Teichmüller Spaces and Bundles with Negatively Curved Fibers
aa r X i v : . [ m a t h . DG ] M a r Teichm¨uller Spaces and Bundles with NegativelyCurved Fibers
F. T. Farrell and P. Ontaneda ∗ Abstract
In the first part of the paper we introduce the theory of bundles withnegatively curved fibers. For a space X there is a forgetful map F X between bundle theories over X , which assigns to a bundle with neg-atively curved fibers over X its subjacent smooth bundle. Our Mainresult states that, for certain k -spheres S k , the forgetful map F S k is notone-to-one. This result follows from Theorem A, which proves that thequotient map MET sec < ( M ) → T sec < ( M ) is not trivial at some homotopylevels, provided the hyperbolic manifold M satisfies certain conditions.Here MET sec < ( M ) is the space of negatively curved metrics on M and T sec < ( M ) = MET sec < ( M ) /DIF F ( M ) is, as defined in [12], the Teichm¨ullerspace of negatively curved metrics on M . Two remarks: (1) the nontriv-ial elements in π k MET sec < ( M ) constructed in [13] have trivial image bythe map induced by MET sec < ( M ) → T sec < ( M ) (2) the nonzero classes in π k T sec < ( M ) constructed in [12] are not in the image of the map inducedby MET sec < ( M ) → T sec < ( M ) ; the nontrivial classes in π k T sec < ( M ) givenhere, besides coming from MET sec < ( M ) and being harder to construct,have a different nature and genesis: the former classes -given in [12]- comefrom the existence of exotic spheres, while the latter classes -given here-arise from the non triviality and structure of certain homotopy groupsof the space of pseudoisotopies of the circle S . The strength of the newtechniques used here allowed us to prove also a homology version of The-orem A, which is given in Theorem B. ∗ Both authors were partially supported by NSF grants. ection 0. Introduction. Let M be a closed smooth manifold. We will denote the group of all selfdiffeomorphisms of M , with the smooth topology, by DIF F ( M ). By a smoothbundle over X , with fiber M , we mean a locally trivial bundle for which thechange of coordinates between two local sections over, say, U α , U β ⊂ X is givenby a continuous map U α ∩ U β → DIF F ( M ). A smooth bundle map betweentwo such bundles over X is bundle map such that, when expressed in a localchart as U × M → U × M , the induced map U → DIF F ( M ) is continuous.In this case we say that the bundles are smoothly equivalent . Smooth bundlesover a space X , with fiber M , modulo smooth equivalence, are classified by h X, B (cid:16)
DIF F ( M ) (cid:17)i , the set of homotopy classes of (continuous) maps from X to the classifying space B (cid:16) DIF F ( M ) (cid:17) . h In what follows we will be considering everything pointed : X comes with a basepoint x , the bundles come with smooth identifications between the fibers over x and M , and the bundle maps preserve these identifications. Also, classifyingmaps are base point preserving maps. i If we assume that X is simply connected, then we obtain a reduction inthe structural group of these bundles: smooth bundles over a simply con-nected space X , with fiber M , modulo smooth equivalence, are classified by h X, B (cid:16)
DIF F ( M ) (cid:17)i , where DIF F ( M ) is the space of all self diffeomorphismsof M that are homotopic to the identity 1 M . In what follows we assume X tobe simply connected. If we assume in addition that M is aspherical with π ( M )centerless (e.g. admits a negatively curved metric) then old results of Borel[1], Conner-Raymond [6] say that, by pushing forward metrics, DIF F ( M ) actsfreely on ME T ( M ), the space of smooth Riemannian metrics on M (whichwe consider with the smooth topology). Moreover, Ebin’s Slice Theorem [8]assures us that DIF F ( M ) → ME T ( M ) → (cid:16) ME T ( M ) /DIF F ( M ) (cid:17) is alocally trivial bundle. Hence, since ME T ( M ) is contractible, we can write B (cid:16) DIF F ( M ) (cid:17) = ME T ( M ) /DIF F . In [12] we called T ( M ) = ME T ( M ) .(cid:16) R + × DIF F ( M ) (cid:17) the Teichm¨uller Space of Riemannian Metrics on M , where the R + factor acts on ME T ( M ) by scalar multiplication. Since T ( M ) is homo-topy equivalent to ME T ( M ) /DIF F ( M ) we can also write B (cid:16) DIF F ( M ) (cid:17) = T ( M ). Therefore smooth bundles over a simply connected space X , with as-pherical fiber M and π ( M ) centerless, modulo smooth equivalence, are classi-fied by h X, T ( M ) i . 2et S be a complete collection of local sections of the bundle ME T ( M ) →T ( M ). Using S and a given map f : X → T ( M ) we can explicitly construct asmooth bundle E over X , with fiber M . Yet, with these data we seem to get alittle more: we get a Riemannian metric on each fiber E x of the bundle E . Thiscollection of Riemannian metrics does depend on S , but it is uniquely defined(i.e. independent of the choice of S ) up to smooth equivalence.Of course, any bundle with fiber M admits such a fiberwise collection of Rie-mannian metrics because ME T ( M ) is contractible, so we seem to have gainednothing. On the other hand, in the presence a geometric condition we do get ameaningful notion. We explain this next.Denote by ME T sec < ( M ) the space of all Riemannian metrics on M withnegative sectional curvatures and by T sec < ( M ) the image of ME T sec < ( M ) in T ( M ) by the quotient map ME T ( M ) → T ( M ). In [12] we called T sec < ( M )the Teichm¨uller Space of negatively Curved Riemannian Metrics on M . If we arenow given a map X → T sec < ( M ), we get a smooth bundle E with fiber M , andin addition, as mentioned before, we get a collection of Riemannian metrics, oneon each fiber E x , x ∈ X . And, since now the target space is T sec < ( M ), theseRiemannian metrics are all negatively curved. We call such a bundle a bundlewith negatively curved fibers . Still, to get a bona fide bundle theory we have tointroduce the following concept. We say that two bundles E , E over X , withnegatively curved fibers, are negatively curved equivalent if there is a bundle E over X × [0 , E | X ×{ i } is smoothlyequivalent to E i , i = 0 ,
1, via bundle maps that are isometries between fibers.Then, bundles with negatively curved fibers over a (simply connected) space X ,modulo negatively curved equivalence, are classified by h X, T sec < ( M ) i . Andthe inclusion map F : T sec < ( M ) ֒ → T ( M ) gives a relationship between thetwo bundle theories: h X, T sec < ( M ) i F X −→ h X, T ( M ) i and the map F X is the “forget the negatively curved structure” map. The “ker-nel” K X of this map between the two bundle theories is given by bundles over X ,with negatively curved fibers, that are smoothly trivial. Every bundle in K X canbe represented by the choice of a negatively curved metric on each fiber of thetrivial bundle X × M , that is, by a map X → ME T sec < ( M ). Note that thisrepresentation is not unique, because smoothly equivalent representations give3ise to the same bundle with negatively curved fibers. In any case, we have that K X is the image of h X, ME T sec < ( M ) i by the map h X, ME T sec < ( M ) i −→ h X, T sec < ( M ) i , induced by the quotient map ME T sec < ( M ) → T sec < ( M ).Note that we can think of h X, ME T sec < ( M ) i as a bundle theory: the “bun-dles” here are choices of negatively curved metrics, one for each fiber of thetrivial bundle X × M , modulo the following weak version of negatively curvedequivalence. Two “bundles” E , E , here are equivalent if there is a “bundle” E over X × I such that E | X ×{ i } = E i , i = 0 ,
1. Summarizing, we get the followingexact sequence of bundle theories:( ∗ ) h X, ME T sec < ( M ) i R X −→ h X, T sec < ( M ) i F X −→ h X, T ( M ) i where the map R X is the “representation map”, which, to each smoothly trivialbundle with negatively curved fibers E ∈ K X , assigns the set R − X ( E ) of repre-sentations of E of the form X → ME T sec < ( M ).It is natural to inquire about the characteristics of these maps. For instance,are they non constant? are they one-to-one? are they onto? If, in (*), we spec-ify X = S k , k > π k ( ME T sec < ( M )) → π k ( T sec < ( M )) → π k ( T ( M )). Some information aboutthese maps between homotopy groups was given in [12] and [13]:1. It was proved in [13] that π ( ME T sec < ( M )) is never trivial, provided ME T sec < ( M ) = ∅ and dim M >
13. But the nonzero elements in π ( ME T sec < ( M )), constructed in [13], are mapped to zero by the map π ( ME T sec < ( M )) → π ( T sec < ( M )). Therefore the representation map R S in (*) is never one-to-one, provided ME T sec < ( M ) = ∅ and dim M > Remark.
It was also proved in [13] (assuming
ME T sec < ( M ) = ∅ ) that π ( ME T sec < ( M )) contains the infinite sum ( Z ) ∞ = (cid:16) Z / Z (cid:17) ∞ as a subgroup,thus π ( ME T sec < ( M )) is not finitely generated. Moreover, it was provedthat the same is true for π k ( ME T sec < ( M )), for k = 2 p − p > Z p ) ∞ instead of ( Z ) ∞ ) , provided dim M is large (how large dependingon k ). Furthermore, π ( ME T sec < ( M )) contains ( Z ) ∞ , provided dim M > π ( ME T sec < ( M ))also proves that the forget structure map F S is not onto. To see thisjust glue two copies of D × M along S using a nontrivial element in π ( ME T sec < ( M )). Thus, there are (nontrivial) smooth bundles E over S which do not admit a collection of negatively curved Riemannian met-rics on the fibers of E . Using the remark, the same is true for S k , k = 2 p − p > π k ( T sec < ( M )) is nonzero. Here M depends on k and always k >
0. In [12] no conclusion was reached on the case k = 0 (i.e. aboutthe connectedness of T sec < ( M )). Also, the images of these elements bythe inclusion map T sec < ( M ) → T ( M ) are not zero. Hence the forgetstructure map F S k is, in general, not trivial. This means also that thereare bundles with negatively curved fibers that are not smoothly trivial,i.e. the representation map R S k is not onto in these cases. Remark.
In all the discussion above we can replace “negatively curved met-rics” by “ ǫ -pinched negatively curved metrics”.Our main result here is the following: Main Theorem.
The forget structure map F S k is, in general, not one-to-one,for k = 2 p − , p prime. The Main Theorem follows from Theorems A, B and C, which actuallyprove more. Theorems A and C together show that for “sufficiently large”closed hyperbolic n -manifolds the quotient map ME T sec < ( M ) → T sec < ( M )is not trivial at the homotopy group level. That is, π k ( ME T sec < ( M )) → π k ( T sec < ( M )) is nonzero, provided a certain condition is satisfied by k and n . In particular this condition is satisfied for k = 0 and n >
9. Thereforewe obtain as Corollary that for sufficiently large closed hyperbolic n -manifolds, n > T sec < ( M ) is disconnected. This solves the question left open in [12]whether T sec < ( M ) can ever be disconnected (see item 3 above). Also, the case k = 1 is included here.Theorem B and C together show that for “sufficiently large” closed hy-perbolic n -manifolds the quotient map ME T sec < ( M ) → T sec < ( M ) is nottrivial at the homology level. That is, H k ( ME T sec < ( M )) → H k ( T sec < ( M ))is nonzero, again provided a certain condition is satisfied by k and n . This is5nteresting because it gives characteristic classes (mod a prime p ) for the bun-dle theory. Finally, the case k = 1 is also included here. All results mentionedabove also hold if we replace “negative sectional curvature” by “ ǫ -pinched to -1sectional curvature”. To give more detailed statements of our results we needsome notation.Let M be a closed hyperbolic manifold and let γ be an embedded closedgeodesic in M . We denote by ω ( γ ) the width of its normal geodesic neighbor-hood. Given any r > γ in M it is possibleto find a finite sheeted cover N of M such that γ lifts to a geodesic γ N in N and ω ( γ N ) > r (see [10], Cor.3.3).For a smooth closed manifold N let P ( N ) be the space of topological ra-dioisotopes of N , that is, the space of all homeomorphisms N × I → N × I , I = [0 , N ×{ } . We consider P ( N ) with the compactopen topology. Also, P s ( N ) is the space of all smooth pseudoisotopies on N , withthe smooth topology. Let T OP ( N ) be the group of self homeomorphisms of N ,with the compact open topology. Note that DIF F ( N ) ⊂ T OP ( N ). We havethe “take top” map τ : P s ( N ) → T OP ( N ), given by τ ( f ) = f | N ×{ } : N → N .We will use the following notation. Let X be a space. The geometric real-ization | S ( X ) | of the singular simplicial set S ( X ) of X will be denoted by X • .If f : X → Y then we get the induced map f • : X • → Y • Let L ⊂ T OP ( S × S n − )) be the subgroup of all “orthogonal” self homeo-morphisms of S × S n − , that is f : S × S n − → S × S n − belongs to L if f ( z, u ) =( e iθ z, A ( z ) u ), for some e iθ ∈ S , and A : S → SO ( n − T OP ( S × S n − )) /L but the quotient map T OP ( S × S n − )) → T OP ( S × S n − )) /L is not a fibration. Instead we consider the simplicial quo-tient T OP ( S × S n − )) //L := (cid:12)(cid:12)(cid:12) S (cid:16) T OP ( S × S n − ) (cid:17). S ( L ) (cid:12)(cid:12)(cid:12) (see Section 1.3).Now, define the map Υ n, k : π k ( P s ( S × S n − ) • ) → π k ( T OP ( S × S n − ) // L ) asbeing the following composite: π k ( P s ( S × S n − ) • ) π k ( τ • ) −→ π k ( T OP ( S × S n − ) • ) −→ π k ( T OP ( S × S n − ) //L )where all homotopy groups have the corresponding identities as base points. Wewill say that Υ n, k is strongly nonzero if π k ( P s ( S × S n − ) • ) ∼ = π k ( P s ( S × S n − ) )6ontains an infinite torsion subgroup T such that Υ n, k | T is injective. Theorem A.
