have built. This presents no real problem. Similar to the proof of Theorem 2, the purpose of the model Z if it were orientable, would house a (framed) link L ⊂ Z so that Z/S (L) ∼ = V where the isomorphisms extend the identity on ∂ Z = ∂V (and the surgery S is the standard-not homological-one). Since Z is non-orientable, there is no link L ⊂ Z so that surgery will produce an oriented manifold V , but it is easily shown that framed surgery on some L ⊂ Z will yield V # K where K is the nonorientable 2-sphere bundle over S 1 . This is just as useful. Because crossing an index 2 critical point takes V # K to V . Thus if we can find a link L ⊂ Z with Z/S (L) = ∼ = V # K, we will be able to build V ∼ = Z\(S (L), 2-surgery) and then build V * from V by attaching cobordism models C 1 from the proof of theorem 2. The outer boundaries C 1 of these models may be designed as in Theorem 2 to propogate the topologically flat connection from Z across the homological model to a topologically flat connect on V * = V \ N(L)∪ ⊥ ⊥ C 1 's. The 2-surgery, on an (either of two) essential 2-spheres in V # K results in V (or a V component). Since π 1 (S 2 ) ∼ = 0, there is no obstruction to carrying through the topoloigcally flat connection across the surgery. So in passing from Z to V we do first surgery on L and then a 2-surgery. The latter allows the propagation of a topologically flat connection. The former does not-but this issue is resolved using the homology model C 1 and a topologically flat extension.
Let us briefly recall how to construct the link L. The vanishing of the unoriented bordism group Ω 3 ∼ = 0 guarantees the existence of L. L is obtained by taking the 4-manifold Q bounding Z ∂ Z=∂V V (see Figure 10 ) and giving it a Morse function f , a product over ∂ Z × I. Remove 0-and 4-handles, then trade 1-handles and 3-handles for trivial 2-handles; a single orientation reversing 1-handle cannot be traded and corresponds to the # K factor. The set of attaching curves of the resulting 2-handles constitute the link L.
To summarize, the reason for writing the integer above as 2k j is that we need the factor 2 for the preceeding pairing, 2 j to put the root ρ(l 0 ) 1 2 near id, and k, as in section 2 line (16) (where the constant (n + 1)m is now simplified in notation to a new variable k) is the number of commutators of (products of conjugates of) some model element h needed to realize an arbitrary element α ∈ N δ (id) ⊂ G ⊂ Homeo 0 (G), the δ -neighborhood of the identity of Homeo(M) is in the norm metric.
The Lie group G was presumed to possess a bi-invariant metric so that ρ(l 0 ) lies in a 1parameter subgroup Γ. Taking 2 j th roots in Γ, we define for sufficiently large, α := (ρ(l 0 )) 1/2 j ∈ N δ (id) ⊂ G ⊂ Homeo 0 (G). The large root was needed to put α in N δ (id). Now using line (16) and the corresponding geometric realization of the identity, shown in Figures 6 and 7 , we may extend ρ over π 1 (V ) sequentially accounting for the relations on line (22). To satisfy the first relation
we may take ρ(m i ) and ρ(m ′ i ) each to have the form of the terms in the commutator on line (16); products of conjugates of our small model homeomorphism, h. The last two equations on line (22) can now be used as definitions of ρ(l i ) and ρ(l ′ i ), 1 ≤ i ≤ k. This extends ρ over π 1 (Z) and completes the proof of Theorem 5 ′ in the case Σ ∼ = T 2 . Now we return to complete the proof of Lemma 1 in the case where Σ is a closed oriented surface of genus g > 1. We define first a 3-manifold Y with ∂Y = Σ ∪ (∪ 2n tori), Σ union a large even number of tori. Our strategy is to extend ρ on π 1 (Σ) over π 1 (Y ) as a representation into Homeo 0 (G) with two properties:
(1) The 2n tori can be paired up so that within pairs the restriction of the extension ρ, ρ| π 1 (torus) and ρ| π 1 (paired torus) are identical.
(2) For any ε > 0, the geometric condition ρ(S) ⊂ N ε (I 0 ) is respected for some symmetric generating set S for π 1 (Y ).
From Y , Z is again formed by giving paired tori in the standard manner (but inconsistent with orientations). This preserves the condition on the generating set and allows the proof to follow the previously considered case.
Similar to the case Σ ∼ = T 2 , Y will be the complement of a 1-complex K embedded in S 3 . One component of K is a wedge of g circles (giving rise to the boundary component Σ) and the rest of K consists of a link of 2n circles lying in the complement of the wedge. Its form is shown in Figure  11 for g = 2 and from a different perspective the complement is drawn in Figure 12 . The higher genus cases, g ≥ 3, proceed in a similar fashion and are left to the reader. π 1 (V ) has a presentation very similar to line (22). The new feature is captured by loops such as γ in Figure 12 which bound a separating disk in the handle body complementary to the wedge. Such scc on Σ are products of second order commutators of the form
The key is to proceed in order. The relations involving l ′ 0 and γ are simultaneously solved by assigning appropriate values {ρ(m ′ ) and ρ(m ′ )} to the meridians in "batch A" of the Bing pairs. Referring now to Figure 13 , this gives some fairly random contribution to the r.h.s. ρ(l 2 0 )
which is, in general, inconsistent. But now this "error" can be corrected by choosing ρ correctly on the B-batch of meridians. Now, as before, any sufficiently small element of Homeo(G) to be constructed as a bounded product commutators of (bounded products of controlled conjugates of) h. This completes the g > 1 case of Lemma 1 and thus the proof of Theorem 5.
