Estimates of covering type and the number of vertices of minimal triangulations

The covering type of a space $X$ is defined as the minimal cardinality of a good cover of a space that is homotopy equivalent to $X$. We derive estimates for the covering type of $X$ in terms of other invariants of $X$, namely the ranks of the homology groups, the multiplicative structure of the cohomology ring and the Lusternik-Schnirelmann category of $X$. By relating the covering type to the number of vertices of minimal triangulations of complexes and combinatorial manifolds, we obtain, within a unified framework, several estimates which are either new or extensions of results that have been previously obtained by ad hoc combinatorial arguments. Moreover, our methods give results that are valid for entire homotopy classes of spaces.


Introduction
Many concepts in topology and homotopy theory are related to the size and the structure of the covers that a given space admits. Typical examples that spring in mind are (Lebesgue) dimension and Lusternik-Schnirelmann category. M. Karoubi and C. Weibel [7] have recently introduced another interesting measure for the complexity of a space based on the size of its good covers.
Recall that an open cover U of X is said to be a good cover if all elements of U and all their non-empty finite intersections are contractible. Karoubi and Weibel defined sct(X), the strict covering type of a given space X, as the minimal cardinality of a good cover for X. Note that sct(X) can be infinite (e.g., if X is an infinite discrete space) or even undefined, if the space does not admit any good covers (e.g. the Hawaiian earring). In what follows we will always tacitly assume that the spaces under consideration admit finite good covers.
Strict covering type is a geometric notion and is not homotopy invariant, which led Karoubi and Weibel to define the covering type of X as the minimal size of a good cover of spaces that are homotopy equivalent to X: The covering type is a homotopy invariant of the space and is often strictly smaller than the strict covering type, even for simple spaces like wedges of circles ([7, Example 1.3]). Karoubi and Weibel also proved a useful result ( [7,Theorem 2.5]) that the covering type of a finite CW complex is equal to the size of a good closed cover of some CW complex that is homotopy equivalent to X. Furthermore, they computed exactly the covering type for finite graphs (i.e., finite wedges of circles) and for some closed surfaces (the sphere, torus, projective space), while giving estimates for the covering type of other surfaces. Finally, they estimated the covering type of mapping cones, suspensions and covering spaces. Most of these estimates are quite coarse and some are even incorrect (e.g., for projective spaces).
Good covers arise naturally in many situations, e.g., as geodesically convex neighbourhoods in Riemannian manifolds or as locally convex covers of polyhedra. Their main feature is that the pattern of intersections of sets of a good cover capture the homotopy type of a space. Specifically, let N (U ) denote the nerve of the open cover U of X, and let |N (U )| be its geometric realization. We may identify the vertices of |N (U )| with the elements of U and the points of |N (U )| with the convex combinations of elements of U . If U is numerable, that is, if U admits a subordinated partition of unity {ϕ U : X → [0, 1] | U ∈ U }, then the formula ϕ(x) := U ∈U ϕ U (x) · U determines the so called Aleksandroff map ϕ : X → |N (U )|, which has many remarkable properties. In particular we have the following classical result, whose discovery is variously attributed to J. Leray, K. Borsuk and A. Weil (see [6,Corollary 4G.3] for a modern proof). Theorem 1.1 (Nerve Theorem). If U is a numerable good cover of X, then the Aleksandroff map ϕ : X → |N (U )| is a homotopy equivalence.
As a consequence, a paracompact space admits a finite good cover if, and only if it is homotopy equivalent to a finite (simplicial or CW) complex. In the literature one can find many variants of the Nerve theorem, which under different sets of assumptions show that the Alexandroff map is a homotopy equivalence, or a weak homotopy equivalence, or a homology equivalence, etc.
The idea of covering type provides an important link between good covers and minimal triangulations. In general, given a polyhedron P , one often looks for triangulations of P with the minimal number of vertices. Again, there are many variants and aspects of the problem as one can ask for the minimal number of vertices needed to accommodate a specific triangulation, or allow different triangulations for the same polyhedron, and so on.