Let k and n > max { k + 8 , k + 9 } be such that Υ n, k is stronglynonzero. Given ℓ > there is a constant r = r ( n, k, ℓ ) such that the fol-lowing holds. If M is a closed hyperbolic n -manifold that contains an embed-ded closed geodesic γ with trivial normal bundle, length ≤ ℓ and ω ( γ ) > r ,then the map π k ( ME T sec < ( M )) → π k ( T sec < ( M )) is nonzero. In particular π k ( T sec < ( M )) is not trivial .In Theorem A all homotopy groups are based at (the class of) the givenhyperbolic metric. For k = 0 the word “nonzero” in the conclusion of TheoremA should be read as “not constant”. And the last sentence of Theorem A, when k = 0, should be read as: “ T sec < ( M ) is not connected”. Also, note that if M is orientable, then condition “ γ has trivial normal bundle” is redundant.We have a homology version of Theorem A. Let h : π k ( T OP ( S × S n − ) //L ) → H k ( T OP ( S × S n − ) // L ) denote the Hurewicz map. As before, we will say that h Υ n, k is strongly nonzero if π k ( P s ( S × S n − ) ) contains an infinite torsion sub-group T such that h Υ n, k | T is injective. Theorem B.
Let k > and n > max { k + 8 , k + 9 } be such that h Υ n, k is strongly nonzero. Given ℓ > there is a constant r = r ( n, k, ℓ ) such thatthe following holds. If M is a closed hyperbolic n -manifold that contains anembedded closed geodesic γ with trivial normal bundle, length ≤ ℓ and ω ( γ ) > r ,then the map H k ( ME T sec < ( M )) → H k ( T sec < ( M )) is nonzero. In particular H k ( T sec < ( M )) is not trivial .The statements of Theorems A and B hold also for ǫ -pinched negativelycurved metrics: Addendum to Theorems A and B.
The statements of Theorems A and Bremain true if we replace the decoration “sec <
0” on both
ME T sec < ( M ) and T sec < ( M ) by “-1- ǫ < sec ≤ -1”. But now r also depends on ǫ , i.e. r = r ( n, k, ℓ, ǫ ) . As mentioned before, the condition “ γ has a trivial normal bundle” in The-orems A and B can be obtained after taking, if necessary, a two sheeted cover.The condition ω ( γ ) > r can also be obtained after taking a big enough finitesheeted cover. To see this just take r = r ( n, k, ℓ ), ℓ = length ( γ ), and apply theresult mentioned after the the definition of ω ( γ ): given any r > it is possible o find a finite sheeted cover N of M such that γ lifts to a geodesic γ N in N and ω ( γ N ) > r (see [10], Cor.3.3). These facts imply the following results. Corollary 1.
Let k > and n > max { k + 8 , k + 9 } be such that h Υ n, k isstrongly nonzero. Then for every closed hyperbolic n -manifold M there is a finitesheeted cover N of M such that the maps π k ( ME T sec < ( N )) → π k ( T sec < ( N )) , H k ( ME T sec < ( M )) → H k ( T sec < ( M )) are nonzero. And taking k = 0 in Theorem A and Theorem C (see below) we have: Corollary 2.
Let M be a closed hyperbolic n -manifold, n > . Then M admitsa finite sheeted cover N such that T sec < ( N ) is disconnected. Remark.
The Addendum to Theorem A implies that the Corollaries remaintrue if we replace the decoration “ sec < ǫ < sec ≤ -1”. In this case N depends not just on n and k but also on ǫ > h Υ n, k is strongly not zero. We de-note the cyclic group of order p by Z p , and ( Z p ) ∞ is the (countably) infinite sumof Z p ’s. Theorem C.
Consider the map h Υ n, k . We have the following cases: k=0 the group π ( P s ( S × S n − )) contains a subgroup ( Z ) ∞ and h Υ n, restrictedto this ( Z ) ∞ is injective, provided n ≥ . k=1 the group π ( P s ( S × S n − )) contains a subgroup ( Z ) ∞ and h Υ n, restrictedto this ( Z ) ∞ is injective, provided n ≥ . k=2p-4, p > the group π k ( P s ( S × S n − )) contains a subgroup ( Z p ) ∞ and h Υ n, k restrictedto this ( Z p ) ∞ is injective, provided n ≥ k + 8 . Remark.
Of course if h Υ is strongly nonzero, then Υ is strongly nonzero, sowe have a statement similar to Theorem C for Υ. In fact, this is Theorem Dand it is used to prove Theorem C (see Section 3.)8 ection 1. Preliminaries.
For a Riemannian manifold Q , with metric g , and a submanifold P we de-note by ⊥ p P the orthogonal complement of T p P in T p Q , with respect to themetric g . As usual the exponential map T Q → Q is denoted by exp and, toavoid complicating our notation, the normal (to P ) exponential map will also bedenoted: exp : ⊥ p P → Q . If we need to show the dependence of these objectson g we shall write ⊥ g P and exp g . Recall that if E is a subbundle of T Q | P such that E ⊕ T P = T Q | P then exp | E : E → Q (here exp : T Q | P → Q ) is adiffeomorphism near P , is the identity on P , and the derivative at any point p ∈ P is the identity (after the obvious identification of T E | P with T Q | P ). Wewill need the following result. Proposition 1.1.1.
Let M be smooth n -manifold without boundary, and g , g two Riemannian metrics on M . Let also P be a closed smooth k -submanifold of M , k + 3 ≤ n with trivial normal bundle and η : P → M a smooth embeddinghomotopic to the inclusion ι : P ֒ → M . Then there is a smooth isotopy h t : M → M , ≤ t ≤ , h = 1 M , such that (write h = h , and g = h ∗ g ): η = hι . ⊥ gp P = ⊥ g p P , for all p ∈ P . g ( u, v ) = g ( u, v ) , for all u, v ∈⊥ gp P . There is ǫ > such that exp gp ( v ) = exp g p ( v ) , for all p ∈ P , v ∈⊥ gp P with g ( v, v ) ≤ ǫ . Proof. If ι = η and g = g we are done. If not let H : P × [0 , → M bea homotopy between ι and η . Since 2 k + 3 ≤ n we can assume that H is anembedding. Hence ι and η are isotopic. This isotopy can be extended in theusual way (using vector fields) to an ambient isotopy of M . In this way obtainan isotopy that satisfies (1). We will construct other isotopies to obtain (2)-(4)(these remaining isotopies will fix P ). Hence we can assume that ι = η .Let V = { v , ..., v l } , l = n − k , be an orthonormal framing of the bundle ⊥ g P ⊂ T M | P . Let also V ′ = { v ′ , ..., v ′ l } be the projection of V in ⊥ g P , thatis v ′ i = u i + v i , u i ∈ T P and v ′ i ∈⊥ g P . Since ⊥ g P ⊕ T P = T M we have that V ′ is a framing of ⊥ g P . Denote by Φ t : ⊥ g P → T M | P the bundle map given9y Φ t ( v i ) = tu i + v i , and let E t be the subbundle of T M | P generated by the tu i + v i , that is E t = Φ t ( ⊥ g P ). Then E t ⊕ T P = T M | P . Let exp t : E t → M be the restriction of exp : T M | P → M . Then H t = exp t ◦ Φ t ◦ ( exp ) − is anisotopy defined on a neighborhood of P , starting at the identity. And, since thederivative of exp t at a p ∈ P is the identity we have that the derivative of H at a p ∈ P sends ⊥ g P to ⊥ g P . Extend H t to the whole M . It is not difficultto show that ( H ) ∗ g satisfies item (2) (with ( H ) ∗ g instead of g ). Hence wecan suppose now that ⊥ g p P = ⊥ g p P , for all p ∈ P .We now further change g by an isotopy so as to obtain (3). Note that if V is also orthonormal with respect to g we are done. If not let V t be a pathof framings of ⊥ g P = ⊥ g P with V = V and V orthonormal with respectto g (for this just apply the canonical Gram-Schmidt orthonormalization pro-cess). Let Φ ′ t : ⊥ g P →⊥ g P be the bundle map that sends V to V t . Let H ′ t = exp ◦ Φ ′ t ◦ ( exp ) − (here exp : ⊥ g P → M ) is an isotopy defined on aneighborhood of P , starting at the identity. And the derivative of H at a p ∈ P sends V to V . Extend H t to the whole M . It is not difficult to show that( H ′ ) ∗ g satisfies item (3) (with ( H ) ∗ g instead of g ). Hence we can supposenow that g ( u, v ) = g ( u, v ), for all u, v ∈⊥ gp P .Now, note that the map exp g ◦ ( exp g ) − is a diffeomorphism defined on aneighborhood of P , and its derivative at a p ∈ P is the identity. Hence a fiberversion of Alexander’s trick (see Appendix) gives an isotopy H ′′ t that deforms exp g ◦ ( exp g ) − to the identity (near P ). Extending this isotopy to the whole M , we have that ( H ′′ ) ∗ g satisfies item (4). This proves the Proposition. Remarks.1.
From the proof of the Proposition we see that the isotopy h t in the statementof the Proposition can be chosen to have support in a small (as small as we want)neighborhood of the image of the embedding H mentioned in the first paragraphof the proof. It can be checked from the proof above that a (local) parametrized version ofProp. 1.1.1 also holds: fix g and suppose that we are given a C -neighborhood U of some g in ME T ( M ), and a continuous map g ′ η g ′ ∈ Emb ∞ ( P, M ), g ′ ∈ U . Then the proof of Prop. 1.1.1 above gives us a method to construct amap h g ′ such that the map g ′ h g ′ (and hence the map g ′ h ∗ g ′ ( g ′ )) is welldefined and continuous on some C -open W ⊂ U , g ∈ W . (Here ǫ will dependon W .) Moreover, if g ′ already satisfies (1)-(4) of the Proposition then this mapleaves g ′ invariant, that is, g ′ = h ∗ g ′ ( g ′ ).10 .2. The map Ω γ We will need the following construction, which have some similarities to theone given in [11].Write S ( ℓ ) = { ( x, y ) ∈ R , x + y = ( ℓ/ π ) } . Let M be a hyperbolic n -manifold, with metric g . Let γ : S ( ℓ ) → M be an embedded closed geodesicof length ℓ . Sometimes we will denote the image of γ just by γ . We assumethat the normal bundle of γ is orientable, hence trivial. Let r > r is less than the width of the normal geodesic tubular neighborhood of γ and denote by U the normal geodesic tubular neighborhood of γ of width 6 r .Write V = U \ γ . Using the exponential map of geodesics orthogonal to γ andparallel translation along γ we get that V (with the given hyperbolic metric ρ ) is isometric to the quotient of R × S n − × (0 , r ], equipped with the doublywarped Riemannian metric: ρ ′ ( s, u, t ) = cosh ( t ) ds + sinh ( t ) σ S n − ( u ) + dt , by the action of an isometry A : R × S n − × (0 , r ] → R × S n − × (0 , r ] of theform ( s, u, t ) ( s + ℓ, T u, t ) for some T ∈ SO ( n − σ S k is the canonicalround Riemannian metric on the k -sphere S k . Remark.