Proof of Theorem 6 ′ . Finally we use our second set of models C 2 and C 2 . The starting point is the presumed extension of the flat g-connection over V ′ , ∂V ′ = Σ. As is now familiar, there is some framed linke L ′ ⊂ V ′ so that V = V ′ /S (L ′ ). Because we need to propogate A across the surgeries, they should not be standard but will be based on the model (C 2 ,C 2 ).
From line (10) by using the composition:
It is well known that every compact, connected, semi-simple Lie group G contains either a copy of SU(2) or SO(3) (see [Hum98] ), thus for any such G line (26) may be extend to either line 
In both cases α(m) maps to a nontrivial element g ∈ G. In the second case this is because BI proj − − → A 5 kills only the central element of order 2. We need to find inclusions:
which normally generates G, so that g, above, will also normally generate G. To do this let G be the universal covering group. G admits a decomposition, canonical up to permutation, into a direct sum of J simple Lie groups. Pick an SU(2) subgroup for each factor of G and let ∆ be the diagonal within Cartesian product of these subgroups:
The normal closure of ∆ in G does not lie in any product factor, as ∆ does not. This means that the normal closure ∆ G = G.
Let ∆ be the image of ∆ under the finite-to-one projection G → G. ∆ must be isomorphic to SU(2) or SO(3). Clearly ∆ inherits the property that its normal closure is all of G. This yields line (29) with the inclusions normally generating.
But α(m) normally generates any SU(2) in which it lies and similarly its image in A 5 normally generates any SO(3) in which it lies. Thus under the composition π 1 (C 2 ) → π 1 (P 2 ) → BI → ∆ → Ĝ m g ∈ ∈ m maps to g normally generating G. Now the simplicity trick, essential to Theorem 2, allows us to build C 2 from C 2 along the pattern illustrated in Figures 6 and 7 . Since G is a semi-simple Lie group, it is perfect so the general element f ∈ G can be written first as a product of commuters:
and then a's (and b's) can be further written as products of conjugates of g and g −1 . This leads the pattern of Fig. 6 (with ramification = m) and of Fig. 7 with the multiplicities of longitudinal boundary connected sums being the number of conjugates of g and g −1 need to express a i , b i , 1 ≤ i ≤ m. These representation extensions ensure the flat extension of A over V * . The semi-s-cobordisms W 0 and W are now constructed precisely as in the proof of Theorem 2.
Proof of Theorem 7. Following [Mil58] and [Woo71] , up to sign, Euler class χ(B) may be written as: where we are using part of the fibration sequence for a universal covering space 1 → Z ∼ = π 0 (fiber) p − → PSL(2, R) → PSL(2, R) → 1 and Γ 1 , . . . , Γ 2g are (arbitrary) lifts of γ 1 = ρ(a 1 ), γ 2 = ρ(b 2 ), . . ., γ 2g = ρ(b g )
images of the usual generators for π 1 (Σ g ) under a suitable representation ρ : π 1 (Σ g ) → PSL(2, R). These generators satisfy the familiar relation
The idea is that in the Lie algebra sl(2, R) we have generators and bracket structure constants: ρ(a i ) = e εE , ρ(b i ) = e εF , 1 ≤ i ≤ g This is close, but will not actually obey the relation (33) because of the error term on line 35, and a less serious integrality issue. So this choice of ρ is not a representation.
To fix this problem, we would like to make some small adjustment to e εE and e εF so that their (group) commutator is exactly a rotation quite close to e ε 2 H . This kind of problem is typically approached by an application of the inverse function theorem. As a first step, we collect this information. Of course, the bracket on sl(2, R), which we write as a bilinear R 3 (since R 3 ∼ = SL(2, R) as a vector space), has zero differential: Thus for sufficiently small ε > 0, every element z of SL(2, R) within the operator norm ball B cε 2 (id), for some constant c > 0, is realized as
This is already enough information to complete the proof if 2π were replaced in Theorem 7 by an arbitrary positive constant. But to obtain the more refined statement we need the following identity from line (95) of [CW13] which follows from the Baker-Campbell-Hausdorff formula. We write it for skew-Hermitian matrices where in the original context it was for Hermitian. We will use line (40) to explicitly solve the problem of writing an ε 2 -small rotation as a commutator of small ε-small (near) boosts up to an error of order ε 5 . This tiny error can then be patched using line (39). An alternate approach would be to use an untruncated version of line (40) and solve exactly for near boosts commuting to a rotation. This surely can be done, but some technical work would be needed to show that the resulting asymptotic expansion for the near boosts has a positive radius of convergence. The present course avoids this technicality.
As an ansatz, write the near boosts in the form E ′ = E + bH and F ′ = F + bH. Plugging into line (40) we can compute the coefficients for the r.h.s. of (40) = ε 2 c E E + ε 2 c F F + ε 2 c H H as:
The total differential of c E (42) dc E = 18b 2 − 5 3 ε 2 + 2ε + 2 db + 12b 3 − 13 3 b ε + 1 dε where α ε 2 is the rotation of ε 2 radians cos(ε 2 ) sin(ε 2 ) − sin(ε 2 ) cos(ε 2 ) ∈ U(1) ⊂ SL(2, R). It is possible to find the requisite w using line (39) because: (1) the number of products g = O(ε −2 ) (2) each factor deviates from a rotation by O(ε 5 ), meaning α ′ ε 2 = β α ε 2 = α ε 2 (α −1 ε 2 β α ε 2 ) where ||β ||= O(ε 5 ) and (3) since α ε 2 is an isometry of S 1 , ||α −1 ε 2 β α ε 2 ||= ||β ||= O(ε 5 ). By (3) we may pull all errors β through to the left side of the composition to obtain O(ε −2 ) error factors each of size O(ε 5 ) resulting in a total error of size O(ε 3 ), small enough to be corrected using line (39) by the proper choice of w ∈ B ε (id). This completes the proof of Theorem 7.