Let us introduce the following systematic notation: for a (finite) simplicial complex define ∆(K) := min{ n | K ≤ ∆ n }, where ∆ n is the standard n simplex and ≤ denotes a subcomplex. The geometric realization of K may admit smaller triangulations, so one can consider ∆ ≈ (K) := min{∆(L)||L| ≈ |K|} , (where ≈ denotes homeomorphism between geometric realizations). In practice, the most interesting cases are triangulations of manifolds, where one is principally interested in combinatorial (or PL-) triangulations, for which the links of vertices are combinatorial spheres. Let Computing ∆(K) and its variants is a hard and intensively studied problem of combinatorial topology -see Datta [5] and Lutz [8] for surveys of the vast body of work related to this question.
Every triangulation of a space X gives rise to a good cover of X given by stars of vertices in the triangulation. On the other hand, if U is a good cover of X, then |N (U )| is homotopy equivalent to X by the Nerve Theorem. We may thus introduce a homotopy analogue of ∆(K) as ∆ ≃ (K) := min{∆(L) | |L| ≃ |K|} , (where ≃ denotes homotopy equivalence between geometric realizations). Clearly, ∆ ≃ (K) is a lower bound for other invariants, since On the other hand ∆ ≃ (K) is directly related to the covering type.
Proof. Let U be a good cover of |K| of cardinality ct(|K|). The nerve N (U ) of U has ct(|K|) vertices, so it is a subcomplex of ∆ ct(|K|)−1 . By Nerve Conversely, if L is a subcomplex of ∆ ∆ ≃ (K) , such that |L| ≃ |K|, then the cover of |L| by stars of vertices is a good cover of |L| by (∆ ≃ (K) + 1) elements, hence ∆ ≃ (K) ≥ ct(K) − 1 .
As a consequence, it is of great interest to find good lower estimates for ct(X) as they in turn give lower bounds for the size of minimal triangulations. On the other hand, it is easy to find examples where ct(|K|) is strictly smaller than ∆ ≈ (K) + 1 (cf. [7, Example 1.3]). As a consequence, the upper estimates for ct(X) are less relevant as a tool for the study of minimal triangulations. In practice upper estimates of ct(X) are usually obtained by finding explicit triangulations of X, while the lower estimates are based on certain obstructions. The latter is a natural setting for the methods of homotopy theory, and is one of the reasons why the relation with ct(X) is so useful.
However, the situation may be more favourable in the case of closed manifolds. We are not aware of any closed triangulable manifold M for which ct(M ) is strictly smaller than ∆ P L (M ) + 1. Any such example would correspond to a closed manifold M that has a minimal triangulation with n vertices but is homotopy equivalent to the geometric realization of a simplicial complex with less than n vertices. It is hard to envisage how this could happen, which leads us to the following: We should also mention another set of problems that are closely linked to the covering type, which is the study of intersection patterns of convex and contractible sets and the related Helly-type theorems -see [10] for more information.
The paper is organized as follows. In the next two sections we relate the Lusternik-Schnirelmann category and the cohomology ring of a space to the covering type and derive a series of lower estimates for the covering type. In the last section we study the effect that suspensions and wedge-sums have on the covering type, and give some useful upper and lower estimates for the covering type of a Moore space.

LS-category estimates
Recall the definition of the Lusternik-Schnirelmann (LS-)category of a space X. A subset A ⊆ X is categorical if the inclusion A ֒→ X is homotopic to the constant map. Then LS-category of X, denoted cat(X), is the minimal n, for which X can be covered by n open categorical subsets. A standard reference is [4] Remark 2.1. Contractible subsets of X are clearly categorical, but the converse is not true -e.g., the sphere is a categorical subset of the ball. There is a related concept called geometric category, defined as the minimal cardinality of a cover of X by open contractible sets (see [4,Chapter 3]). Like the strict covering type, the geometric category is not a homotopy invariant of X, so one defines the strong category, Cat(X), as the minimum of geometric categories of spaces that are homotopy equivalent to X. Although the categorical sets may be very different from contractible ones, the following remarkable relation holds: cat(X) ≤ Cat(X) ≤ cat(X) + 1 (see [4,Proposition 3.15]).
Little is known about analogous relationships between the covering type and the strict covering type. For the wedge on n circles W n we have sct(W n ) = n + 2, while ct(W n ) = 3+ √ 1+8n 2 (see [7,Proposition 4.1]), so the difference between the two can be arbitrarily large. On the other side, we do not know whether the covering type of a manifold can be strictly smaller than its strict covering type.