Note that V is diffeomorphic to S × S n − × (0 , r ]. For a compactnessargument that will be used later we need some canonical ways of identifying V with S × S n − × (0 , r ]. We do this by choosing certain A -invariant trivializationsof the bundle R × R n − → R . To do this consider SO ( n −
1) with its bi-invariantmetric and let B , ..., B k be closed geodesic balls that cover SO ( n −
1) and suchthat: for each S ∈ B i there is a path α S, i ( u ), u ∈ [0 , S . Also, for each i fixed, α S, i varies continuously (in thesmooth topology) with S . We require also that α S, i be constant near 0 and 1.(For instance, α S, i could be a properly rescaled geodesic.) Then our canonical A -invariant trivializations are constructed in the following way. For T ∈ B i define v ij ( t ) = α T, i ( t/ℓ ) . e j , j = 1 , ..., n −
1, for t ∈ [0 , ℓ ] (where the e j ’s form thecanonical base of R n − ). Note that A (0 , v ij (0)) = A (0 , e j ) = ( ℓ, T e j ) = ( ℓ, v ij ( ℓ )),thus we can extend the v ij ( t ) periodically to all t ∈ R . Therefore, for each i such that T ∈ B i , n v i , ..., v in − o is a A -invariant trivialization. Hence for each T we get finitely many “canonical” trivializations, one for each i such that T ∈ B i . Caveat:
We are giving two identifications of the universal cover of V with R × S n − × (0 , r ]: (1) V is the quotient of R × S n − × (0 , r ] by the action11f the isometry A , and (2) using the canonical trivializations mentioned in thisremark. These two identifications do not necessarily coincide.Let δ : [0 , → [0 ,
1] be a smooth map such that δ (0) = 0, δ (1) = 1 andwhich is constant near 0 and 1. Define the metric ρ ′′ on R × S n − × (0 , r ] in thefollowing way: • ρ ′′ = ρ ′ outside R × S n − × [2 r, r ]. • On R × S n − × [2 r, r ] we have: ρ ′′ ( s, u, t ) = cosh ( t ) ds + 14 (cid:20) e t + (cid:18) δ ( t − rr ) − (cid:19) e − t (cid:21) σ S n − ( u ) + dt Note that for t ≤ r , and near 3 r , we have ρ ′′ ( s, u, t ) = cosh ( t )[ ds + σ S n − ( u )] + dt . That is, ρ ′′ is a simply warped metric in this case. • On R × S n − × [3 r, r ] define: ρ ′′ ( s, u, t ) = cosh ( t )[ ds + σ S n − ( u )] + dt . • On R × S n − × [4 r, r ] define: ρ ′′ ( s, u, t ) = cosh ( t ) ds + 14 (cid:20) e t + (cid:18) − δ ( t − rr ) (cid:19) e − t (cid:21) σ S n − ( u ) + dt Note that ρ ′′ is also invariant by A , hence induces a Riemannian metric ρ on the quotient V . Lemma 1.2.1.
Given ǫ > , we have that all sectional curvatures of ρ ′′ and ρ lie in the interval ( − − ǫ, − ǫ ) , provided r is sufficiently large (how largedepending solely on ǫ and n ). For a proof see Lemma 1.2.1 in [11].
Remark.
In the next Section we will need a canonical way of deforming ρ ′′ to ρ ′ . To do this we assume in addition that δ (1 − v ) = 1 − δ ( v ), for v ∈ [0 , ρ ′′ v in the following way. Define ρ ′′ v = ρ ′′ on R × S n − × (cid:16) [2 r, r + vr ] ∪ [5 r − vr, r ] (cid:17) and ρ ′′ v ( s, u, t ) = cosh ( t ) ds + h e t + (2 δ ( v ) − e − t i σ S n − ( u ) + dt on R × S n − × [2 r + vr, r − vr ]. (And also let ρ ′′ v = ρ ′′ = ρ ′ outside R × S n − × [2 r, r ].)Then ρ ′′ = ρ ′ and ρ ′′ = ρ ′′ . Also ρ ′′ v is invariant by A , hence induces a deforma-tion ( ρ ) v from ρ to ρ . Furthermore, Lemma 1.2.1 holds for all ρ ′′ v and ( ρ ) v ,12rovided r is sufficiently large.We now define a map Ω γ : P s ( S × S n − ) → ME T ( M ). Identify V with S × S n − × (0 , r ] (via one of the finitely many canonical ways mentioned inthe remark above). Note that for the metric ρ on V = S × S n − × (0 , r ] wehave that: (1) ρ = ρ outside S × S n − × [2 r, r ] and (2) on S × S n − × [3 r, r ]we have ρ ( z, u, t ) = cosh ( t ) σ ( z, u ) + dt for some metric σ on S × S n − . For ϕ ∈ P s ( S × S n − ) write µ = τ ϕ × [0 , ∈ DIF F ( S × S n − × [0 , ω = Ω γ ( ϕ ) be the metric defined as follows: a. ω is the given hyperbolic metric ρ (or ρ ) outside S × S n − × [ r, r ] ⊂ M .And ω = ρ on S × S n − × [4 r, r ] ⊂ M b. ω = [ λ − ϕλ ] ∗ ρ = [ λ − ϕλ ] ∗ ρ , where λ ( z, u, t ) = ( z, u, t − rr ), for ( z, u, t ) ∈ S × S n − × [ r, r ]. c. ω = [ λ − µλ ] ∗ ρ , where λ ( z, u, t ) = ( z, u, t − rr ), for ( z, u, t ) ∈ S × S n − × [2 r, r ]. Note that for t ≤ r , and t near 3 r , ω ( z, u, t ) = cosh ( t )[( τ ϕ ) ∗ σ ( z, u )]+ dt d. On S × S n − × [3 r, r ] define: ω ( z, u, t ) = cosh ( t ) σ t ( u ) + dt where σ t = (cid:16) − δ ( t − rr ) (cid:17) ( τ ϕ ) ∗ σ ( z, u ) + δ ( t − rr ) σ ( z, u ). Remark.
We are assuming that all pseudoisotopies are products near 0 and 1.Hence items (a)-(d) give a well defined Riemannian metric ω on M . Lemma 1.2.2.
Given ǫ > , ℓ > and a compact set K ⊂ P s ( S × S n − ) , thereis r > such that the sectional curvatures of Ω γ ( ϕ ) lie in ( − − ǫ, − ǫ ) , forall ϕ ∈ K , provided γ has length ℓ . Proof.
Outside S × S n − × [3 r, r ], ω = Ω γ ( ϕ ) coincides with ρ or a pull-back of it. Hence Lemma 1.2.1 implies in this case all sectional curvatures ofΩ γ ( ϕ ) lie in ( − − ǫ, − ǫ ). For S × S n − × [3 r, r ] we apply Lemma 2.2 of[21]. How large r needs to be in this case depends only on ǫ , ( τ ϕ ) ∗ σ and theirderivatives up to order 2. Since K is compact ( τ ϕ ) ∗ σ and their derivatives up toorder 2 are bounded provided that all possible metrics σ on S × S n − and theirderivatives up to order 2 are bounded. Recall that these metrics σ are obtainedin the following way. For T ∈ SO ( n −
1) let α T, i : [0 , → SO ( n − T ∈ B i be as in the remark above. Let f : [0 , × S n − → [0 , ℓ ] × S n − defined13y f ( t, u ) = ( ℓt, α T, i ( t ) .u ). Let σ ′ = f ∗ ( ds + σ S n − ), where ds + σ S n − is thecanonical product metric on [0 , ℓ ] × S n − . Gluing { } × S n − to { } × S n − themetric σ ′ gives a metric σ on S × S n − . Since ℓ is fixed and the set of all α T, i is compact in
DIF F ([0 , , SO ( n − σ on S × S n − and their derivatives up to order 2 arebounded. This proves the Lemma. Remark.
A subtle point here. In Lemma 1.2.2 the number r depends only on ǫ , ℓ , K and the dimension n of the manifold, but not on the particular manifold M .The independence from M stems from the canonical identifications mentionedin the remark at the beginning of this Section. Let X be a space and S ( X ) be its singular simplicial set. Recall that the q -simplices of S ( X ) are the singular q -simplices on X , i.e maps ∆ q → X . Write X • = | S ( X ) | where the bars denote “geometric realization”. There is a canoni-cal map h X : X • → X which is a weak homotopy equivalence. If f : X → Y is amap then the simplicial map S ( f ) : S ( X ) → S ( Y ) defines a map f • : X • → Y • and clearly X • f • → Y • ↓ ↓ X f → Y commutes.Let G be a topological group acting freely on X . Then S ( G ) is a simpli-cial group acting simplicially on S ( X ) and we get a simplicial set S ( X ) /S ( G ).We define the simplicial quotient as X//G = | S ( X ) /S ( G ) | . The map S ( X ) → S ( X ) /S ( G ) defines a map X • → X//G . We will use the following facts. We have that X • → X//G is a fibration with fiber L • . Let G and H act freely on X and Y , respectively. Let f : X → Y becontinuous and D : G → H be a homomorphism (or anti homomorphism)of topological groups, and assume that f is D -equivariant, that is, f ( gx ) =14 ( g ) f ( x ), for all x ∈ X and g ∈ G . Then f defines a map F : X//G → Y //H and X • f • → Y • ↓ ↓ X//G F → Y //H is commutative. If q : X → X/G is a (locally trivial) fiber bundle then the simplicial map S ( X ) → S ( X/G ) is onto. Furthermore, two singular simplices in X have thesame image in S ( X/G ) iff they differ by an element in S ( G ). Hence the simplicialmap S ( X ) /S ( G ) → S ( X/G ) is a bijection. It follows that
X//G → ( X/G ) • isa homeomorphism. ME T ( Q, g ) . We have considered the space of Riemannian metrics of a closed manifold.We now mention some facts and give a few definitions related to the non compactcase. Let Q be a complete Riemannian manifold, with metric g . We consider ME T ( Q ) to be the set of complete Riemannian metrics on Q with the smoothtopology, which is the union, for all k , of the topologies of C k -convergence oncompact sets. Similar topology is given to DIF F ( Q ).Let f : ( X , d ) → ( X , d ) be surjective map between metric spaces. Re-call that, in this particular case, f is a ( λ, δ )-quasi-isometry if λ d ( x, y ) − δ ≤ d ( f ( x ) , f ( y )) ≤ λ d ( x, y ) + δ , for all x, y ∈ X . Let g ′ , g ′′ ∈ ME T ( Q ). Wesay that g ′ and g ′′ are ( λ, δ )-quasi-isometric if the identity ( Q, d g ′ ) → ( Q, d g ′′ ) isa ( λ, δ )-quasi-isometry, where d g ′ , d g ′′ are the intrinsic metrics induced by theRiemannian metrics g ′ and g ′′ , respectively. A useful way to prove that twometrics are quasi-isometric is the following: Let g, g ′ be two complete Riemannian metrics on the manifold Q . Suppose there are constants a, b > such that a ≤ g ′ ( w, w ) ≤ b for every w ∈ T Q with g ( w, w ) = 1 . Then g and g ′ are ( λ, -quasi-isometric, where λ = max { a , b } . The proof is straightforward (see Lemma 2.1 of [13]).15ere is a variation of the space
ME T ( Q ). We define ME T ( Q, g ) to be theset of complete Riemannian metrics on Q that are quasi-isometric to g . Wegive ME T ( Q, g ) the smooth quasi-isometry topology: basic neighborhoods ofa g ′ ∈ ME T ( Q, g ) are intersections of open neighborhoods (in
ME T ( Q )) of g ′ with the quasi-geodesic balls B λ,δ ( g ′ ) = { h ∈ ME T ( Q, g ) / h is ( λ ′ , δ ′ ) − quasi-isometric to g ′ , λ ′ < λ, δ ′ < δ } Then the inclusion
ME T ( Q, g ) ֒ → ME T ( Q ) is continuous, but the topology of ME T ( Q, g ) is strictly finer than the one induced by
ME T ( Q ).Let M be closed and let p : Q → M be a covering map. Let g ∈ ME T ( Q )be such that g is quasi-isometric to a mertric of the form p ∗ ( g ′ ) for some (henceall) g ′ ∈ ME T ( M ). Then the map ME T ( M ) lift −→ ME T ( Q, g )given by g ′ p ∗ ( g ′ ) is well defined and clearly continuous. For instance we cantake g = p ∗ ( g ′ ), g ′ ∈ ME T ( M ). Note that the topology of ME T ( Q, p ∗ ( g ′ )) isindependent of the choice of g ′ .Let DIF F ( Q, g ) be the subset of
DIF F ( Q ) of all self-diffeomorphisms φ that are at bounded g -distance from the identity 1 Q ; that is d g ( φ, Q ) = sup { d g ( φ ( x ) , x ) / x ∈ Q } is finite. We give DIF F ( Q, g ) the smooth quasi-isometry topology: the open basic sets of the identity 1 Q are intersections ofopen neighborhoods of 1 Q in DIF F ( Q ) with the sets { φ / d g ( φ, Q ) < ǫ } . Wedefine DIF F ( Q, g ) to be the subspace of
DIF F ( Q, g ) of self-diffeomorphismsof Q that are g -boundedly homotopic to 1 Q .We have that the action of DIF F ( Q, g ) on
ME T ( Q, g ) is continuous.Of course if Q is closed then ME T ( Q ) coincides with ME T ( Q, g ), for any g .Finally, define T sec < ( Q, g ) =
ME T sec < ( M, g ) / R + × DIF F ( Q, g ).16 .5. The Space at Infinity.
