HIGHER HOMOTOPY OF GROUPS DEFINABLE IN O-MINIMAL STRUCTURES

ALESSANDRO BERARDUCCI, MARCELLO MAMINO, AND MARGARITA OTERO

Abstract. It is known that a definably compact group G is an extension of a compact Lie group L by a divisible torsion-free normal subgroup. We show that the o-minimal higher homotopy groups of G are isomorphic to the corresponding higher homotopy groups of L. As a consequence, we obtain that all abelian definably compact groups of a given dimension are definably homotopy equivalent, and that their universal cover are contractible.

1. Introduction

Definable groups in o-minimal structures have been studied by several authors. The class of such groups includes all algebraic groups over either a real closed field or an algebraically closed field, but is not limited to such groups. Any compact Lie group is isomorphic to a real algebraic subgroup of the general linear group GL(n, ℝ) and as such it also falls within the o-minimal category. On the other hand a non-compact Lie group may or may not be definable (e.g. the integers are not definable in any o-minimal structure). Working in the o-minimal category can have some advantages even if one is only interested in the compact Lie groups. Indeed some proofs can be simplified due to the presence of a flexible set of tools like the cell decomposition theorem, the triangulation theorem (holding in the presence of field operations), and a model theoretic notion of “Euler characteristic” invariant under definable bijections, not necessarily continuous. The presence of these tools, combined with the general lack of analytical tools (like the exponential map of Lie group theory), has prompted some researchers in the field to adapt to the o-minimal category some of the usual devices of algebraic topology (homology, cohomology, homotopy) in order to investigate the definable groups. A fundamental result obtained in this way is that a definably compact abelian group G of (ominimal) dimension d has the same torsion subgroup of a classical d-dimensional torus [10]. By the work of several authors on the so called ”Pillay’s conjecture” of [16] every definably compact group G has a canonical type-definable divisible subgroup G^{00} such that G/G^{00} with the “logic topology” is a compact Lie group [7]. In the abelian case G/G^{00} is a torus and its dimension can be computed by studying the structure of the torsion subgroup. This can be determined as follows. In [11] it is proved that in the abelian case G^{00} is torsion free and therefore G and G/G^{00} have isomorphic torsion subgroups. By the quoted result of [10], one then deduces that the o-minimal dimension of G coincides with the dimension of G/G^{00} as a Lie group. (This continues to hold also in the non-abelian case [11]. Also the fact that G^{00} is torsion free can be extended to the non-abelian case [3].) The proof in [10] proceeds by studying the (o-minimal) fundamental group π_{1}(G) of G, showing that it is isomorphic to ℤ^{d}. In the light of Pillay’s conjecture this can be equivalently expressed in the form

Date: Sep. 29, 2008.

2000 Mathematics Subject Classification. 03C64, 57T20, 55P45.

Key words and phrases. definable group, Lie group, o-minimal homotopy.

First author is partially supported by GEOR MTM2005-02568.

Third author is partially supported by GEOR MTM2005-02568 and Grupos UCM 910444.

π_{1}(G) ≅ π_{1}(G/G^{00})  (1)

where the latter is the fundamental group of the Lie group G/G^{00}. The nice thing of this reformulation is that it makes sense to ask whether (1) continues to hold in the non-abelian case. The paper [4] deals with the corresponding question for the cohomology groups, showing that for every definably compact group G, every i, and any constant coefficient group,

H^{i}(G) ≅ H^{i}(G/G^{00})  (2)

where the left-hand side denotes the the i-th o-minimal (singular or sheaf) cohomology group. Taking i = 1 we easily deduce the validity of (1) in the non-abelian case by the o-minimal Poincaré-Hurewicz theorem in [10] and the corresponding classical result. We actually show in this paper that

π_{n}(G) ≅ π_{n}(G/G^{00})  (3)

for all n ≥ 1 and all definably compact groups G (where the left-hand side is the o-minimal n-th homotopy group of G). The proof of (3) is non-trivial even in the abelian case, where it amounts to show that

π_{n}(G) = 0  (4)

for all n > 1 (recalling that the case n = 1 has already been dealt with). This would be easy to prove if one could show (by analogy with the torus) that G factors definably into one-dimensional subgroups (as π_{n}(G) would also factor), but in general this is not the case (here we measure the effect of the lack of the exponential maps). Since π_{n}(G) can be proved to be divisible for n > 1, to prove (4), it suffices to show that π_{n}(G) is finitely generated. Clearly we cannot make a direct appeal to the triangulation theorem since a finite simplicial complex does not need to have a finitely generated π_{n} (although it has finitely generated homology and cohomology groups). What we need is to use the structure of o-minimal H-space of (the triangulated copy of) G. Recall that any topological group is an H-space and it is known that an H-space with finitely generated homology groups has finitely generated higher homotopy groups. This however holds in the classical setting, and we need to transfer the latter to the o-minimal category. To do so we exploit some results of [1] which allow to transfer the o-minimal H-space structure (but not the group structure) to the classical setting – so we can apply the above classical result – and then go back to the o-minimal setting thanks again to a result in [1] which establishes a group isomorphism between the classical and o-minimal higher homotopy groups of the relevant spaces. As a consequence of (4), we first obtain that all the abelian definably compact definably connected groups of the same dimension are definably homotopy equivalent and therefore their o-minimal homology groups (over ℤ) are torsion free. This is done using results of [13] and the o-minimal Whitehead theorem proved in [1]. Then, we apply results in [2] to get that the universal covering group of our abelian definable group is contractible (in the category of locally definable spaces).

