SMOOTH CASE FOR T or Z

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1 SMOOTH CASE FOR T or Z Preliminaries. Theorem [DDK] Let F be a simply finite connected CW-complex and E 0 (F ) be the connected component of identity of the space of all homotopy equivalences F F. Then E 0 (F ) is homotopy equivalent to a CW-complex with finite number of cells in each dimension. Corollary. Given a finite simply connected CW-complex, there is natural number n = n(f ) such that: If κ : S 1 F be a one parameter of homotopy equivalences, such that for some N N we have κ N := κ(z N ) is homotopic to constant along z identity map of F, then κ n is homotopic to constant along z identity map of F. Technical Lemma. Let F be simply connected manifold and h : D 2 F F be D 2 -family of homothopy equivalences F F such that for x D we have h(x, ) = id F. Then β x : D 2 F defined by β x : u h(u, x) for a fixed point x F. Then [β] T or(π 2 (F )). Moreover, given a map f : D 2 F which is eqaul x F on the boundary we have [h f] [f] = [β] T or(π 2 (F )). Proof. Assume it is not, then it means that correspondent derivation D h : H k+2 (F ) H k (F ) has nontrivial element in the second cohomology, i.e. for some θ H 2 (F ) we have H 0 (F ) D h θ 0 then D h θ k = kθ k 1 D h θ so we get that [θ k 1 ] 0 implyes that [θ k ] 0 therefore for any k we have [θ k ] 0, therefore F has infinite dimension, a contradiction. Part 1. Claim A. Let S 1 E B be an orented S 1 bundle, the action of π 1 (B) on π 2 (B) is trivial. Then the action of π 1 (E) on π 2 (E) is trivial. Proof. Let γ π 1 (E, p) and α π 2 (E, p). Let us denote by γ π 1 (B, p) and ᾱ π 2 (B, p) their projection to B. From the assumptons we have that there is a homotopy h : S 2 S 1 B such that h(u, v) = p, h(u, ) γ and h(, v) ᾱ. The induced bundle S 1 E S 2 S 1 is trivial, i.e. E = S 2 S 1 S 1, therefore the α γ = α for any element in γ π 1 (E ). Claim B. Let F E B be a fiber bundle with simply connected fiber F, which admits a trivialization on 1-skeleton of B, π 1 (B) is almost nilpotent 1

2 and the action of π 1 (B) on π 2 (B) is almost trivial. Then the action of π 1 (E) on π 2 (E) is almost trivial. Proof. Let γ π 1 (E, p) and α π 2 (E, p). Let us denote by γ π 1 (B, p) and ᾱ π 2 (B, p) their projection to B. From the assumptons we have that there is a homotopy h : S 2 S 1 B such that h(u, v) = p, h(u, ) γ and h(, v) ᾱ. The induced bundle F E S 2 S 1 admits a trivialization over u S 1. Let E be induced bundle F E S 2 v S 2 S 1. The monodromy map (arround S 1 ) f : E E induces identity map on the base S 2 and due the trivialization over u S 1, it can be chousen to be identity on F v. Let us consider map m : D 2 S 2 which is homeomorphism inside D 2 and m( D 2 ) = v and fix a trivialization F D 2 on the induced bundle, let m : F D 2 E be correspondent mapping, clearly m (, x) is an homeomorphism F x and the fiber F m(x). Then the map f : E E is completely described by the map f : F D 2 F D 2 uniquely described by identity m f = f m. Clearly f (, x) = id F x for any x D 2. Now applying the Technical Lemma we get that there is element β T or(π 2 (F )) such that if i(β) = α γ α. Sinse the bundle is trivial over S 1 we get that i(nβ) = α γn α and therefore for n(f ) = T or(π 2 (F )) we have α γn(f ) α = 0. Since π 1 (B) is almost nilpotent we have that its subgroup < γ n(f ) γ π 1 (B) > is of finite index. Part 2. Now we start to prove that T or(π 1 (M)) Z(π 1 (M), it is easely follows from the following property: Definition. A connected manifold M is called reliable if for any τ T or(π 1 (M, p)) and for any other element γ π 1 (M, p) there is a map of h : S 1 S 1 M such that h(u, v) = p, h(u, ) γ and h(, v) τ. and more over there is a finite cover ι : T 2 : S 1 S 1 such that there is a map of solid torus f : D 2 S 1 M and an isomorphism i : T 2 D 2 S 1 such that f i = h ι. Claim AA. Let S 1 E B be an oriented S 1 -bundle, the base B is reliable and the action of π 1 (B) on π 2 (B) is almost trivial. Then E is reliable. 2