Let H be a complete, simply connected manifold of nonpositive curvature,that is, H is a Hadamard manifold. Recall that the space at infinity ∂ ∞ H is defined as the quotient of the set of geodesic rays by the relation: “finiteHausdorff distance” (see, for instance, [2]). In this definition “geodesic rays”can be replaced by “quasi-geodesic rays”, provided H has sectional curvatures ≤ c <
0. The compactification H = H ∪ ∂ ∞ H is given the “cone topology”. Wemention three useful facts:1. The definition of the cone topology implies the following. If β is a geodesicray between p ∈ H and q ∈ ∂ ∞ H , and V is a neighborhood of q in H ,then there is T > β ′ such that the distance between β ( t ) and β ′ ( t ) is ≤
1, for t ∈ [0 , T ],we have that β ′ ( t ) ∈ V , for all t ∈ [ T, ∞ ].2. If we assume that H has sectional curvatures ≤ c <
0, then we get thefollowing quasi-geodesic version of item 1. Let β be a geodesic ray between p ∈ H and q ∈ ∂ ∞ H , V is a neighborhood of q in H and λ > δ ≥ T > λ, δ )-quasi-geodesic ray β ′ forwhich the distance between β ( t ) and β ′ ( t ) is ≤
1, for t ∈ [0 , T ], we havethat β ′ ( t ) ∈ V , for all t ∈ [ T, ∞ ].3. If g and g are quasi-isometric complete Riemannian metrics on the sim-ply connected manifold H , with sectional curvatures ≤ c <
0, then thespace at infinity and the compactification of H are the same (as topologicalspaces) if taken with respect to g or g .We can generalize most of the concepts mentioned above to the following nonsimply connected case (see Section 2 of [13]). Let Q be a complete Riemannianmanifold (with metric g ) with sectional curvatures ≤ c <
0. Let also S bea closed totally geodesic submanifold of Q such that π ( S ) → π ( Q ) is anisomorphism. Then Q is diffeomorphic to the total space of the normal of S in Q , via the normal (to S ) exponential map. A geodesic ray (i.e. a local isometry[0 , ∞ ) → Q ) either diverges from S or stays at bounded distance from S . Thenthe space at infinity of Q can be defined as before: the space at infinity ∂ ∞ Q isdefined as the quotient of the set of geodesic rays that diverge from S , by therelation: “finite Hausdorff distance”. In this definition we can replace “geodesicrays that diverge from S ” by “ quasi-geodesic rays”. The compactification is Q = Q ∪ ∂ ∞ Q and is given the “quotient cone topology”. Then, in this context,we also get (almost) exact versions 1’,2’ and 3’ of items 1, 2, and 3 above. Wewill use only use 2’ and 3’. Here they are (see Section 2 of [13]).17’. Let β be a geodesic ray between p ∈ Q and q ∈ ∂ ∞ Q , V a neighborhoodof q in Q and λ > δ ≥
0. Then there is
T > λ, δ )-quasi-geodesic ray β ′ for which the distance between β ( t ) and β ′ ( t )is ≤
1, for t ∈ [0 , T ], we have that β ′ ( t ) ∈ V , for all t ∈ [ T, ∞ ].3’. If g is another Riemannian metric on Q with sectional curvatures ≤ c <
0, and it is quasi-isometric to g then the space at infinity and thecompactification of Q are the same (as topological spaces) if taken withrespect to g or g . (Note that we do not need S to be totally geodesicwith respect to g .) Section 2. Proof of Theorems A and B.
We will say that two Riemannian metrics g , g on a manifold M are homo-topic (or isotopic ) if there is a homotopy (or isotopy) h t : M → M , h = 1 M such that ( h ) ∗ g = g . We will also use the notation given at the beginningof section 1. In what follows M will denote a complete hyperbolic manifoldwith dim M = n ≥
5. The given hyperbolic metric will be denoted by g .In what follows in this section if M is non compact it is understood that allspace of metrics considered are with respect to g . For instance ME T ( M ) and ME T sec < ( M ) mean ME T ( M, g ) and ME T sec < ( M, g ) respectively. Fur-thermore DIF F ( M ) = DIF F ( M, g ) and DIF F ( M ) = DIF F ( M, g ). Also,in this non compact case, g , g ′ in ME T ( M ) = ME T ( M, g ) being homotopicmeans “boundedly homotopic”, that is, the homotopy h t is such that all h t areat bounded g -distance from the identity (the bound independent of t ). Λ ǫγ : ME T ǫγ ( M ) → P ( S × S n − ) . Let’s assume that there is an embedded closed geodesic γ : S ( ℓ ) → M in M of length ℓ , with orientable (hence trivial) normal bundle. We definethe subspace ME T ǫγ ( M ) ⊂ ME T sec < ( M ) as the space of all metrics g ∈ME T sec < ( M ) such that: 18 . The closed geodesic in (
M, g ) representing the homotopy class of γ coin-cides as a set with γ . Moreover, the identity map ( γ, g | γ ) → ( γ, g | γ )) is ahomothety i.e. there is c > g ( v, v ) = cg ( v, v ), for all v ∈ T γ . ⊥ gz γ = ⊥ g z γ , for all z ∈ γ . g ( u, v ) = g ( u, v ), for all u, v ∈⊥ gz γ and z ∈ γ . exp gz ( v ) = exp g z ( v ), for all z ∈ γ , v ∈⊥ gz γ with g ( v, v ) ≤ ǫ .Define ME T γ ( M ) = S ǫ> ME T ǫγ ( M ) and define T γ ( M ) to be the image of ME T γ ( M ) by the quotient map ME T ( M ) → T ( M ).Recall that R + × DIF F ( M ) acts on ME T sec < ( M ), where R + acts byscalar multiplication. Let D γ ( M ) be the isotropy group of ME T γ ( M ), that is: D γ ( M ) = ( ( λ, φ ) ∈ R + × DIF F ( M ) : λφ (cid:16) ME T γ ( M ) (cid:17) = ME T γ ( M ) ) Lemma 2.1.1.
Let ( λ, φ ) ∈ R + × DIF F ( M ) . The following statements areequivalent.
1. ( λ, φ ) ∈ D γ ( M ).2. λφg ∈ ME T γ ( M ) , for some g ∈ ME T γ ( M ).3. φ ( γ ) = γ , the derivative √ λ Dφ z : ( T z M, g ) → ( T φ ( z ) M, g ) is an isom-etry, for all z ∈ γ , and there is an ǫ > such that φ ( exp g ( v )) = exp g ( Dφ ( v )) , for all v ∈⊥ g γ , g ( v, v ) ≤ ǫ . Proof.
Clearly 1 implies 2. Also, an inspection of items 1-4 in the definitionof
ME T γ ( M ) above shows that 2 implies 3, and 3 implies 1. This proves theLemma. Lemma 2.1.2.
Assuming M is closed then the map ME T γ ( M ) → T sec < γ ( M ) is a principal D γ ( M ) -bundle. Proof.
Since the action of
DIF F ( M ) on ME T sec < ( M ) is free and M is closed we have that the action of R + × DIF F ( M ) on ME T sec < ( M )is also free. This, together with Ebin’s Slice Theorem [8] implies that q :19 E T sec < ( M ) → T sec < ( M ) is a principal (cid:16) R + × DIF F ( M ) (cid:17) -bundle. Let g ∈ ME T γ ( M ), then g ∈ ME T δγ ( M ), for some δ >
0. It follows from Remark2 after the proof of Prop. 1.1.1 that there is a C -open neighborhood W of g in ME T ( M ) and a continuous map r : W → ME T ǫγ ( M ), for some ǫ >
0, suchthat r ( g ′ ) is isotopic to g ′ , for every g ′ ∈ W , that is, r ( g ′ ) = φ ∗ g ′ , for some φ isotopic to the identity. Hence qr = q . h Here to be able to apply Remark 2 we have to use the fact, due to Sampson[24] and Eells-Lemaire ([9], Prop. 5.5), that the the map g η g is continuous,where η g : S → M is the g -geodesic freely homotopic to γ . i Since q : ME T sec < ( M ) → T sec < ( M ) is a locally trivial (cid:16) R + × DIF F ( M ) (cid:17) -bundle we can choose W to be a local product, i.e. there is an open set V in T sec < ( M ), an open neighborhood U of (1 , M ) in (cid:16) R + × DIF F ( M ) (cid:17) and amap (a section) s : V → ME T sec < ( M ) with qs = 1 V and W = U.s ( V ) = { λφ ∗ s ( a ) : a ∈ V , ( λ, φ ) ∈ U } . Note that q − ( V ) = (cid:16) R + × DIF F ( M ) (cid:17) .s ( V ).Then rs : V → ME T sec < ( M ) is also a section (i.e. qrs = qs = 1 V ) andnote that the image of rs lies in ME T γ ( M ). It is straightforward to verifythat q | − MET γ ( M ) ( V ) = q − ( V ) ∩ ME T γ ( M ) = D γ ( M ) .rs ( V ). This togetherwith Lemma 2.1.1 and the fact that q is principal (cid:16) R + × DIF F (cid:17) -bundle implythat the map V × D γ ( M ) → q | − MET γ ( M ) ( V ) given by ( v, ( λ, ϕ )) λϕs ( v ) is ahomeomorphism. This completes the proof of the Lemma.We shall now define a map:Λ ǫγ : ME T ǫγ ( M ) → P ( S × S n − )Let g ∈ ME T ǫγ ( M ). Thus g satisfies items 1-4 above. Write also N = N ǫ = { exp g ( v ) : v ∈ ⊥ γ, g ( v, v ) ≤ ǫ } . We have that ⊥ g γ = ⊥ g γ and we justwrite ⊥ γ . Let Q be the covering space of M corresponding to the infinitecyclic subgroup of π ( M, γ ( ℓ π , γ . (Here ( ℓ π , ∈ S ( ℓ ) ⊂ R .)Denote also by g and g the pullbacks to Q of the hyperbolic metric g and themetric g . Note that γ and N lift to Q and we denote these liftings also by γ and N , respectively. Hence 1-4 above also hold true if we replace M by Q .We have that the normal exponential map exp g : ⊥ γ → Q is a diffeomor-phism, and since we are assuming that the normal bundle of γ is orientable(hence trivial) we have that Q is diffeomorphic to S × R n − . Therefore we will20dentify the following objects: • Identify (
Q, g ) with ( S × R n − , ρ ) using one of the identifications givenin the remark before 1.2.1, section 1. • Identify γ ⊂ Q = S × R n − with S = S × { } ⊂ S × R n − . • Identify ⊥ γ = ⊥ S also with S × R n − . Hence the exponential map exp g is just the identity. • With all these identifications we have that N = S × S n − × [0 , ǫ ].Then 1-4 above (with Q instead of M ) can be written in the following way: The unique closed geodesic in ( S × R n − , g ) (representing the homotopyclass of S ) coincides with S . Moreover, S ( ℓ g ) → ( S , g | S ) is an isometry,where ℓ g = length g ( S ). ⊥ gz S = { z } × R n − , for all z ∈ γ . g ( u, v ) = h u, v i R n − , for all u, v ∈ R n − and z ∈ S . exp gz ( v ) = ( z, v ), for all ( z, v ) ∈ N with h v, v i R n − ≤ ǫ .Define ϕ ′ g ∈ DIF F ( S × S n − × [ ǫ, ∞ )) as ϕ ′ = exp g . Fix a diffeomorphism λ : [ ǫ, ∞ ) → [0 ,
1) and with it identify these two intervals to obtain ϕ g ∈ DIF F ( S × S n − × [0 , ϕ g ( z, v, t ) = ( z ′ , v ′ , t, ) where ( z ′ , v ′ , t ′ ) = exp g ( z, λ − ( t ) v ) and t = λ ( t ′ ).) By (4’) we have ϕ g ( z, v,
0) = ( z, v, ϕ g ∈ ( DIF F ( S × S n − × [0 , , ∂ ). We now extend ϕ g to DIF F ( S × S n − × [0 , z, v ) ∈ S × S n − . Then β ( t ) = exp gz ( tv ), t >
0, is a g -geodesicray in S × R n − . Hence it is a ρ -quasi-geodesic ray. Therefore it determinesa point at infinity (¯ z, ¯ v ) ∈ ∂ ∞ ( S × R n − ) = S × S n − . (Equivalently, theHausdorff ρ -distance between the g -geodesic exp gz ( tv ) and the ρ -geodesic ray(¯ z, t ¯ v ), t ≥
0, is finite.) We define then ϕ g ( z, v,
1) = (¯ z, ¯ v, ϕ g : ( S × S n − ) × I → ( S × S n − ) × I is continuous (this can also be provedusing item 2’ of Section 1.5). So, we get ϕ g ∈ P ( S × S n − ). We define thenΛ ǫγ ( g ) = ϕ g ∈ P ( S × S n − ). Remark.