Having proved (3) in the abelian case it remains to deal with the non abelian case. We have already discussed the case n = 1. The case n > 1 is a reduction to the abelian and to the semisimple centreless cases making use of the long exact homotopy sequence of a fibration and again results in [1]. The whole of section 2 is devoted to prove that a surjective definable morphism G → G/H of definable groups is a definable fibration (in analogy with the corresponding result for Lie groups). Once we have a definable fibration we obtain a long exact homotopy sequence by [1]. Of course to be able to compare π_{n}(G) and π_{n}(G/G^{00}) we need to work with homotopy sequences both in the o-minimal and in the Lie category. To be able to match the two sides, we exploit the fact that the correspondence G ↦ G/G^{00} gives rise to a functor from definable groups (and definable homomorphisms) to compact Lie groups (and continuous homomorphisms) which, by [3], preserves exact sequences.

For a general reference on definable groups see [12], and for Lie groups see [8].

1.1. Notations. We work over an o-minimal expansion M of a real closed field. Definable means definable in M with parameters. We take the order topology in M and the product topology in the M^{n}’s. We denote by I the interval [0, 1] ⊂ M. If G is a definable group, π_{n}(G) denotes the o-minimal n-th homotopy group of the pointed space (G, e) (see [1, section 4]), where e denotes the identity of G. If G is a Lie group π_{n}(G) is the classical n-th homotopy group of the pointed space (G, e), where e denotes the identity of G. Similar notation for homology and cohomology. Except in section 2, M is a sufficiently saturated structure.

2. Definable fibre bundles

Definition 2.1. Let E and B be definable sets. Let p : E → B be a definable continuous map.

We say that p is a definable fibre bundle with fibre a definable set F if there is a definable finite open definable covering of B = ⋃_{i=1}^{k} U_{i} and for each i, a definable homeomorphism φ_{i}: U_{i} × F → p^{−1}(U_{i}) such that p ∘ φ_{i}: U_{i} × F → U_{i} is the projection onto the first factor.

We say that p is a definable fibration if p has the homotopy lifting property with respect to all definable sets i.e., for every definable set X, for every definable homotopy f : X × I → B and for every definable map g : X → E such that p ∘ g = f(−, 0) there is a definable homotopy h : X × I → E such that p ∘ h = f (h is a lifting of f) and h(−, 0) = g(−).

A definable covering map is an example of a definable fibre bundle in which the fibre is finite, while a projection p : B × F → B is an example of a trivial fibre bundle with arbitrary fibre. As the reader may verify, both maps are also definable fibrations (the first one by uniqueness of path lifting, see [1, proposition 4.10]). A less trivial example of definable fibre bundle is provided by the following.

Lemma 2.2. Let H be a definable subgroup of a definable group G. Then the natural map p : G → G/H is a definable fibre bundle (with the quotient topology on G/H).

Proof. Consider a (discontinuous) definable section s : G/H → G. Let X be a large subset of G/H (where large means that that dim((G/H) \ X) < dim(G/H)) such that s is continuous on X. Without loss of generality, we can assume X to be open in G/H, for, observe that the interior of X is large, since the dimension of the boundary of X is strictly less than the dimension of G/H. By [14, claim 2.12] finitely many left translates X_{1}, …, X_{m} of X cover G/H. Fix an i ≤ m. We have an induced section s_{i} : G/H → G (a conjugate of s) that is continuous on X_{i}. Therefore, the map φ_{i}: X_{i} × H → p^{−1}(X_{i}) defined by φ_{i}(gH, h) = s_{i}(gH)h is a definable homeomorphism with its inverse sending g to (gH, (s_{i}(gH))^{−1}g). Finally, note that p ∘ φ_{i} = p_{1}, where p_{1} : X_{i} × H → X_{i} is the projection onto the first coordinate. □

This section will be devoted to prove the following.

Theorem 2.3. Every definable fibre bundle is a definable fibration.

Which we need, in particular, for the following.

Corollary 2.4. Let H be a definable subgroup of a definable group G. Then p: G → G/H is a definable fibration (with the quotient topology on G/H).

The proof of theorem 2.3 is an adaptation of section 7 in chapter 2 of [17]. The main difference lies in that we can not work with the path-space (which is not even a definable object) and its compact-open topology.

In the rest of this section, p : E → B is a fixed continuous definable map between definable sets E and B.

