DIFFERENTIAL GEOMETRY AND APPLICATIONS Proc. Conf., Aug. 28 – Sept. 1, 1995, Brno, Czech Republic Masaryk University, Brno 1996, 239–244

SOME SITUATIONS WHEN THE TOPOLOGICAL φ-CATEGORY IS INFINITE

DORIN ANDRICA AND CORNEL PINTEA

Abstract. The topological φ-category φ_{top}(M, N) of a pair of manifolds (M, N) is introduced and studied in the authors’ paper [3]. The main result of this present paper gives sufficient conditions in terms of fundamental groups of the compact manifolds M^{m}, N^{n} (m ≥ n ≥ 2), in order that φ_{top}(M, N) = +∞. Some applications and examples are given.

1. Preliminaries

Let M^{m}, N^{n} be two topological manifolds and let f : M → N be a continuous map. The point x_{0} ∈ M is called topological regular point of f if f has a local representation f_{φψ} = ψ ∘ f ∘ φ^{−1} : φ(U) → ψ(V) at x_{0} such that for any z = (z^{1}, … z^{m}) ∈ φ(U) ⊆ R^{m}

f_{φψ}(z) = \{ \begin{matrix}(z^{1}, …, z^{m}, \underbrace{0, …, 0}_{n−m}) & if m ≤ n \\ (z^{1}, …, z^{n}) & if m ≥ n \end{matrix} (1)

There are different definitions of regular points in the case of real functions which are defined on a topological manifold. We mention here the papers of N.H. Kuiper [15], W. Smith [17], F. Wille [19] and B. Doubrovine, a.o.[12, pp 186-187].

If the point x_{0} is not topological regular, it is called topological critical point of the map f. This notion was introduced and studied in our paper D. Andrica, C. Pintea [3]. An interesting problem is to compare our definition with those of critical points obtained from the above mentioned papers, but this is not the intention of the present paper.

Let us denote by C_{top}(f) the set of all such above defined points and let us call it the topological critical set of f.

When M^{m}, N^{n} are differentiable manifolds and f : M → N is a C^{1}-differentiable mapping, recall that the critical set of f is given by

C(f) = {x ∈ M : rank_{x}f < min(m, n)}

1991 Mathematics Subject Classification. 58E05, 58C27.

Key words and phrases. Topological critical points, topological φ-category, φ-category.

This paper is in final form and no version of it will be submitted for publication elsewhere.

where rank_{x}f = dim_{R}Im(T f)_{x}, (T f)_{x} : T_{x}M → T_{f(x)}N being the tangent map of f at x. According to the well-known rank’s theorem (see for instance Th. Bröker, L. Lander [6, pp. 2-5]) it follows the relation:

C_{top}(f) ⊆ C(f) (2)

Generally, the equality in (2) not holds (see Proposition 1.3 and Example in [3]).

For a continuous map f : M → N, where M, N are topological manifolds, let us consider μ_{top}(f) = |C_{top}(f)|, the cardinal number of topological critical set of f. Let us denote φ(M, N) = min{μ_{top}(f) : f ∈ C(M, N)} the topological φ-category of the pair (M, N).

If M^{m}, N^{n} are smooth differentiable manifolds we define the φ−category of the pair (M, N) by φ(M, N) = min{μ(f) : f ∈ C^{∞}(M, N)}, where μ(f) denotes the cardinal number of C(f). Note that when N = R, φ(M, R) is even the φ-category of M (or the functional category of M), it is simply denoted φ(M) and it was investigated by F. Takens [18] (see also D. Andrica [1], [2]).

φ(M, N) points out a class of mappings f ∈ C^{∞}(M, N) having the critical set C(f) minimal. Let us remark that in a series of papers P.T. Church, [8], [9] and P.T. Church, J. G. Timourian, [10], [11] using results from dimension theory, obtain informations concerning mappings f ∈ C^{∞}(M, N) with ”small” critical set.

Using inclusion (2), it follows that if M, N are smooth differentible manifolds and f ∈ C^{∞}(M, N) the relation μ_{top} ≤ μ(f) holds and consequently one obtains the inequlity

φ_{top}(M, N) ≤ φ(M, N) (3)

2. The main result

The principal result of our paper [3] is the following

Theorem 2.1. Let M^{m} be a compact topological manifold and let k be an integer number with m ≥ k ≥ 2. Then

φ_{top}(M, R^{k}) = +∞ (4)

Taking into account the inequality (3) one obtains (see [3, Corollary 3.2]) that if M^{m} is a compact differentiable manifold and if k is an integer with m ≥ k ≥ 2, then the relation

φ(M, R^{k}) = +∞ (5)

is satisfied.

The main purpose of this paper is to obtain sufficient conditions for the manifolds M, N such that φ(M, N) = +∞, i.e. in the differential case φ(M, N) = +∞, via the inequality (2). Let us mention that in the first author’s paper [2] are obtained such results for φ(M, N), but the conditions are given in terms of Stiefel-Whitney characteristic classes.