Taking M = Q we also obtain a map Λ ǫγ : ME T ǫγ ( Q ) → P ( S × S n − ),which is essentially the map g exp g , where exp g is the normal exponentialmap. And we get the following commutative diagram (see Section 1.4):21 E T ǫγ ( M ) Λ ǫγ −→ P ( S × S n − )lift ↓ Λ ǫγ րME T ǫγ ( Q ) Lemma 2.1.3 If M is closed the map Λ ǫγ : ME T ǫγ ( M ) → P ( S × S n − ) is con-tinuous. Proof.
It is enough to prove that the map F : ME T ǫγ ( M ) × (cid:16) ( S × S n − ) × [ ǫ, ∞ ] (cid:17) −→ ( S × S n − ) × [ ǫ, ∞ ] (cid:16) g, ( z, v ) , t (cid:17) exp gz ( tv )is continuous. Choose q = (¯ z, ¯ v, ¯ t ) ∈ ( S × S n − ) × ( ǫ, ∞ ] and a neighborhood of q of the form ¯ Z × ¯ V × ( ¯ T , T ] (we can have T = ∞ ). Let also F ( g, z, v, t ) = q . If¯ t < ∞ we can clearly find an open neighborhood W of ( z, v ) in S × S n − and a C -neighborhood U of g in ME T ( M ) such that F ( g ′ , z ′ , v ′ ) ∈ ¯ Z × ¯ V × ( ¯ T , T ],for all ( z ′ , v ′ ) ∈ W and g ′ ∈ U ∩ ME T ǫγ ( M ).Let ¯ t = ∞ . Then also t = ∞ and T = ∞ . Since t exp gz ( tv ) is a g -geodesic ray, it is a ( λ, δ ) ρ -quasi-geodesic ray, for some λ > δ ≥
0. Now,given
T > W of ( z, v ) in S × S n − and a C -neighborhood U of g in ME T ( M ) such that: a. the ρ -distance between exp gz ( tv ) and exp g ′ z ′ ( tv ′ ) is less than, say, one, forall t < T , ( z ′ , v ′ ) ∈ W and g ′ ∈ U . b. the g ′ -geodesic t exp g ′ z ′ ( tv ′ ) is a (2 λ, δ + 1) ρ -quasi-geodesic ray, forevery g ′ ∈ U ∩ ME T ǫγ ( M ) and ( z ′ , v ′ ) ∈ W .But item 2’ of Section 1.5 allows us to choose T large enough so that wecan ensure that exp g ′ z ′ ( tv ′ ) ∈ ¯ Z × ¯ V × ( ¯ T , ∞ ], for all ( z ′ , v ′ , t ) ∈ W × [ T, ∞ ] and g ′ ∈ U ∩ ME T ǫγ ( M ). (Item 3’ of section 1.5 is also used here because 2’ refersto the topology generated by g , not ρ . But these two topologies coincide, byitem 3’.) This proves the Lemma. 22 ddendum to Lemma 2.1.3 If M = Q then the map Λ ǫγ : ME T ǫγ ( Q ) → P ( S × S n − ) is continuous. The proof of the Addendum is the same as the proof of lemma 2.1.3. Justrecall that by
ME T ( Q ) here we mean ME T ( Q, g ) (or ME T ( Q, ρ )). Hencethe C -neighborhood U is really a C -neighborhood in (the true) ME T ( Q ) in-tersected with a quasi-geodesic ball B λ,δ ( g ) (see section 1.4).Now, note that the definition of Λ ǫγ ( g ) = ϕ g depends on ǫ because λ :[ ǫ, ∞ ) → [0 ,
1) depends on ǫ . But τ ϕ g does not depend on ǫ , hence we geta well defined map Λ γ : ME T γ ( M ) → T OP ( S × S n − ) defined by Λ γ ( g ) = τ Λ ǫγ ( g ) = τ ϕ g . Lemma 2.1.4 If M is closed, or M = Q , then Λ γ : ME T γ ( M ) → T OP ( S × S n − ) is continuous. Proof.
This follows from the fact that in the proof of Lemma 2.1.3, the C -neighborhood U of g in ME T ( M ) does not depend on ǫ . This proves Lemma2.1.4.Let j > j -sheeted cover S × S n − → S × S n − ,( z, u ) ( z j , u ) induces a continuous map ν j : P s ( S × S n − ) → P s ( S × S n − )obtained simply by pulling back (lifting) smooth pseudoisotopies using the j -sheeted cover. Let K ⊂ P s ( S × S n − ) be a compact subset and write ι forthe inclusion of K in P s ( S × S n − ) ⊂ P ( S × S n − ). Also, let α : S → K be a map, with S compact. Let r > γ ( K ) ⊂ ME T sec < ( M ). We will also assume that r > ǫ . Hence, by thedefinition of Ω γ (see section 1.2) we haveΩ γ ( K ) ⊂ ME T ǫγ ( M ) ⊂ ME T γ ( M ) ⊂ ME T sec < ( M )Consider the following diagram: S α −→ K Ω γ −→ ME T ǫγ ( M ) i ֒ → ME T γ ( M ) ι ց Λ ǫγ ↓ Λ γ ↓ P ( S × S n − ) τ −→ T OP ( S × S n − )23here i denotes the inclusion. The square on the right is commutative, bythe definition of Λ γ . Let c : S → P s ( S × S n − ) denote the constant map c ( ϕ ) = 1 S × S n − × [0 , . Proposition 2.1.5
If, for some integer j > , we have ν j α ≃ c then ι α ≃ Λ ǫγ Ω γ α , provided r is large enough (how large depending on ℓ , K , n and thehomotopy between ν j α and c ). Therefore, the following diagram homotopy com-mutes S ι α −→ P ( S × S n − ) i Ω γ α ↓ ↓ τ ME T γ ( M ) Λ γ −→ T OP ( S × S n − )The proof of this Proposition is given in Section 6. ∆ γ : T sec < ( M ) • → T OP ( S × S n − ) //L . Recall from the introduction that L ⊂ T OP ( S × S n − ) is the subgroup of all“orthogonal” self homeomorphisms of S × S n − . That is, f : S × S n − → S × S n − belongs to L if f ( z, u ) = ( e iθ z, A ( z ) u ), for some e iθ ∈ S , and A : S → SO ( n − τ : P ( S × S n − ) → T OP ( S × S n − ) denoted the “take top” map. Proposition 2.2.1.
Let M be closed, or M = Q , and g , g ∈ ME T γ ( M ) behomotopic. Then Λ γ ( g ) = Λ γ ( g ) f , for some f ∈ L . Remark.
Recall that for M = Q we have that “homotopic” means “boundedlyhomotopic”. Proof.
Let h : ( M, g ) → ( M, g ) be an isometry homotopic to the identity 1 M .Lifting h to S × R n − we obtain an isometry h (we use the same letter) between( S × R n − , g ) and ( S × R n − , g ) such that h is at bounded ρ -distance from theidentity 1 S × R n − . (Since g , g , ρ are all quasi-isometric, the same is true if we24se the g i -distance.) Let Dh be the derivative of h . Since g , g ∈ ME T γ ( M )we have that:(i) h ( γ ) = γ . Moreover, there is e iθ ∈ S such that h ( z ) = e iθ z , for z ∈ S .(ii) h ( exp g ( z, v )) = exp g ( e iθ z, Dh ( z ) v ), for all ( z, v ) ∈ S × R n − .(iii) Dh ( z ) : { z } × R n − → { e iθ z } × R n − is orthogonal with respect to ρ , g and g . (Recall that all these metrics coincide on ⊥ γ = S × R n − , seeitems 2 and 3 in the definition of ME T γ ( M ).) Therefore Dh : S × S n − → S × S n − and Dh ∈ L . Claim. Λ γ ( g )( z, v ) = Λ γ ( g )( e iθ z, Dh ( z ) v ) . Proof of the Claim . By item (ii) above and the fact that h is at boundeddistance from the identity we have that the quasi-geodesics t exp g ( z, tv )and t exp g ( e iθ z, tDh ( z ) v ) are a bounded distance apart. Hence, by the def-inition of Λ γ ( g )( z, v,
1) we have Λ γ ( g )( z, v,
1) = Λ γ ( g )( e iθ z, Dh ( z ) v, f : S × S n − → S × S n − is defined as f ( z, v ) = ( e iθ z, Dh ( z ) v ), then fromthe claim we have that Λ γ ( g ) = Λ γ ( g ) f . This proves the Proposition.It follows from the proof of Proposition 2.2.1 that the function f is just thederivative Dh . By modifying the proof of the Proposition in a straightforwardway we obtain the following addition this Proposition: Addendum to Proposition 2.2.1.
Let M be closed, or M = Q , and g , g ∈ME T γ ( M ) such that λϕg = g , for some ( λ, ϕ ) ∈ D γ ( M ) . Then Λ γ ( g ) =Λ γ ( g ) f , where f = √ λD ( ϕ ) ∈ L . Therefore we obtain a continuous group homomorphism D : D γ ( M ) → T OP ( S × S n − ), D ( λ, ϕ ) = √ λD ( ϕ ), such that Λ γ ( ϕg ) = Λ γ ( g ) D ( ϕ ), for all g ∈ ME T γ ( M ). This together with Lemma 2.1.2 and items 2 and 3 of Section1.3 imply the following Proposition. Proposition 2.2.2.
The following diagram commutes, where the lower hori-zontal arrow is the ‘orbit map’ induced by Λ γ . E T γ ( M ) • Λ • γ −→ T OP ( S × S n − ) • ↓ ↓ME T γ ( M ) // D γ ( M ) −→ T OP ( S × S n − ) //L Remark. If M is closed item 3 of Section 1.3 together with Lemma 2.1.2 givea canonical identification of T sec < γ ( M ) • with ME T γ ( M ) // D γ ( M ). Lemma 2.2.3.
We have that
ME T γ ( Q ) // D γ ( Q ) = ME T sec < ( Q ) // R + × DIF F ( Q ) Proof.
Since D γ ( Q ) is the isotropy group of ME T γ ( Q ) we have that thesemi-simplicial map S ( ME T γ ( Q )) / S ( D γ ( Q )) → S ( ME T sec < ( Q )) / S ( R + × DIF F ( Q )) is injective. Surjectivity follows from the following three facts: Any closed g -geodesic representing γ has to be embedded. Hence Proposition1.1.1 implies that T γ ( Q ) // D γ ( Q ) = T sec < ( Q ). k -simplices are contractible. The fact, due to Sampson [24] and Eells-Lemaire ([9], Prop. 5.5), that thethe map g η g is continuous, where η g : S → M is the g -geodesic freelyhomotopic to γ . This proves the Lemma.Now, let M be closed. Lifting metrics gives us a map MET sec < ( M ) // R + × DIF F ( M ) → MET sec < ( Q ) // R + × DIF F ( Q ) But
ME T sec < ( M ) → T sec < ( M ) is a (locally trivial) bundle, hence the do-main of the above map is canonically identified with T sec < ( M ) • (see section1.3). Denote by ∆ γ the composition map T sec < ( M ) • → MET sec < ( Q ) // R + × DIF F ( Q ) = MET γ ( Q ) // D γ ( Q ) → T OP ( S × S n − ) //L and by chasing diagrams around we obtain the following commutative diagram26 E T γ ( M ) • Λ • γ −→ T OP ( S × S n − ) • ↓ ↓T sec < ( M ) • ∆ γ −→ T OP ( S × S n − ) //L Let n and k be such that the map Υ = Υ n, k : π k ( P s ( S × S n − ) • ) → π k ( T OP ( S × S n − ) // L ) is strongly nonzero. Let β : S k → P s ( S × S n − ) • rep-resent a class such that Υ([ β ]) = 0. Using the fact that the map P s ( S × S n − ) • → P s ( S × S n − ) is a weak homotopy equivalence we can assume that β has theform S k −→ ( S k ) • α • −→ P s ( S × S n − ) • , for some α : S k → P s ( S × S n − ), andsome (fixed) homotopy equivalence S k → ( S k ) • . We intend to use Proposition2.1.5, so we need to know that we can choose α so that there is an integer j > ν j α is nullhomotopic, provided r is large enough. This followsfrom Lemma 2.3.1 below, that shows, together with the definition of ‘stronglynonzero’ (see sentence before Theorem A), that in fact we have infinitely manychoices for α . Lemma 2.3.1.