Now, we are going to consider a definable subset X of the path-space of B. More precisely, we will consider a family of paths parametrized by some definable set X, which we will represent using a continuous definable function from X × I to B. The reader may think of any x ∈ X as a path itself, and, by abuse of notation, of f(x, −) as x(−). From this point of view f is a continuous map from X to the path-space of B with the compact-open topology. For each such object, we will call lifting function any continuous definable map sending each pair (e, x), where x is an element of X and e is a point in p^{−1}(x(0)), to a lifting of x starting at e. It is easily shown (lemma 2.6) that the existence of a lifting function for any X is equivalent to p being a definable fibration. Moreover, in lemma 2.9, we will show how, given lifting functions for each element of a finite definable open covering of X, we can get a lifting function for X (basically blending the pieces via a definable partition of unity). Finally, through lemmata 2.10 and 2.11, we will use the local triviality of p to get such a covering for any subset X of the pathspace of B.

Definition 2.5. For any continuous definable function f : X × I → B, let B̄_{f} be the definable set

B̄_{f} = {(e, x) ∈ E × X | p(e) = f(x, 0)} .

We will call a continuous definable function λ : B̄_{f} × I → E a lifting function for f if for all (e, x) ∈ B̄_{f},

p ∘ λ ((e, x), −) = f(x, −) and λ ((e, x), 0) = e.

Lemma 2.6. The map p is a definable fibration if and only if for any definable set X and for any continuous definable function f : X × I → B there is a lifting function for f.

Proof. To prove the if part, consider functions f : X × I → B and g : X → E such that f(−, 0) = p ∘ g(−). Let λ be a lifting function for f. Then h = λ((g × Id)(−), −) : X × I → E is a lifting of f and h(−, 0) = g(−).

To prove the only if part, consider the homotopy h : B̄_{f} × I → B defined by h((e, x), t) = f(x, t). Since h(−, 0) = p ∘ p_{1}, where p_{1} : B̄_{f} → E is the projection on the first component, we may lift h to a map λ : B̄_{f} × I → E such that λ(−, 0) = p_{1}, which is a lifting function for f. □

Definition 2.7. For any continuous definable function f : X × I → B and for any definable subset W ⊂ X, let W̃_{f} be the definable set

W̃_{f} = {(e, x, s) ∈ E × W × I | p(e) = f(x, s)} .

We will call a continuous definable function Λ : W̃_{f} × I → E an extended lifting function for f over W if for all (e, x, s) ∈ W̃_{f},

p ∘ Λ ((e, x, s), −) = f(x, −) and Λ ((e, x, s), s) = e.

The idea behind an extended lifting function is that it maps any triple (e, x, s), where x is a path, s is an element of I, and e is a point in p^{−1}(x(s)), to a lifting of x passing through e at time s. As we will see in the next lemma, an extended lifting function is nothing but a lifting function in disguise (basically because, using appropriate lifting functions, we can lift separately the two halves of any path which are before and after s); in fact, extended lifting functions were introduced for merely technical reasons.

Lemma 2.8. For any definable set X and for any definable continuous function f : X × I → B there is lifting function for f if and only if for any definable set X and for any definable continuous function f : X × I → B there is an extended lifting function for f over X.

Proof. One direction is trivial. For the other one, let Y = X × I and let g′, g′′ : Y × I → B be defined by

g′((x, s), t) = \begin{cases}f(x, s − t) & for t ≤ s \\ f(x, 0) & for t > s \end{cases}

g′′((x, s), t) = \begin{cases}f(x, s + t) & for t ≤ 1 − s \\ f(x, 1) & for t > 1 − s \end{cases}

By hypothesis there are lifting functions λ′ and λ′′ for g′ and g′′ respectively. Now, an extended lifting function Λ : X̃_{f} × I → E for f over X is given by

Λ((e, x, s), t) = \begin{cases}λ′((e, (x, s)), s − t) & for t ≤ s \\ λ′′((e, (x, s)), t − s) & for t > s \end{cases}

which is easily proved continuous, moreover for t ≤ s we have

p ∘ Λ((e, x, s), t) = p ∘ λ′((e, (x, s)), s − t) = g′((x, s), s − t) = f(x, t)

and similarly for s < t; and finally

Λ((e, x, s), s) = λ′((e, (x, s)), 0) = e.
□

Lemma 2.9. Given a definable set X and a definable continuous function f : X × I → B, if there is a finite open covering W of X such that for each element W of W there is an extended lifting function for f over W, then there is a lifting function for f.

Proof. Let W = {W_{j}}_{j∈J} and consider a definable partition of unity {g_{j}}_{j∈J} on X (i.e. g_{j} : X → M^{≥0} and ∑_{j} g_{j} ≡ 1) such that cl (W′_{j}) ⊂ W_{j} for each j, where W′_{j} = {x ∈ X | g_{j}(x) > 0}. For each subset κ of J define

W′_{κ} = ⋃_{j∈κ} W′_{j}  f_{κ} = f|_{W′_{κ}×I}  B̄_{κ} = B̄_{f_{κ}}  g_{κ} = ∑_{j∈κ} g_{j}

Consider a maximal subset α of J such that there is a lifting function λ_{α} for f_{α}. For the sake of contradiction, suppose there is a j_{0} ∈ J \ α. We shall construct a lifting function for f_{β}, where β = α ∪ \{j_{0}\}. Let h = \frac{g_{α}}{g_{β}} : W_{β} → I. Clearly h is the constant 1 on W′_{α} \ W′_{j_{0}}, the constant 0 on W′_{j_{0}} \ W′_{α}, and takes values in (0, 1) on W′_{α} ∩ W′_{j_{0}}. Define μ : B̄_{{j_{0}}} → E by:

μ(e, x) = \begin{cases}e & for 0 ≤ h(x) < \frac{1}{2} \\ λ_{α}((e, x), 2h(x) − 1) & for \frac{1}{2} ≤ h(x) < 1 \end{cases}

Consider an extended lifting function Λ for f over W_{j_{0}}, then λ_{β} : B̄_{β} × I → E defined by:

λ_{β}((e, x), t) = \begin{cases}Λ((e, x, 0), t) & for 0 ≤ h(x) < \frac{1}{2} \\ λ_{α}((e, x), t) & for \frac{1}{2} ≤ h(x) ≤ 1 and 0 ≤ t ≤ 2h(x) − 1 \\ Λ((μ(e, x), x, 2h(x) − 1), t) & for \frac{1}{2} ≤ h(x) ≤ 1 and 2h(x) − 1 < t ≤ 1 \end{cases}

is a lifting function for f_{β}. Note that λ_{β} is well defined, and continuous since on \{x ∈ W_{β} | h(x) ≥ \frac{1}{2}\}

λ_{α}((e, x), 2h(x) − 1) = Λ((μ(e, x), x, 2h(x) − 1), 2h(x) − 1)

and μ(e, x) = e on \{x ∈ W_{β} | h(x) = \frac{1}{2}\} (in particular, here we used the fact that Λ is defined on cl (W′_{j_{0}}) ⊂ W_{j_{0}}). Moreover p ∘ λ_{β}((e, x), t) = f(x, t) and λ_{β}((e, x), 0) = e because Λ and λ_{α} are (extended) lifting functions. □

Lemma 2.10. Let f be a continuous function f : X × I → B and let U be a finite definable open covering of B. Then, there are continuous definable functions 0 ≡ g_{1} ≤ … ≤ g_{k+1} ≡ 1 : X → I such that for each x ∈ X and each i ∈ {1, …, k}, the set f(x, [g_{i}(x), g_{i+1}(x)]) is entirely contained in some element of U.

Proof. Caveat: The fact that if the lemma holds for an ordered set of continuous definable functions G then it also holds for any superset of G will be used tacitly and pervasively. The proof is by induction on the dimension of X. If dim(X) = 0 then X is finite and the statement is trivial.

Suppose the statement is true for dim(X) ≤ n, then we will prove it for dim(X) = n + 1. Observe that for each x ∈ X there are 0 = l_{1} < … < l_{h(x)+1} = 1 such that for each i ∈ {1, …, h(x)} the set f(x, [l_{i}, l_{i+1}]) is entirely contained in some element of U. Without loss of generality we may suppose h(x) to be minimal for each x, then h = sup_{x∈X} h(x) < ∞ by uniform finiteness. By definable choice we have definable functions 0 ≡ l_{1} ≤ … ≤ l_{h+1} ≡ 1 : X → I, not necessarily continuous, such that for each i ∈ {1, …, h} the set f(x, [l_{i}(x), l_{i+1}(x)]) is entirely contained in some element of U. Let X be a finite partition of X into definable sets such that each of the functions l_{i} is continuous on each of the sets of X. Now consider the set X′ of the interiors of the sets in X, and let Y = X \ ⋃ X′: clearly dim(Y) ≤ n. By the induction hypothesis applied to f_{|Y ×I} and U we get definable continuous functions 0 ≡ g_{1} ≤ … ≤ g_{k+1} ≡ 1 : Y → I such that for each x ∈ Y and each i ∈ {1, …, k} the set f(x, [g_{i}(x), g_{i+1}(x)]) is entirely contained in some element of U. Consider a definable open neighbourhood Z of Y in X such that there is a retraction r : Z → Y. We claim that, by continuity of the functions g_{i} ∘ r, we can find in Z a new definable open neighbourhood Z′ of Y such that for each x ∈ Z′ and each i ∈ {1, …, k} the set f(x, [g_{i} ∘ r(x), g_{i+1} ∘ r(x)]) is entirely contained in some element of U. In fact, let U = {U_{j}}_{j}: working in Z, we observe that, since I is closed and bounded, the projection p_{1} : Z × I → Z on the first component is a closed function, hence, for each i and j, the set

Z_{i,j} = {x ∈ X | f(x, [g_{i} ∘ r(x), g_{i+1} ∘ r(x)]) ⊂ U_{j}}
= Z \ p_{1} ({(x, t) ∈ Z × I | t ∈ [g_{i} ∘ r(x), g_{i+1} ∘ r(x)]} \ f^{−1}(U_{j}))

is open, and so is Z′ = ⋃_{i,j} Z_{i,j}, which contains Y by hypothesis.