3 Proof. Let γ π 1 (E, p) and τ T or(π 1 (E, p)). Let us denote by γ π 1 (B, p) and τ π 1 (B, p) their projection to B. From the assumptons we have that there is a map h : S 1 S 1 B such that which satisfy the definition of relable manifold. Note that sinse B is realabale we have [h] = 0 H 2 (B, Q). Therefore the induced bundle S 1 E S 1 S 1 is trivial, in particular E = T 3. Therefore there is a lifting h : S 1 S 1 E of map h such that h(u, v) = p, h(u, ) γ and h(, v) τ, One can consider the covering map h ι : T 2 E and existance of map f : D 2 S 1 B shows that h ι is homotopic to a map h : T 2 E which projection to B is the central S 1 of D 2 S 1. Therefore there is element ψ π 1 (B) and φ π 1 (S 1 ) the fundamental group of fiber, such that h ι is homotopic to a torus with these generators. If ψ = 0 π 1 (B) we can homotopy h to single S 1 -fiber, and it gives the needed map of solid torus. Otherwise, we have i(φ) T or(π 1 (E)) therefore passing to a finite cover of T 2 if nessesury we get a map of disc D 2 E whic contracts one of generators (kφ) of T 2. The projection of this disc to B gives an element α π 2 (B) and sinse π 1 (B) acting on π 2 (B) almost trivially we get that passing to finite cover of T 2 again (along ψ) we have a map of ḡ : S 2 S 1 B with ḡ(u, ) ψ l and ḡ(, v) α. We can lift this map to a map g : D 2 S 1 E on such a way that g i = h ι. Which together with homothopy above gives the needed map of D 2 S 1 E. Claim BB. There is n = n(f, dim(b)) such that if F E B be a fiber bundle with simply connected fiber F, reliable base B with nilpotent fundamental group, which admits a trivialization on a 1-skeleton of B then there is a cover Ẽ of the total space E with at most n folds which is relaeble. Proof. Let us first describe the plan of the proof: 1. Use the corollary to prove that there is a finite cover B of B with at most n = n(f, dim(b)) folders such that for any two elements γ π 1 (B, p) and τ π 1 (B, p) there is a map h : S 1 S 1 B such that which satisfy the definition of relable manifold and such that the induced bundle h over S 1 S 1 and fiber F is homotopy trivial. 3

4 2. Using techical Lemma we prove that a section over h : S 1 S 1 B, after finite cover (whci is infact a cover of T 2 ) can be cotructed by solid torus (i.e. Ẽ is reliable. Step 1. Let γ π 1 (E, p) and τ T or(π 1 (E, p)). Let us denote by γ π 1 (B, p) and τ π 1 (B, p) their projection to B. From the assumptons we have that there is a map h : S 1 S 1 B such that which satisfy the definition of relable manifold. The induces bundle f over D 2 S 1 is trivial sinse the bundle over S 1 is. Consider induced bundle h over S 1 S 1. After finite cover ι : T 2 S 1 S 1 the induced bundle ( h ι) over T 2 becomes trivial, sinse it is also can be induced induced bundle for i : T 2 D 2 S 1. The bundle over S 1 S 1 admits a trivialization on 1-skeleton for the stadard CW-structure. Therefore it can be induced from a bundle on S 2 and a map of degree 1 which sends 1-skeleton of S 1 S 1 to a single point of S 2. Such a bundle can be described by a one-parameter family of homeomorphisms κ : S 1 F F. Note that homotopy type of this map does NOT depend on the choice of trivialization over 1-skeleton of S 1 S 1. Since the induced bundle over T 2 is trivial we have that if ι : T 2 S 1 S 1 is an N-fold cover then κ N : S 1 F F is homotopic to map (z, x) x therefore applying the Corollary we get that infact κ n(f ) is homotopic to constant along (z, x) x. From this it is clear that there ia a finite cover B of B with at most n = n (F, dim(b)) folders such that for any two elements γ π 1 (B, p) and τ π 1 (B, p) there is a map h : S 1 S 1 B such that and such that the induced bundle h over S 1 S 1 and fiber F is homotopy trivial. Let us pass to this covering (i.e. now B := B) and continue. Step 2. Let us summarize what we have: 1. The induced bundle f over D 2 S 1 is trivial sinse the bundle over S 1 is. 2. There is yet an other homotopic trivialization over D 2 S 1 = T 2 which is induced from homotopic trivialization on S 1 S 1. Such a pair of trivializations can be charcterised by homotopy type of ĥ : D 2 F F, a D 2 -family of homothopy equivalences F F such that for x D we have ĥ(x, ) = id F, it can be constructed by lifting homotopy D 2 S 1 B to a homotopy to m : D 2 F R E (here S 1 = R/Z on such a way that on its boundary D 2 F it goes along the trivialization over D 2 S 1 = T 2. Then defining ĥ(u, x) = π F m(u, x, 1) for all u D 2 and x F. 4

5 Let us take a horisontal section over S 1 S 1 it is covered by a section over T 2. First lets us extend this section over D 2 u for u S 1, it is possible sinse F is simply connected. According to the Technical Lemma we have that [s(d 2, 2π)] [m(d 2, 0)] = β T or(π 1 (F )) therefore [m(d 2, 2πn)] [m(d 2, 0)] = [m(d 2, 2π)] n [m(d 2, 0)] = nβ = 0 T or(π 1 (F )) for some n. In particular after n-fold cover of T 2 it has a section which can be cotructed by solid torus. References [DDK] Dror, E. and Dwyer, W. G. and Kan, D. M., Self-homotopy equivalences of virtually nilpotent spaces, Comment. Math. Helv., 56, 1981, 4, pp , 5