Let T be an infinite torsion subgroup of π k (cid:16) P s ( S × S n − ) (cid:17) andassume that n >> k . Then the subgroup T of T consisting of all elements whichvanish under some ( ν j ) ∗ is also infinite. The proof of Lemma 2.3.1 is given in section 5.
Remarks.
1. Recall that Theorem C, proved in section 3, shows that such subgroups T do exist in our relevant cases.2. That T is a subgroup follows easily from the fact that ν j ν k = ν k ν j for allpositive integers j and k .3. n >> k in Lemma 2.3.1 and the rest of Section 2 refers to Igusa’s stablerange, namely n > max { k + 8 , k + 9 } (see [17], p.6).Write K = α ( S k ). Then K is compact. It follows from Lemma 2.3.1 above,Lemma 1.2.2 and Proposition 2.1.5 that we can choose r > γ ( K ) ⊂ ME T sec < ( M ) is well defined and the following diagramcommutes, up to homotopy S k ι α −→ P ( S × S n − ) i Ω γ α ↓ ↓ τ ME T γ ( M ) Λ γ −→ T OP ( S × S n − )Therefore, the upper right square of the following diagram homotopy commutes: S k ι • β −→ P ( S × S n − ) • ( i Ω γ ) • β ↓ τ • ↓ME T γ ( M ) ← ME T γ ( M ) • Λ • γ −→ T OP ( S × S n − ) • ↓ ↓ ↓T sec < ( M ) ← T sec < ( M ) • ∆ γ −→ T OP ( S × S n − ) //L The left square obviously commutes and the bottom right square is the com-mutative diagram at the end of section 2.2.Hence, applying the π k functor to the diagram above we obtain a commu-tative diagram of groups, and, using the facts that Υ([ β ]) = 0, and π k ( X • ) → π k ( X ) is an isomorphism for any X , we get that the map π k ( ME T γ ( M )) → π k ( T sec < ( M )) is nonzero. But this map factors through π k ( ME T sec < ( M ))) → π k ( T sec < ( M )). This proves Theorem A.To prove Theorem B apply the functor H k instead of π k to the above diagramand use the fact that h Υ([ β ]) = 0. The rest of the proof is similar. This provesTheorem B. 28 ection 3. Proof of Theorem C. Throughout this and next Sections we will use the following notation. Foran abelian group A , τ ( A ) is the torsion subgroup of A . Also, for a prime p , τ p ( A ) is the subgroup of A consisting of all elements of order a power of p . Then τ ( A ) = L p prime τ p ( A ) and A/τ ( A ) is torsion free.For a prime p , let Z ( p ) be the ring Z localized at p , i.e. Z ( p ) = Z [ , , ..., ˆ p , ... ] = { rs ∈ Q : ( r, s ) = 1 , p s } . We denote by C p the class of all abelian groups A for which A ⊗ Z ( p ) is finitely generated as a Z ( p ) -module. We remark that if A isin C p then τ p ( A ) is finitely generated, hence finite (see Section 4). Also, recallthat if A is a class of abelian groups, a group homomorphism f : G → G isan A -isomorphism if kerf, cokerf are in A .Theorem C for the case k = 2 p − π i ( ), i >
0, is the one corresponding to theidentity homeomorphism.
Theorem D.
Consider the map Υ n, k . We have the following cases: k=0 the group π ( P s ( S × S n − )) contains a subgroup ( Z ) ∞ and Υ n, restrictedto this ( Z ) ∞ is injective, provided n ≥ . k=1 the group π ( P s ( S × S n − )) contains a subgroup ( Z ) ∞ and Υ n, restrictedto this ( Z ) ∞ is injective, provided n ≥ . k=2p-4, p > the group π k ( P s ( S × S n − )) contains a subgroup ( Z p ) ∞ and Υ n, k restrictedto this ( Z p ) ∞ is injective, provided n ≥ k + 8 . Theorem E.
For p > prime, and n > k + 8 , the Hurewicz map h : π p − ( T OP ( S × S n − ) //L ) → H p − ( T OP ( S × S n − ) //L ) is a C p -isomorphism. Note that Theorem C for the case k = 0, follows directly from TheoremD above (case k = 0) because h : π ( T OP ( S × S n − ) //L ) → H ( T OP ( S × n − ) //L ) is one-to-one. The case k = 1 of Theorem C follows also from The-orem D (case k = 1) because π ( T OP ( S × S n − ) //L ) is abelian (therefore h : π ( T OP ( S × S n − ) //L ) → H ( T OP ( S × S n − ) //L ) is an isomorphism).And π ( T OP ( S × S n − ) //L ) is abelian because π ( T OP ( S × S n − )) is abelianand π ( T OP ( S × S n − )) → π ( T OP ( S × S n − ) //L ) is onto (for this last factsee the proof of Proposition 3.2 below, after the proof of Prop. 3.4). Proof of Theorem D.
Write N = S × S n − and let F : P s ( N ) → P ( N )be the “forget structure map”. We first treat the case k=2p-4, p > Note that L is homeomorphic to S × SO ( n − × Ω (cid:16) SO ( n − (cid:17) , thus itshomotopy groups are all finitely generated. In particular, the natural map π k ( T OP ( N )) = π k ( T OP ( N ) • ) → π k ( T OP ( N ) //L ) is a C p -isomorphism. SinceΥ n, k is the composition of π k ( τ ) ◦ π k ( F ) with the natural map, to prove The-orem D for the case k=2p-4, p > , it is enough to prove that there isa subgroup ( Z p ) ∞ of π k ( P s ( N )) such that π k ( τ ) ◦ π k ( F ) restricted to ( Z p ) ∞ isone-to-one. We shall prove this.There is a spectral sequence (see [16]) with E st = π t ( P ( N × I s )) convergingto π s + t +1 (cid:16) g T OP ( N ) //T OP ( N ) (cid:17) . Remark.
With our notation we have g T OP ( N ) //T OP ( N ) = (cid:12)(cid:12)(cid:12) g T OP ( N ) /S (cid:16) T OP ( N ) (cid:17)(cid:12)(cid:12)(cid:12) ,where g T OP ( N ) is the space of block topological automorphisms of N . This isthe simplicial set whose k -simplices are the automorphisms of N × ∆ k whichleave invariant each N × (cid:16) face of ∆ k (cid:17) . The space of block self homotopy equiv-alences e G ( N ) is defined analogously and we have that e G ( N ) ≃ G ( N ), where G ( N ) is the H-space of self homotopy equivalences of N . The correspondingquotients are similarly defined (see [16]).In particular we have E t = π t ( P ( N )). Consider the composite map π t ( P ( N )) = E t onto −→ E ∞ t −→ π t +1 (cid:16) g T OP ( N ) //T OP ( N ) (cid:17) −→ π t ( T OP ( N )) onto ց ր onto E t π t ( τ ) : π t ( P ( N )) → π t ( T OP ( N )). In Igusa’sstable range we can identify E s t as H s (cid:16) Z ; π t P ( N ) (cid:17) , where P ( ) is the sta-ble pseudo-isotopy functor. Then E s t is a 2-torsion group when s > s + n ≥ t + 8. And the Igusa stable condition holds for both E k and E s t suchthat both s + t = k + 1 and s ≥
2. Consequently the surjective map E k → E ∞ k is C p -injective (i.e. its kernel is in C p ). Claim.
The map π t +1 (cid:16) g T OP ( N ) //T OP ( N ) (cid:17) −→ π t ( T OP ( N )) is a C p -isomorphism. Proof of the Claim.
To see this first observe that π i ( G ( N )) is finitely gen-erated, hence we need only to show that π i (cid:16) e G ( N ) // g T OP ( N ) (cid:17) is finitely gen-erated. For this we use the functional space approach to surgery theory devel-oped by Quinn in his thesis [22] and exposed in [27], pp. 240-241. In particular,there is a fibration (up to homotopy) e G ( N ) // g T OP ( N ) → ( G/T OP ) N → L ( N ),where the homotopy groups of L ( N ) are the Wall surgery groups of π ( N ) = Z and hence finitely generated due to Browder [3]. The Homotopy groups of( G/T OP ) N are finitely generated due to Kirby-Siebenmann [19]. This provesthe claim.It follows from the Claim and the discussion above that the composite E k onto −→ E ∞ k −→ π k +1 (cid:16) g T OP ( N ) //T OP ( N ) (cid:17) −→ π k ( T OP ( N ))is C p -injective. Now, the surjective map π k ( P ( N )) = E k → E k can be identi-fied as the quotient map π k ( P ( N )) → H (cid:16) Z ; π k P ( N ) (cid:17) where Z acts on P ( N ) via the “turning upside down” involution − on P ( N ).Therefore, to prove Theorem D, for the case k=2p-4, p > , it is enoughto prove that there is a subgroup ( Z p ) ∞ of π k ( P s ( N )) such that the map π k ( P s ( N )) π k ( F ) −→ π k ( P ( N )) → H (cid:16) Z ; π k P ( N ) (cid:17) restricted to ( Z p ) ∞ is one-to-one. But observe that the map π k ( F ) : π k P s ( N ) → π k P ( N ) is a Z -module map which is a C p -isomorphism (see Lemma 4.1 of [13]).Consequently, the right hand vertical arrow in the following diagram is also a C p -isomorphism: 31 k ( P s ( N )) −→ H (cid:16) Z ; π k ( P s ( N )) (cid:17) π k ( F ) ↓ ↓ π k ( P ( N )) −→ H (cid:16) Z ; π k ( P ( N )) (cid:17) Consequently, to prove Theorem D, for the case k=2p-4, p > , itis enough to prove that there is a subgroup ( Z p ) ∞ of π k ( P s ( N )) such that themap π k ( P s ( N )) → H (cid:16) Z ; π k P s ( N ) (cid:17) restricted to ( Z p ) ∞ is one-to-one. We will prove this.There is a Z -module map π k +2 ( A ( N )) → π k ( P ( N )) which is both an epi-morphism and a C p -isomorphism, where A ( ) is Waldhausen’s functor. (Seesection 4 of [13] for more details.) Therefore we have a commutative diagram π k +2 ( A ( N )) −→ H (cid:16) Z ; π k +2 ( A ( N )) (cid:17) ↓ ↓ π k ( P s ( N )) −→ H (cid:16) Z ; π k ( P s ( N )) (cid:17) such that the right hand vertical arrow is also a C p -isomorphism. But an obvi-ous modification of the argument proving Prop. 4.6 of [13] yields a subgroup( Z p ) ∞ of π k +2 ( A ( N )) which maps monomorphically into H (cid:16) Z ; π k +2 ( A ( N )) (cid:17) ,and therefore the same is true for π k ( P s ( N )) and the map π k ( P s ( N )) → H (cid:16) Z ; π k ( P s ( N )) (cid:17) , which is what we wanted to prove. This proves TheoremD for the case k=2p-4, p > .It can readily be checked that for the case k=0 no changes are needed andthe whole argument goes through, except for the part in which we prove thatthe map E k onto −→ E ∞ k is C p -injective. Since we are working with prime p=2 inthis case, we can not use the fact that the terms E st are 2-torsion, for s > π j ( P ( N )) = 0 for j <
0, hence the spectral sequence is a firstquadrant spectral sequence and it follows that E = E ∞ .Similarly, for the case k=1 the only problem appears in the proof of the C p -injectivity of the map E k onto −→ E ∞ k . Again, since this is a first quadrant32pectral sequence, we have that E = E ∞ and we obtain the following exactsequence H (cid:16) Z ; π P ( N ) (cid:17) = E → E → E = E ∞ → E = H (cid:16) Z ; π P ( N ) (cid:17) is C -isomorphic to H (cid:16) Z ; π A ( N ) (cid:17) ,because the composite Z -module map π A ( N ) → π P s ( N ) → π P ( N ) is a C -isomorphism. Furthermore, we have (see discussion in the last Section of [13]) π A ( N ) ∼ = π A ( S ) = π ( A ( ∗ )) ⊕ π ( A ( ∗ )) ⊕ π ( N − A ( ∗ )) ⊕ π ( N + A ( ∗ ))and the conjugation leaves invariant the first two terms and interchanges the lasttwo. But π ( A ( ∗ )) and π ( A ( ∗ )) are both finitely generated, hence H (cid:16) Z ; π A ( S ) (cid:17) is finitely generated. Consequently E = H (cid:16) Z ; π P ( N ) (cid:17) is in C . Therefore E → E is C -injective. This concludes the proof of Theorem D.To prove Theorem E we want to use the following general version of Hurewicz’sTheorem (see Spanier [25], p. 510): Theorem.