Finally, we have a definable open covering X′′ = X′ ∪ {Z′} of X such that the statement of the lemma holds for each element of X′′: we will show that this implies the statement for X. For, let X′′ = {V_{1}, …, V_{m}} and suppose that for each i the statement holds for V_{i} and definable functions g_{i,j} : V_{i} → I. Consider a shrinking {V′_{1}, …, V′_{m}} of X′′ such that cl (V′_{i}) ⊂ V_{i} for each i. Since I is contractible, for each i and j we have a continuous extension g′_{i,j} : X → I of g_{i,j}_{| cl(V′_{i})}. Hence, the functions

g_{r}(x) = the r-th element of {g′_{i,j}(x)}_{i,j} in ascending order

satisfy the statement, since they do on each of the sets cl (V′_{i}) which cover X. □

Lemma 2.11. Given definable subsets U_{1}, …, U_{k} of B, if for any definable set X, for any definable continuous function f : X × I → B, and for any i, there is an extended lifting function for f over {x ∈ X | f(x, I) ⊂ U_{i}}; then for any definable set X, for any definable continuous function f : X × I → B, and for any definable continuous functions 0 ≡ g_{1} ≤ … ≤ g_{k+1} ≡ 1 : X → I, there is an extended lifting function for f over W, where

W = {x ∈ X | f(x, [g_{i}(x), g_{i+1}(x)]) ⊂ U_{i} for all i = 1, …, k}.

Proof. The idea behind the following convoluted formulæ is that we can lift any path x in W for times in the interval [g_{i}(x), g_{i+1}(x)] using the extended lifting function for U_{i}, which we have by hypothesis. Hence, all we have to do, is to lift such paths interval by interval, taking care that they fit together properly.

Let Y = {(x, a, b) ∈ X × I^{2} | a < b}, and let h : Y × I → B be defined by

h((x, a, b), t) = \begin{cases}f(x, a) & for t ∈ [0, a) \\ f(x, t) & for t ∈ [a, b] \\ f(x, b) & for t ∈ (b, 1] \end{cases}

By hypothesis, for each i, we have an extended lifting function Λ_{i} for h over Y_{i} = {y ∈ Y | h(y, I) ⊂ U_{i}}. Let

L_{i,j} = {((e, x, s), t) ∈ W̃_{f} × I | s ∈ [g_{i}(x), g_{i+1}(x)] and t ∈ [g_{j}(x), g_{j+1}(x)]}

and define Λ′_{i,j} : L_{i,j} → E by

Λ′_{i,j}((e, x, s), t) = \begin{cases}Λ_{i}((e, (x, g_{i}(x), g_{i+1}(x)), s), t) & for i = j \\ Λ_{j}((Λ′_{i,j−1}((e, x, s), g_{j}(x)), (x, g_{j}(x), g_{j+1}(x)), g_{j}(x)), t) & for j > i \\ Λ_{j}((Λ′_{i,j+1}((e, x, s), g_{j+1}(x)), (x, g_{j}(x), g_{j+1}(x)), g_{j+1}(x)), t) & for j < i \end{cases}

It is routine to prove simultaneously by induction on |j − i| that these are welldefined continuous functions and

(⋆)  ∀((e, x, s), t) ∈ L_{i,j} f(x, t) = p ∘ Λ′_{i,j}((e, x, s), t)

Now, let

l : X × I → {1, …, k}
l(x, s) = i < k iff g_{i}(x) < g_{i+1}(x) and s ∈ [g_{i}(x), g_{i+1}(x))
l(x, s) = k iff s ∈ [g_{k}(x), g_{k+1}(x)]

and define

Λ′ : W̃_{f} × I → E
Λ′((e, x, s), t) = Λ′_{l(x,s),l(x,t)}((e, x, s), t)

We have the continuity of Λ′ since the functions Λ′_{i,j} coincide on the intersections of their domains, which are closed. That, in turn, may be proved easily observing that

Λ′_{i,i}((e, x, s), s) = e

by definition,

Λ′_{i,j}((e, x, g_{i}(x)), t) = Λ′_{i−1,j}((e, x, g_{i}(x)), t)

by induction on |j − i|, and

Λ′_{i,j}((e, x, s), g_{j}(x)) = Λ′_{i,j−1}((e, x, s), g_{j}(x))

by definition. Since Λ′((e, x, s), s) = e, by (⋆) we can conclude that Λ′ is an extended lifting function for f over W. □

Theorem 2.12. Given a definable map p : E → B, if there is a finite definable open covering U of B, such that for each U ∈ U the map p|_{p^{−1}(U)} : p^{−1}(U) → U is a definable fibration, then p is a definable fibration.

Proof. We apply lemma 2.6. Fix a definable set X and a continuous definable function f : X × I → B, we have to show that there is a lifting function for f over X. By lemma 2.9, it suffices to find a definable finite open covering W of X and for each W ∈ W an extended lifting function for f over W.

On the other hand, by lemma 2.10 there are continuous functions 0 ≡ g_{1} ≤ … ≤ g_{k+1} ≡ 1 : X → I such that for each x ∈ X and each 1 ≤ i ≤ k the set f(x, [g_{i}(x), g_{i+1}(x)]) is entirely contained in some element of U. Let U = {U_{j}}_{j=1}^{h}, and for each map σ : {1, …, k} → {1, …, h} let

W_{σ} = {x ∈ X | f(x, [g_{i}(x), g_{i+1}(x)]) ⊂ U_{σ(i)}, for all i = 1, …, k}.