We begin with two useful results in the proof of main theorem. Let us consider the topological manifolds M, N

Lemma 2.2. If there exists a local homeomorphism Ñ → N, then φ_{top}(M, N) ≤ φ_{top}(M, Ñ)

SOME SITUATIONS WHEN THE TOPOLOGICAL φ-CATEGORY IS INFINITE 241

Proof. Let p : Ñ → N be the asserted local homeomorphism. If f ∈ C(M, Ñ) consider g ∈ C(M, N) so that g = p ∘ f. It is not difficult to note that C_{top}(f) = C_{top}(g) (see also [3, Proposition 1.4 (ii)]), that is μ_{top}(f) = μ_{top}(g) ≥ φ_{top}(M, N). Because the inequality holds for any map f ∈ C(M, N), one obtains the desired relation. □

Lemma 2.3. Let p : Ñ → N be the universal covering of manifold N. If Hom(π_{1}(M), π_{1}(N)) = 0, then the relation

φ_{top}(M, N) = φ_{top}(M, Ñ) (6)

is satisfied.

Proof. For the begining let us recall that the manifold N has an universal covering p : Ñ → N, and Ñ is also a manifold (see M.J. Greenberg, J.R. Harper [14, Corollary 6.8]).

Taking into account the result of Lemma 2.2 it follows that it is sufficient to prove the inequality φ_{top}(M, Ñ) ≤ φ_{top}(M, N). Let us consider f ∈ C(M, N). Then f_{∗} : π_{1}(M) → π_{1}(N) is the null homomorphism, thus the lifting crterion (see M.J. Greenberg, J.R. Harper [14, Theorem 6.1] or C. Godbillon [13, Théoreème 3.2]) is verified. One obtains the continuous map f̃ : M → Ñ such that p ∘ f̃ = f. Because the projection p is a local homeomorphism it implies the relation C_{top}(f) = C_{top}(f̃) (see [3, Proposition 1.4 (ii)]), i.e. μ_{top}(f) = μ_{top}(f̃) ≥ φ_{top}(M, Ñ), and the desired inequality φ_{top}(M, N) ≥ φ_{top}(M, Ñ) is proved. □

Our main result is contained in the following theorem.

Theorem 2.4. Let M^{m}, N^{n} be two connected compact manifolds with m ≥ n ≥ 2. Suppose that:

(i) Ñ is homeomorphic with R^{n}, where p : Ñ → N is the universal covering of N.

(ii) Hom(π_{1}(M), π_{1}(N)) = 0

Then the following relation holds

φ_{top}(M, N) = +∞ (7)

Proof. Let h : Ñ → R^{n} be the asserted homeomorphism and let us put H : (M, Ñ) → (M, R^{n}), H = (1_{M}, h). For the continuous map f ∈ C(M, Ñ) consider the map f′ ∈ C(M, R^{n}) such that the following diagram is commutative

M ⟶^{f} Ñ
1_{M} ↓  ↓h
M ⟶^{f′} R^{n}

that is f′ = h ∘ f. One obtains C_{top}(f′) = C_{top}(f) (see [3, Proposition 1.4 (ii)]) so μ_{top}(Λ(f)) = μ_{top}(f), where Λ : C(M, Ñ) → C(M, R^{n}) is defined by Λ(f) = f′. The map Λ is bijective and Λ^{−1} is given by Λ^{−1}(f′) = h^{−1} ∘ f′. So we have succesively φ_{top}(M, Ñ) = min{μ_{top}(f) : f ∈ C(M, Ñ)} = min{μ_{top}(Λ(M)) : f ∈ C(M, Ñ)} = min{μ_{top}(f′) : f′ ∈ C(M, R^{n}} = φ_{top}(M, R^{n}).

On the other hand from Lemma 2.3 one obtains φ_{top}(M, N) = φ_{top}(M, Ñ) = φ_{top}(M, R^{n}). Using the result contained in Theorem 2.1 we get the relation (7). □

Remark For any topological manifold N the total space Ñ of its universal covering p : Ñ → N is simply connected (see C. Godbillon [13, Théorème 5.3 and Théorème 4.8]) but it is not neccesary that Ñ to be homeomorphic with R^{n}. For example, we have π_{1}(S^{3}) = 0, π_{1}(S^{2}) = 0 so the condition (ii) in Theorem 2.4 is satisfied. The universal covering of S^{2} is 1_{S^{2}} : S^{2} → S^{2}, i.e. the hypothesis (i) is not verified. On the other hand φ(S^{3}, S^{2}) = 0, because of the well-known Hopf fibration S^{3} → S^{2}.

It is an interesting problem to characterize the manifolds N whose universal space Ñ satisfies condition (i) in Theorem 2.4.