Let X be a strongly simple space and A an acyclic Serre ring ofabelian groups. If π j ( X ) ∈ A , ≤ j < k then H j ( X ) ∈ A , ≤ j < k and theHurewicz map h : π k ( X ) → H k ( X ) is an A -isomorphism. For the definition of an acyclic Serre ring of abelian groups and stronglysimple space see Spanier [25], Chap.9, Sec. 6. Using the general version ofHurewicz’s Theorem given above, Theorem E reduces to the following threePropositions.
Proposition 3.1.
The class C p is an acyclic Serre rings, for any prime p . Proposition 3.2.
The space
T OP ( S × S n − ) //L is strongly simple. Proposition 3.3.
For p > prime, we have that π j ( T OP ( S × S n − ) //L ) ∈ C p , ≤ j < p − . The proof of Proposition 3.1 is given in Section 4. The proof of Proposition3.3 is given at the end of this Section. Before we prove Proposition 3.2, we givefirst a somewhat more general result: 33 roposition 3.4.
Let G be a topological group and H a subgroup of G . Let p : G • → G//H be the projection and assume that π ( p ) : π ( G • ) → π ( G//L ) is onto. Then G//H is strongly simple.
Proof.
The identity of G will be denoted by e and the corresponding vertex in G • will also be denoted by e . According to Example 18 of [25] (p. 510), it isenough to to prove the following: for each a ∈ π ( G//H, p ( e )) there is a map ω a : S × G//H → G//H such that ω a | S ×{ p ( e ) } represents a and ω a | { }× G//H ishomotopic to the identity.Let a ∈ π ( G//H, p ( e )). Since π ( p ) is onto there is a loop α ′ : S → G • suchthat p α ′ represents a . After composing α ′ with the projection map q : G • → G ,we get a loop α = qα ′ : S → G .Identify S with the boundary ∂ ∆ of the canonical 2-simplex, with its canon-ical simplicial complex structure, that is, the one with three vertices: e , e , e ,and three 1-simplices [ e , e ], [ e , e ], [ e , e ]. Let Σ be the simplicial set inducedby this structure. Note that all n -simplices of Σ, n >
1, are degenerate. Notealso that the geometric realization | Σ | is canonically homeomorphic to S andwe just write | Σ | = S . The set of n -simplices that form Σ n are sequences of n + 1 vertices of the form e i ...e i e j ...e j , i ≤ j , i, j ∈ { , , } . Such an object isdetermined by three integers i, j, k where i, j are as before and k is the num-ber of times e i appears in the sequence (hence e j appears ( n + 1) − k times).We denote this n -simplex by τ n ( i, j, k ). For each n -simplex τ = τ n ( i, j, k ) de-note by ¯ τ : ∆ n → [ e i , e j ] the simplicial map that sends the first k vertices of∆ n = [ e , ..., e n ] to e i and the last ( n + 1) − k to e j .Consider the simplicial set Σ × S ( G ) and recall that (Σ × S ( G )) n = Σ n × S ( G ) n . Since | Σ | = S is a CW-complex we have that | Σ × S ( G ) | = | Σ | ×| S ( G ) | = S × G • (see [20], p. 97). We now define a simplicial map Ω ′ = Ω ′ a :Σ × S ( G ) → S ( G ) in the following way. For ( τ, σ ) ∈ Σ n × S ( G ) n = (Σ × S ( G )) n ,define Ω ′ ( τ, σ ) : ∆ n → G as Ω ′ ( τ, σ )( v ) = α (¯ τ ( v )) . σ ( v ), v ∈ ∆ n . Applying thegeometric realization functor we obtain a map | Ω ′ | = | Ω ′ a | : S × G • → G • . Write ω ′ = ω ′ a = | Ω ′ | . Claim 1.
The map ω ′ restricted to S × { e } represents α ′ . Proof of Claim 1.
The inclusion ι : { e } ֒ → G induces the simplicial map S ( ι ) : S ( { e } ) → S ( G ). Note that S ( { e } ) has exactly one n -simplex: σ ne ∆ n → e } . Consider the following sequence of simplicial maps:Σ → Σ × S ( { e } ) S (1 Σ ) × S ( ι ) −→ Σ × S ( G ) Ω ′ → S ( G )where the first map is given by τ ( τ, σ ne ). It can be easily verified that theimage of a τ ∈ Σ in S ( G ) after applying this sequence of simplicial maps is α ¯ τ . Hence the image of [ e i , e j ] ∈ Σ by this sequence of maps is the singular 1-simplex ∆ = [ e , e ] → [ e i , e j ] α → G . It follows that after applying the geometricrealization functor to the sequence above and composing with q : G • → G atthe end we obtain S → S × { e } S × i −→ S × G • ω ′ → G • q → G (here i = | S ( ι ) | is the inclusion) and this composition is just α . Since π ( q ) : π ( G • ) → π ( G ) is an isomorphism, the claim follows. Claim 2.
The map ω ′ restricted to { } × G • is the identity. Proof of Claim 2.
The proof is similar to the proof of claim 1. Just considerthe sequence of obvious simplicial maps S ( G ) → { } × S ( G ) → Σ × S ( G ) Ω ′ → S ( G )and a simple calculation shows that this composition is the identity 1 S ( G ) . Thisproves claim 2.Now consider the simplicial group S ( H ) acting on the right on S ( G ) andtrivially on Σ. The the map Ω ′ is S ( H )-equivariant, hence we obtain a simplicialmap Ω = Ω a : Σ × S ( G ) /S ( H ) → S ( G ) /S ( H ) and the following diagram ofsimplicial maps commutes:Σ × S ( G ) Ω ′ → S ( G ) ↓ ↓ Σ × S ( G ) /S ( H ) Ω → S ( G ) /S ( H )Write ω a = | Ω a | . Applying the geometric realization functor to the diagramabove we have the following commutative diagram: S × G • ω ′ a → G • ↓ ↓ S × G//H ω a → G//H p : G • → G//H is onto and pα ′ represents a , using the diagram above weconclude that ω a satisfies the required properties. This proves Proposition 3.4. Proof of Proposition 3.2.
We just have to verify that the map π ( T OP ( S × S n − ) • ) → π ( T OP ( S × S n − ) //L ) is onto. Since this map is a fibration it isenough to prove that π ( L • ) → π ( T OP ( S × S n − ) • ) is one-to-one. Equivalentlywe have to prove that π ( L ) → π ( T OP ( S × S n − )) is one-to-one. Note that π ( L ) can be identified with π ( SO ( n − α ] ∈ π ( SO ( n − L containing ˆ α ∈ L defined by ˆ α ( z, u ) = ( z, α ( z ) .u ). Let p : S × S n − → S n − denote projection onto the second factor. Recall thatthe Hopf construction associates to each map f : S × S n − → S n − a map H ( f ) : S n → S n − such that homotopic maps go to homotopic maps (see [26],p.112). Also, the J-homomorphism J : π ( SO ( n − → π n ( S n − ) is given by J ([ α ]) = [ H ( p ◦ ˆ α )]. But J is one-to-one (see, for instance, [18], p.512), and J factors through π ( T OP ( S × S n − )) by the map π ( T OP ( S × S n − )) → π n ( S n − ),[ φ ] [ H ( p ◦ φ )]. Therefore the map π ( L ) = π ( SO ( n − → π ( T OP ( S × S n − )) is also one-to-one. This proves Proposition 3.2.Our proof of Proposition 3.3 depends on the following particular case of aresult of Goodwillie [14] which is also a consequence of Grunewald, Klein andMacko’s Theorem 1.2 in [15] (See also the last Section of [13].) Theorem.
Let p be an odd prime. Then π j A ( S ) is in C p , for j < p − . Proof of Proposition 3.3.
As before write N = S × S n − and recall thatin the proof of Theorem D we proved that π j (cid:16) T OP ( N ) //L (cid:17) is C p -isomorphicto π j +1 (cid:16) g T OP ( N ) /T OP ( N ) (cid:17) , for all 1 ≤ j . Also recall that Hatcher’s spectralsequence E rst converges to π s + t +1 (cid:16) g T OP ( N ) /T OP ( N ) (cid:17) , for s + t < k and that E st = H s (cid:16) Z ; π t P ( N ) (cid:17) is a subquotient of π t P ( N ). But π t P ( N ) ∼ = π t P ( S )(when t < k ) and, from the Theorem above, π t P ( S ) is in C p , for all t 4. Therefore E st is in C p for all s, t with s + t < k . Consequently E ∞ st , s + t < k , is in C p because these groups are subquotients of E st . Fi-nally, since π j +1 (cid:16) g T OP ( N ) /T OP ( N ) (cid:17) has a finite length filtration with suc-cessive quotient groups E ∞ st , j = s + t and C p is a Serre class, it follows that π j +1 (cid:16) g T OP ( N ) /T OP ( N ) (cid:17) is in C p , for j < k . This proves Proposition 3.3.36 ection 4. Proof of Proposition 3.1. We will use the following facts about the ring Z ( p ) . (i) If 0 → A → B → C → → A ⊗ Z ( p ) → B ⊗ Z ( p ) → C ⊗ Z ( p ) → Z ( p ) -modules. (This is because Z ( p ) is torsion free.) (ii) A submodule of a finitely generated Z ( p ) -module is finitely generated as a Z ( p ) -module. (This is because Z ( p ) is a principal ideal domain.) (iii) For a prime q = p and any abelian group A , we have τ q ( A ) ⊗ Z ( p ) = 0. (iv) For any abelian group A we have that τ p ( A ) ⊗ Z ( p ) = τ p ( A ). (Proof:Clearly Z p n ⊗ Z ( p ) ∼ = Z p n . Applying this and (i) to the subgroup generatedby a supposed element in the kernel of τ p ( A ) → τ p ( A ) ⊗ Z ( p ) , we get that τ p ( A ) → τ p ( A ) ⊗ Z ( p ) is monic. Any element in τ p ( A ) ⊗ Z ( p ) can be writtenin the form as = a ⊗ s , ( s, p ) = 1, a ∈ A , p n a = 0, for some n . Hence thereare integers λ and µ such that λp n + µs = 1. Then as = ( λp n + µs ) as = µa .Therefore as is in the image of τ p ( A ) → τ p ( A ) ⊗ Z ( p ) .) (v) If C is a finitely generated Z ( p ) -module, then C is isomorphic to a finitesum, where each summand is either Z ( p ) or Z ( p ) /p n Z ( p ) = Z /p n Z = Z p n , forsome n . This is because Z ( p ) is a principal ideal domain.Recall that an abelian group A is in the class C p if A ⊗ Z ( p ) is finitely gen-erated as a Z ( p ) -module. If A is in C p then, by items (i), (ii), (iv) and (v) τ p ( A ) ⊗ Z ( p ) = τ p ( A ) is finitely generated, hence finite. Therefore, since by (i)0 → τ ( A ) ⊗ Z ( p ) → A ⊗ Z ( p ) → (cid:16) A/τ ( A ) (cid:17) ⊗ Z ( p ) → A is in C if and only if τ p ( A ) is finite and (cid:16) A/τ ( A ) (cid:17) ⊗ Z ( p ) is finitely generatedas a Z ( p ) module. Hence, for A a torsion group, A being in C p is equivalentto τ p ( A ) being finite. On the other hand, for A torsion free, A being in C p isequivalent to A being (isomorphic to) a subgroup of ( Z ( p ) ) k = Z ( p ) ⊕ ... ⊕ Z ( p ) , forsome k . This follows from (v) and the fact that A → A ⊗ Z ( p ) is injective. (Themap A → A ⊗ Z ( p ) is injective because Z → Z ( p ) is injective and A is torsion free.)We have to prove (see Spanier [25], chap. 9, sec. 6): (a) C p contains the trivial group. (b) If A is in C p and A ′ is isomorphic to A , then A ′ is in C p . c) If A is in C p and B ⊂ A , then B is in C p . (d) If A is in C p and B ⊂ A , then A/B is in C p . (e) If → A → B → C → is a short exact sequence and A, C are in C p ,then B is in C p . (f ) If A, B are in C p , then A ⊗ B is in C p . (g) If If A, B are in C p , then T or ( A, B ) is in C p . (h) If A is in C p , then H j ( A ) is in C p , j > . Properties (a) and (b) are obviously true. Property (c) follows from facts(i) and (ii) above.We prove (d). Let A be in C p and B ⊂ A . Since 0 → B → A → ( A/B ) → A ⊗ Z ( p ) → (cid:16) A/B (cid:17) ⊗ Z ( p ) is onto. Hence (cid:16) A/B (cid:17) ⊗ Z ( p ) is finitely generated. This proves (d).Property (e) follows directly from fact (i).We prove (f). Let A and B be in C p . Since Z ( p ) ⊗ Z ( p ) ∼ = Z ( p ) we have that( A ⊗ B ) ⊗ Z ( p ) and ( A ⊗ Z ( p ) ) ⊗ ( B ⊗ Z ( p ) ) are isomorphic as Z ( p ) -modules. It followsthat ( A ⊗ B ) ⊗ Z ( p ) ∼ = ( A ⊗ Z ( p ) ) ⊗ ( B ⊗ Z ( p ) ) is finitely generated as a Z ( p ) -module.To prove (g) let A and B be in C p and recall that an abelian group C is in C p if and only if τ p ( C ) is finite and (cid:16) C/τ ( C ) (cid:17) ⊗ Z ( p ) is finitely generated as a Z ( p ) -module. Therefore, since T or ( A, B ) is a torsion group, to prove (g) we just haveto prove that τ p (cid:16) T or ( A, B ) (cid:17) is finite. But T or ( A, B ) = T or (cid:16) τ p ( A ) , τ p ( B ) (cid:17) ⊕ L q = p T or (cid:16) τ q ( A ) , τ q ( B ) (cid:17) . Therefore τ p (cid:16) T or ( A, B ) (cid:17) = T or (cid:16) τ p ( A ) , τ p ( B ) (cid:17) , whichis finite because τ p ( A ) and τ p ( B ) are finite.Finally, we prove (h). First note that, by taking the homology long exactsequence induced by 0 → τ ( A ) → A → (cid:16) A/τ ( A ) (cid:17) → 0, we can see that proving(h) is equivalent to proving (h) in the following two special cases: when A is atorsion group, and when A is torsion free.Let A be a torsion group in C p . Then A = τ p ( A ) ⊕ L q = p τ q ( A ) ! . Write B = L q = p τ q ( A ). Since τ p ( A ) is a finite p -group, it is in the acyclic Serre rings38 and 7 of p. 505 of Spanier [25]. Hence H j (cid:16) τ p ( A ) (cid:17) is also a finite p -group, forall j > 0. Also, since B is a torsion group with no p -torsion, it is in the acyclicSerre ring 8 of p. 505 of Spanier. Therefore H j (cid:16) B ) is also a torsion groupwith no p -torsion. These facts, together with the K¨unneth Formula imply that H j ( A ) = H j (cid:16) τ p ( A ) (cid:17) ⊕ H j ( B ), for all j > 0. Then, by facts (iii) and (iv) above, H j ( A ) ⊗ Z ( p ) = H j (cid:16) τ p ( A ) (cid:17) , which is finite. This proves (h) when A is a torsiongroup.Let A be a torsion free group in C p . Then A is a subgroup of ( Z ( p ) ) k , forsome k . Note that any finitely generated subgroup of ( Z ( p ) ) k is a free groupof rank at most k . Ordering properly the products of powers of primes q , q = p , we can find a sequence of integers s , s , s ... such that s i | s i +1 , p s i ,and in addition satisfying the following condition: for every s , with p s ,there is i such that s | s i . Define the free rank k subgroup B i of ( Z ( p ) ) k as B i = s i Z k . Then ( Z ( p ) ) k = S B i . Therefore ( Z ( p ) ) k = lim → B i . Note that B i +1 /B i = ( Z m ) k , with m = s i +1 s i , thus the finite groups B i +1 /B i have no p -torsion. Hence, we also have that A = lim → (cid:16) A ∩ B i (cid:17) , and (cid:16) A ∩ B i +1 (cid:17) / (cid:16) A ∩ B i (cid:17) has no p -torsion. Since homology and tensor products commute with directlimits we have that H j ( A ) ⊗ Z ( p ) = lim → H j (cid:16) A ∩ B i (cid:17) ⊗ Z ( p ) . We claim the mapsof the direct system n H j (cid:16) A ∩ B i (cid:17) ⊗ Z ( p ) o are all isomorphisms. Consequently H j ( A ) ⊗ Z ( p ) ∼ = H j ( Z ℓ ) ⊗ Z ( p ) , for some ℓ and all j > 0. And this proves (h) oncewe verify our claim. For this note that H j ( A ∩ B i ) ⊗ Z ( p ) = H j ( A ∩ B i , Z ( p ) )by the Universal Coefficient Theorem. Next apply Proposition 9.5 (ii) of [4],p. 82, in which we specify H = A ∩ B i and G = A ∩ B i +1 to conclude that ϕ : H j ( A ∩ B i , Z ( p ) ) → H j ( A ∩ B i +1 , Z ( p ) ) is onto because the index [ G : H ] isinvertible in Z ( p ) . Consequently ϕ is an isomorphism since Z ( p ) is a principalideal domain and the domain and range of ϕ are isomorphic finitely generatedfree Z ( p ) -modules. Section 5. Proof of Lemma 2.3.1. The proof of Lemma 2.3.1 depends on the following facts. Fact 1. For each finite subgroup G of π k (cid:16) P ( S × S n − ) (cid:17) , there exists a positive39nteger j such that (¯ ν j ) ∗ ( G ) = 0 in π k (cid:16) P ( S × S n − ) (cid:17) where ¯ ν j : P ( S × S n − ) → P ( S × S n − ) is induced by the j -sheeted cover S × S n − → S × S n − in the same way ν j is induced. Fact 2. For each positive integer j there is a natural commutative diagram: π k (cid:16) P s ( S × S n − ) (cid:17) F ∗ → π k (cid:16) P ( S × S n − ) (cid:17) ( ν j ) ∗ ↑ ↑ (¯ ν j ) ∗ π k (cid:16) P s ( S × S n − ) (cid:17) F ∗ → π k (cid:16) P ( S × S n − ) (cid:17) where F : P s ( S × S n − ) → P ( S × S n − ) is the forget structure map. Fact 3. ker F ∗ is a finitely generated abelian group provided k << n .Before justifying these facts, we use them to prove Lemma 2.3.1.Because of Fact 3, the torsion subgroup T k of ker F ∗ is finite. Let S be afinite subgroup of T with | S | > | T k | . Note that such S exist with arbitrarilylarge finite cardinality since T is a torsion abelian group of infinite cardinality.(T is obviously abelian when k ≥ . The case k = 0 follows from the fact thatthe stable smooth pseudo-isotopy space P s ( S × S n − ) is an infinite loop space;see [16], Appendix II.)Because of Facts 1 and 2, there exists a positive integer j such that( ν j ) ∗ ( S ) ⊆ T k . Consequently the cardinality of the subgroup ker (cid:16) ( ν j ) ∗ | S (cid:17) of T is at least aslarge as | S | / | T k | . This completes the proof of Lemma 2.3.1 (modulo verifyingFacts 1-3) since | S | / | T k | can be arbitrarily large. Verification of Facts 1-3. Fact 2 needs no justification since it is obvious.Fact 3 is a consequence of results of Burghelea-Lashof [5], see [16], Theorem5.5 and Corollary 5.6, in conjunction with Dywer’s result [7] that π i (cid:16) P s ( D n ) (cid:17) isfinitely generated for i << n .Fact 1 can be deduced from Quinn’s paper [23] by observing that ν j ( f ) be-comes as controlled over S (as needed) as j → ∞ ; then use the result that40 i (cid:16) P ( S n − ) (cid:17) = 0 for i << n which is a consequence of [16], Corollary 5.5, to-gether with the well known fact that P ( D n ) is contractible (via the Alexanderisotopy). Section 6. Proof of Proposition 2.1.5. Consider the following diagram S α −→ K Ω γ −→ ME T ǫγ ( M ) Λ ǫγ −→ P ( S × S n − ) Ω ′ γ ց ↓ lift ր Λ ǫγ ME T ǫγ ( Q )where we wrote a “prime” on Ω ′ γ : K → ME T ǫγ ( Q ) to differentiate it fromΩ γ : K → ME T ǫγ ( M ). Observe that the right hand triangle in the diagramabove is commutative (see Remark before Lemma 2.1.3). Claim 1. Λ ǫγ Ω ′ γ is homotopic to the inclusion ι : K → P ( S × S n − ) . Proof of Claim 1. This is a consequence of the fact that Λ ǫγ Ω ′ γ ( ϕ ) is equalto ϕ , up to rescaling in the t -direction. And this follows from the definitionsof Λ ǫγ and Ω ′ γ and the following fact: “let g t be a family of Riemannian metricon M ( g t ( x ) smooth with respect to ( x, t )) and define the Riemannian metric¯ g ( x, t ) = g t ( x ) + dt on M × R . Then the vertical lines t ( x, t ) are geodesics”.This is a consequence of Koszul’s formula. This proves Claim 1. Claim 2. (lift) ◦ Ω γ ◦ α is homotopic to Ω ′ γ ◦ α . Proof of Claim 2. We prove this by giving two deformations. First deformation. By hypothesis, there is a deformation h v , v ∈ [0 , h ( u ) = ν j ( α ( u )), u ∈ S , and h ( u ) = 1 S × S n − × [0 , . Let K ′ = { h v ( u ) | u ∈ S, v ∈ [0 , } . Then K ′ is compact and let r ′ > ′ and jℓ . We will assume that r > r ′ .Let R denote the normal tubular neighborhood of γ ⊂ M of width 6 r . Thelift of γ ⊂ M to S × R n − gives, as mentioned in Section 2.1, exactly one closedgeodesic, which we call also γ ⊂ S × R n − ; but also gives a countable number ofdisjoint infinite geodesic lines ℓ , ℓ , ... . In the same way lifting the neighborhood R we obtain one copy of R (which contains γ and we also denote by R ) plusdisjoint neighborhoods R i ’s of the ℓ i ’s each diffeomorphic to a cylinder R × D n − .Let u ∈ S and write ω u = Ω γ ( α ( u )). Denote by ¯ ω u the Riemannian metricwhich is the lift of ω u to Q = S × R n − , i.e. ¯ ω u = lift( ω u ). Note that ¯ ω u outside R ∪ S R i coincides with the hyperbolic metric ρ . Also ¯ ω u on R coincides with ω u , and on each R i it is the lifting of ω u on R ⊂ M . Since the cover R i → R canbe written as a composition R i p i → R p → R , where R p → R the j -sheeted cover,we have that ¯ ω u on each R i is also the lifting, by p i , of the metric Ω ′′ γ ( ν j ( α ( u ))),where we put the “two primes” on Ω ′′ γ : ν j ( K ) → ME T ǫγ ( R ), to differentiateit from the other Ω γ ’s. (Here R is considered with the pulled back metric bythe j -sheeted cover R → R . Note that, even though R is not complete, it stillmakes perfect sense to define Ω ′ γ as in section 1.2, provided r is large enough.)Finally, define (cid:16) ¯ ω u (cid:17) v as being equal to ¯ ω u outside S R i and equal to thelifting ( p i ) ∗ Ω ′′ γ ( h v ( u )) on each R i . In this way we can deform ¯ ω u to ˆ ω u , whichis equal to ρ outside R ∪ S R i , equal to ω u on R and equal to ρ ′′ on each R i ,after properly identifying each R i with R × D n − , where D n − is a disc of largeenough radius. (To recall the definition of ρ ′′ see Section 1.2.) This completesthe construction of the first deformation. Second deformation. We deform now ˆ ω u to ω ′ u = Ω ′ γ ( α ( u )). To do this just define (cid:16) ˆ ω u (cid:17) v , v ∈ [0 , ω u outside S R i and on each R i equal to ρ ′′ v , where ρ ′′ v is thedeformation given in the remark after Lemma 1.2.1.To prove that these deformations are continuous and their images lie in ME T ( Q ) (recall that ME T ( Q ) is really ME T ( Q, ρ ) with the smooth quasi-isometry topology, see section 1.4) just note the following two facts: (i). The first deformation only happens on each R i , and all metrics (cid:16) ¯ ω u (cid:17) v to-gether with ρ are invariant by a cocompact action (of translations in the ℓ i direction of certain length) coming from the cover p i : R i → R . Now just apply42emma 1.4.1. (ii). For the second deformation we proceed as in the previous case. Just notethat ρ ′′ v is actually invariant by any translation in the infinite line direction.This completes the construction of the second deformation and the proofsof Claim 2 and Proposition 2.1.5. Appendix The following Lemma is needed in the last part of the proof of Proposition1.1.1. Lemma. Let P be a closed smooth k -manifold and F : P × R m → P × R m asmooth embedding such that F ( p, 0) = ( p, and DF ( p, is the identity, forall p ∈ P . Then F is smoothly isotopic to the identity P × R m , relative to P ×{ } . Proof. Write F = ( f, g ), f : P × R m → P , g : P × R m → R m . Since g ( p, 0) = 0there are smooth maps h i : P × R m → R such that: g ( p, q ) = q h ( p, q ) + ... + q m h m ( p, q ), where q = ( q , ..., q m ) ∈ R m . h i ( p, 0) = e i , where the e i ’s form the canonical basis of R m .Define F t ( p, q ) = (cid:16) f ( p, tq ) , t g ( p, tq ) (cid:17) = f ( p, tq ) , X i q i h i ( p, tq ) ! where the last equality is given by item above. 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