The sets {W_{σ}}_{σ} are a definable finite open covering of X. Therefore, it suffices to prove that for each σ there is an extended lifting function for f over W_{σ}. Fix such a σ and denote U_{σ(i)} by U_{i}^{σ}, for each i = 1, …, k. By hypothesis p_{|p^{−1}(U_{i}^{σ})} : p^{−1}(U_{i}^{σ}) → U_{i}^{σ} is a definable fibration, for each i = 1, …, k. Hence by lemmata 2.6 and 2.8, for each i, each definable Y and each definable continuous h : Y × I → U_{i}^{σ} there is an extended lifting function for h over Y. Therefore, for every i for every definable Z and every definable continuous g : Z × I → B there is an extended lifting function for g over {z ∈ Z | g(z, I) ⊂ U_{i}^{σ}}. That is, we are under the hypothesis of lemma 2.11, and consequently there is an extended lifting function for f over W_{σ}, as required. □

Proof of theorem 2.3. By theorem 2.12, observing that trivial bundles are fibrations. □

3. Homotopy of definable groups

Definition 3.1. A definable H-space is a definable pointed space (X, x_{0}) equipped with a definable continuous map μ : X × X → X such that both μ(−, x_{0}) and μ(x_{0}, −) are definably homotopic to the identity.

It is clear that a definable H-space definable over ℝ is, in particular, an H-space (in the classical sense).

Theorem 3.2. Let G be a definable group. Then π_{n}(G) is finitely generated for each n > 0.

Proof. We can assume, without loss of generality, that G is definably connected and bounded. By the triangulation theorem we can identify G with the realization |K′|(M) in M of a finite simplicial complex K′ with rational vertexes, one of which is the group identity e. Now, let K be a closed subcomplex of K′ such that K(M) is a semialgebraic deformation retract of |K′|(M) containing e. The multiplication on |K′|(M) induces a map ν : |K|(M)×|K|(M) → |K|(M) which gives to (|K|(M), e) the structure of a definable H-space. Moreover, by [1, corollary 3.6], the map ν is definably homotopic to a semialgebraic continuous map

μ: |K|(M) × |K|(M) → |K|(M)

definable without parameters. Of course, μ induces again a definable H-space structure over (|K|(M), e).

Claim 1. Let |K|(ℝ) denote the realization of K in ℝ, then π_{n}(|K|(ℝ)) is finitely generated for each n ≥ 1.

Proof. (|K|(ℝ), e) is endowed with a definable H-space structure by μ, or more precisely by the function defined by the same formula as μ interpreted in ℝ. Therefore (|K|(ℝ), e) is an H-space. Notice that every path-connected H-space is simple (i.e. its fundamental group acts trivially on all homotopy groups, see [17, chapter 7 theorem 3.9]), and for simple spaces the homotopy groups are all finitely generated if the homology groups are so ([18, chapter XIII corollary 7.14]), hence we have the claim since |K|(ℝ) is a finite simplicial complex. □

Now, by [1, corollary 4.3], π_{n} (|K|(M)) is isomorphic to π_{n} (|K|(ℝ)), hence it is a finitely generated abelian group. Since π_{n} (|K|(M)) ≅ π_{n} (G), we get the result. □

Corollary 3.3. Let G be a definable abelian group. Then π_{n}(G) = 0, for each n ≥ 2.

Proof. By theorem 3.2, π_{n}(G) is a finitely generated group; moreover, we know that it is abelian for each n ≥ 2. Since a finitely generated abelian group is divisible if and only if it is trivial, it suffices to show that π_{n} (G) is divisible for each n > 1. We may assume G is definably connected. By [10, corollary 2.12] the maps

p_{k} : G ∋ x → kx ∈ G

with k ∈ ℕ^{+} are definable covering maps, and hence, by [1, corollary 4.11] they induce isomorphisms on the higher homotopy groups. As a consequence, for each [α] ∈ π_{n} (G) and for each k ∈ ℕ^{+} we have some [β] ∈ π_{n} (G) such that [p_{k} ∘β] = [α], and we can conclude that π_{n} (G) is divisible observing that [p_{k} ∘ β] = k[β]. □

Let us denote by 𝕋^{d}(M) the d-dimensional torus defined as the subset [0, 1)^{d} of M^{d} with the sum operation modulo 1.

Theorem 3.4. Let G be a definably connected definably compact d-dimensional abelian group. Then G is definably homotopy equivalent to 𝕋^{d}(M) = [0, 1)^{d}, i.e., there are definable continuous maps g: G → 𝕋^{d}(M) and f: 𝕋^{d}(M) → G, such that f ∘ g and g ∘ f are definably homotopic to the identity.