Corollary 2.5. Suppose that the hypothesis (i) in Theorem 2.4 is satisfied. If π_{1}(M) is a torsion group (i.e. T orπ_{1}(M) = π_{1}(M)) and π_{1}(N) is without torsion (i.e. T orπ_{1}(N) = 0), then the relation (7) holds.

Corollary 2.6. Suppose that the hypothesis (i) in Theorem 2.4 is satisfied. If M is simply-connected, then the relation (7) holds.

For the proof of the above corollaries it is sufficient to observe that the condition Hom(π_{1}(M), π_{1}(N)) = 0 is automatically verified.

3. Applications

Proposition 3.1. Let M^{m} be a compact connected topological manifold which is simply-connected. If the finite group G acts properly discontinuous on M, then

φ_{top}(M/G, T^{n}) = +∞ (8)

where T^{n} = S^{1} ×S^{1} ×⋯×S^{1} (n times) is the n-dimensional torus and m ≥ n ≥ 2.

Proof. Because G acts properly discontinuous on M it follows that p : M → M/G is a covering space of M/G, where p is the canonical map. Taking into account that M is simply-connected one obtains that p : M → M/G is even the universal covering of M/G (see C. Godbillon [13, Théorème 4.8]) and π_{1}(M/G) ≅ G (see G.E. Bredon [5, Corollary 7.3]).

On the other hand the universal covering of the torus T^{n} is R^{n} → R^{n}/Z^{n} = T^{n}, i.e. the condition (i) from Theorem 2.4 is verified. The fundamental group of T^{n} is π_{1}(T^{n}) = Z × ⋯ × Z (n times), therefore π_{1}(T^{n}) is without torsion.

We have Hom(π_{1}(M/G), π_{1}(T^{n})) = 0 since π_{1}(M/G) is a finite group. The desired conclusion is obtain from Theorem 2.4. □

If M_{j}^{m_{j}}, j = \overline{1, k}, are connected and the finite group G_{j} acts properly discontinuous on M_{j}, j = \overline{1, k}, with an analogous argument one obtains

φ_{top}(M_{1}/G_{1} × ⋯ × M_{k}/G_{k}, T^{n}) = +∞ (9)

where m_{1} + m_{2} + ⋯ + m_{k} ≥ n ≥ 2.

Examples 1) Consider S^{2m−1} = {z ∈ C^{m} : ||z|| = 1}. Let ε = e^{2πi/p} be a primitive p^{th} rooth of unity and let q_{1}, …, q_{m} be integers relatively prime to p. Consider the multiplicative group Z_{p} = {1, ε, ε^{2}, …, ε^{p−1}} and let us consider its action on S^{2m−1} given by ε · z = (ε^{q_{1}}z_{1}, …, ε^{q_{m}}z_{m}), where z = (z_{1}, …, z_{m}). The above action is properly discontinuous since it is a free action of a finite group. The orbit space S^{2m−1}/Z_{p} denoted by L(p; q_{1}, …, q_{m}) is called a ”lens space” and it is a (2m-1)-dimensional real manifold. Applying the result of Proposition 2.7 we have

φ_{top}(L(p; q_{1}, …, q_{m}), T^{n}) = +∞ (10)

where 2m − 1 ≥ n ≥ 2.

2) From the relation (9) one obtains for the real projective spaces P^{m_{i}}(R), i = \overline{1, k}, that if m_{1} + ⋯ + m_{k} ≥ n ≥ 2, then

φ_{top}(P^{m_{1}}(R) × ⋯ × P^{m_{k}}(R), T^{n}) = +∞ (11)

Theorem 3.2. Let M^{m} be a compact connected topological manifold and let N^{n} be a compact Riemannian manifold with negative curvature. If π_{1}(M) is commutative torsion group and m ≥ n ≥ 2, then φ_{top}(M, N) = +∞.

Proof. Using the Hadamard theorem (see M.P. doCarmo [7, Theorem 3.1, pp 149]) it follows that the universal covering space Ñ of N is diffeomorphic with R^{n}. Let us show the relation Hom(π_{1}(M), π_{1}(N)) = 0. If F : π_{1}(M) → π_{1}(N) is a homomorphism such that F ≠ 0, one obtains immediately that ImF is a nontrivial commutative torsion subgroup of π_{1}(N). But, taking into account the theorem of Preissman [16] (see also M.P. doCarmo [7, Theorem 3.2, pp 260]) we have that any nontrivial commutative subgroup of π_{1}(N) is infinite cyclic, which is a contradiction with the fact that ImF is a torsion group. Therefore Hom(π_{1}(M), π_{1}(N)) = 0, i.e. the conditions of Theorem 2.4 are satisfied and one can apply it to obtain the desired conclusion. □

Corollary 3.3. Let G be a compact Lie group and let N^{n} be a compact Riemannian manifold with negative curvature. If π_{1}(G) is a torsion group and dim G ≥ n ≥ 2, then φ_{top}(G, N) = +∞.

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