Proof. Consider the following map, defined in the proof of [13, lemma 4.3]:

f : 𝕋^{d}(M) ∋ (t_{1}, …, t_{n}) ↦ γ_{1}(t_{1}) + … + γ_{n}(t_{n}) ∈ G,

where π_{1} (G) is the abelian group freely generated by [γ_{1}], …, [γ_{n}]. We can easily show (following the proof of [13, lemma 4.3]) that f induces the identity between π_{1} (𝕋^{d}(M)) and π_{1} (G). Since by theorem 3.3 all the higher homotopy groups of both G and 𝕋^{d}(M) are 0, f induces the identity on them as well (in a trivial way), hence, by the o-minimal version of Whitehead theorem proved in [1, theorem 5.6], f is an homotopy equivalence. □

By [10, theorem 1.1], and applying duality, we have H_{i} (G; ℚ) ≅ ℚ^{\binom{d}{i}} for each i ≥ 0. Here we improve that result by proving the following.

Corollary 3.5. Let G be a definably connected definably compact d-dimensional abelian group. Then the o-minimal homology group H_{i} (G; ℤ) ≅ ℤ^{\binom{d}{i}}, for each i ≥ 0.

Proof. By theorem 3.4, for each i ≥ 0 we have that H_{i} (G; ℤ) ≅ H_{i} (𝕋^{d}(M); ℤ), and by [6, proposition 3.2] H_{i} (𝕋^{d}(M); ℤ) is isomorphic to H_{i} (𝕋^{d}(ℝ); ℤ), which in turn

is isomorphic to ℤ^{\binom{d}{i}}. □

Next we proceed to study the o-minimal homotopy groups of definably compact (noncommutative) groups. Towards this aim we first prove some results concerning homogeneous spaces of the type G/H, where G is a definable group (not necessary definably compact) and H is a definable subgroup of G. Notice that by [14, corollary 2.14] such G/H can be equipped with a definable manifold topology so that the canonical action of G on G/H is continuous. Moreover, by [5, theorem 4.3], this topology coincides with the quotient topology (inherited by that of G), which in turn coincides with the definable group topology – provided H is normal – of G/H. Notice also that H being closed in G, the coset space G/H is a regular definable space are hence we can consider its definable manifold topology induced by that of the ambient space.

Fact 3.6. Let F be the functor from the category of definably compact groups to the category of compact Lie groups which sends G to G/G^{00}.

(i) F preserves dimension and connectedness (each concept in its category). (ii) F is exact i.e. transforms exact sequences in exact sequences.

Proof. By [11, theorem 8.1] and [3, theorem 5.2], respectively. □

Our next result says that the functor F also preserves homotopy.

Theorem 3.7. Let G be a definably connected definably compact group. Then π_{n}(G) ≅ π_{n}(G/G^{00}).

Proof. Case G abelian. Let d be the dimension of G. By [10, theorem 1.1] and corollary 3.3 above, we have π_{1}(G) ≅ ℤ^{d} and π_{n}(G) = 0 for each n > 1, respectively. On the other hand, since G/G^{00} is an abelian compact Lie group of dimension d, we also have π_{1}(G/G^{00}) ≅ ℤ^{d} and for n > 1, π_{n}(G/G^{00}) = 0.

Case G definably compact and n = 1. First note that by the o-minimal Poincaré-Hurewicz theorem ([10, theorem 5.1]), and the corresponding classical result, it suffices to prove that H_{1}(G) = H_{1}(G/G^{00}). On the other hand, by [4, corollary 5.2] and [3, remark 7.3], the singular cohomology groups H^{1}(G; L) and H^{1}(G/G^{00}; L) are isomorphic for any coefficient group L. Hence, by the universal coefficient theorem (UCT) for cohomology, we are done in this case. The details are as follows: The UCT says that for any chain complex C, for any abelian group L and for any n > 0, if H_{n−1}(C) is free then H^{n}(C; L) ≅ Hom(H_{n}(C); L). Applying the UCT for n = 1 to the (honest) chain complex associated to the definable singular simplexes on our definable group G we get H^{1}(G, L) ≅ Hom(H_{1}(G), L). Hence, by the corresponding result for the singular chain complex of G/G^{00}, Hom(H_{1}(G); L) ≅ Hom(H_{1}(G/G^{00}); L) for any abelian group L. It follows that H_{1}(G) ≅ H_{1}(G/G^{00}), since both groups H_{1}(G) and H_{1}(G/G^{00}) are abelian and finitely generated.

Before we prove the result in the remaining cases we have the following.

Claim 2. Let G and H be two definably connected definably compact groups. Suppose H is a finite extension of G, Then, for every n > 1, if π_{n}(H) ≅ π_{n}(H/H^{00}) then π_{n}(G) ≅ π_{n}(G/G^{00}).

Proof. By [10, proposition 2.11], the onto homomorphism G → H with finite kernel is a definable covering map. Then, π_{n}(G) ≅ π_{n}(H) for any n > 1 by [1, corollary 4.11]. By preservation of exactness, the induced map from G/G^{00} to H/H^{00} is also an onto homomorphism with finite kernel, hence a covering map and so π_{n}(G/G^{00}) ≅ π_{n}(H/H^{00}), for any n > 1. □

Case G definably semisimple and n > 1. Since the centre of G is finite, G is a finite extension of the definably semisimple centreless group G/Z(G). By claim 2, we may consider G is centreless. On the other hand, we may assume G = G(M) is a semialgebraic group over the real algebraic numbers. By [1, corollary 4.3], π_{n}(G(M)) ≅ π_{n}(G(ℝ)), for every n. But G(ℝ) is G/G^{00} by the proof of [16, proposition 3.6].

General case. Let Z be the centre of G. Then, by [15, corollary 5.4], the group G/Z is definably semisimple, hence the result holds for G/Z. By Corollary 2.4 the projection map G → G/Z is a definable fibration. Therefore, for each n ≥ 1, the o-minimal homotopy groups π_{n}(G, Z) and π_{n}(G/Z) are isomorphic; and, hence, the o-minimal homotopy sequence of the pair (G, Z) is the following long exact sequence (see [1, section 4])

⋯ → π_{n+1}(G/Z) → π_{n}(Z) → π_{n}(G) → π_{n}(G/Z) → π_{n−1}(Z) → · · · .

On the other hand, using the exactness of the functor to the Lie category we have that F (Z) is a closed normal subgroup of F (G), so the projection map F (G) → F (G)/F (Z)(≅ F (G/H)) is a fibration and hence we have the sequence

⋯ → π_{n+1}(F (G/Z)) → π_{n}(F (Z)) → π_{n}(F (G)) → π_{n}(F (G/Z)) → π_{n−1}(F (Z)) → ⋯

Since Z and F (Z) are abelian, we have π_{n}(Z) = 0 and π_{n}(F (Z)) = 0 for all n ≥ 2. So for n ≥ 3, π_{n}(G) ≅ π_{n}(G/Z) and π_{n}(F (G)) ≅ π_{n}(F (G/Z)). By the previous case π_{n}(F (G/Z)) ≅ π_{n}(G/Z). For n = 2, recall that the second homotopy group of a compact Lie group is trivial, so π_{2}(F (G/Z)) = 0, and therefore also π_{2}(G/Z) = 0 (by the semisimple case). Since π_{2}(Z) = 0, it follows that also π_{2}(G) = 0. □

Remark 3.8. Let k be a characteristic zero field. Then H_{n}(G; k) ≅ H_{n}(G/G^{00}; k), for each n.

Proof. By a similar argument as the one done in the above proof with the UCT. Indeed, we can use the following facts: (i) the singular cohomology groups H^{1}(G; L) and H^{1}(G/G^{00}; L) are isomorphic for any coefficient group L ([4, corollary 5.2]); (ii) the UCT; (iii) the fact that H_{n}(G; k) are free, and (iv) H_{n}(G; k) are finitedimensional k-vector spaces. □

4. Universal cover

Fact 4.1. Given a definably connected group G there is a universal covering homomorphism p : G̃ → G such that:

(i) p is a locally definable covering map with ker p = π_{1} (G);

(ii) G̃ is a locally definable regular paracompact group;

(iii) G̃ is simply connected, and

(iv) G̃ is unique up to locally definable isomorphisms ( i.e., given p_{1} : G̃_{1} → G satisfying the above conditions, G̃_{1} is isomorphic to G̃ via a locally definable isomorphism).

Proof. See [9] and [2, theorem 4.8]. □

Proposition 4.2. Let G be a definably compact semisimple group. Then, the universal covering group of G is a definably compact definable group.

Proof. It suffices to prove that the fundamental group of G is finite. Indeed, by [9, theorem 3.11] G̃ is a locally definable group and by the above fact G̃/π_{1} (G) ≅ G.

If G is semisimple and centreless, then we may suppose that G = G(M) is a semialgebraic group, then π_{1} (G(M)) ≅ π_{1} (G(ℝ)) by [6, corollary 2.10]. But G(ℝ) is a semisimple compact Lie group and hence, by a theorem of Weyl, it has a finite fundamental group (see e.g., [8, theorem 23.1]).

If G is semisimple then Z(G) is finite and G/Z(G) is semisimple and centreless. By [10, proposition 2.11] the projection p : G → G/Z(G) is a definable covering map, and by [10, corollary 2.8] the map that p induces between the fundamental groups is injective. We have already proved that π_{1} (G/Z(G)) is finite, and hence so it is π_{1} (G). □

Theorem 4.3. Let G be a definably compact abelian group. Then the universal covering group G̃ is contractible in the category of locally definable spaces.

Proof. We first prove that all the homotopy groups of G̃ are trivial. Indeed, by Fact 4.1, π_{1}(G̃) = 0 and p : G̃ → G is a covering map. By [2, corollary 7.2] p induces isomorphism on the higher homotopy groups, hence, by theorem 3.3, we also have that π_{n}(G̃) is trivial for each n ≥ 1. Finally, by the o-minimal version of Whitehead theorem (see [2, theorem 8.2]) G̃ is locally definably contractible. □

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Università di Pisa, Dipartimento di Matematica, Largo Bruno Pontecorvo 5, 56127 Pisa, Italy

E-mail address: berardu@dm.unipi.it

Classe di Scienze - Scuola Normale Superiore, Piazza dei Cavalieri, 7, 56126 Pisa, Italy

E-mail address: m.mamino@sns.it

Departamento de Matemáticas, Universidad Autónoma de Madrid, Francisco Tomás y Valiente 7, 28049 Madrid, Spain

E-mail address: margarita.otero@uam.es