The relation between the category and the covering type of a space is also complicated. For spheres cat(S n ) = 2 while ct(S n ) = n + 2. Neither of them determines the other. We will give below examples of spaces that have the same covering type and yet the difference between respective categories is as big as we want. Nevertheless, if the category of a space is n > 1, then its (homotopy) dimension is at least n − 1 and so its covering type is at least n + 1 (because it is not contractible). Roughly speaking, spaces with big category cannot have small covering type. We are going to make this statement more precise.
The base of our reasoning are the following facts: A) By Nerve theorem, if X admits a good cover U of order ≤ n (i.e., at most n different sets have non-empty intersection), then X is homotopy equivalent to a simplicial complex of dimension n − 1. B) By Nerve theorem, if U 1 , . . . , U n are elements of a good cover that intersect non-trivially, then U 1 ∪ · · · ∪ U n is homotopy equivalent to ∆ n−1 , and therefore contractible. C) If cat(X) ≥ n, X = U ∪ V , where U, V open and U is contractible (or more generally, U is categorical in X), then cat(V ) ≥ n − 1. This is obvious, because cat(V ) < n − 1 would imply cat(U ∪ V ) < n. D) cat(X) ≤ hdim(X)+1, where hdim(X) is the homotopy dimension of X, defined as hdim(X) := min{dim(Y ) | Y ≃ X}. The claim follows from the classical estimate cat(Y ) ≤ dim Y + 1 and the homotopy invariance of LS-category.
Proof. Assume that X has a good cover U of cardinality ct(X). We proceed by induction. If cat(X) = 1 then X is contractible, thus ct(X) = 1 and the inequality reduces to 1 ≥ 1.
Direct application of the theorem gives the following estimates. For spheres cat(S n ) = 2, therefore ct(S n ) ≥ 3, and for surfaces (other than S 2 ) cat(S n ) = 3, therefore ct(S n ) ≥ 6. Furthermore, for real and complex projective spaces cat(RP n ) = cat(CP n ) = n + 1, so that ct A comparison with the results of [7] shows that some of the above estimates are not optimal and can be improved. In fact, we neglected the information about the dimension and connectivity of X, which also have an impact on the covering type. By taking these data into account we obtain much better estimates (except for real projective spaces, which are only 0-connected and the category is directly related to the dimension). Nevertheless it is interesting to observe that the covering type increases (at least) quadratically with the category of the space.
A similar approach can be used to estimate the minimal number of points (vertices) that are required in order to triangulate a given PL-manifold.
Recall that a triangulation of a manifold is combinatorial if the links of all vertices are triangulated spheres. Then we have the following Corollary 2.4. Let K be a combinatorial triangulation of a d-dimensional and c-connected closed manifold M . Then K has at least vertices.
Proof. If c = d − 1 then M ≈ S d , and the corollary correctly states that every triangulation has at least d + 2 vertices. If d = 2 and c = 0, then M is a closed surface other than sphere, hence cat(M ) = 3, and the claim is that at least six vertices are needed, which we already know. Therefore, we may assume d ≥ 3 and c ≤ d − 2.
Observe that the open stars of vertices form a good cover We have seen previously that cat(V ) ≥ cat(M ) − 1, so we use the known inequality (see [4]) vertices.
Observe that this estimate is a strict improvement of Corollary 2.3 for all PLmanifolds which are at least 1-connected and are not spheres. For example, it shows that every triangulation of CP n requires at least 1 2 n(n+7) vertices. Remark 2.5. Substituting the inequlity (1), applied to cat(M ), into the statement of Corolarry 2.4 we get an estimate from below of the number of vertices in a trangulation by This estimate is qudratic in d consequently for a fixed c, and d respectively large, it is better than a corresponding previous result of [2] which states that this number is greater or equal to with the same supposition on M .
We may also reverse the above estimates to obtain upper bounds for the category of a space based on the cardinality of good cover or the number of vertices in a triangulation.
Corollary 2.6. Assume that X admits a good cover with n elements. Then the category of X is bounded above by The estimate can be improved, if the dimension of X is known, as then In particular, if hdim(X) ≥ n − 3, then cat(X) ≤ 2, which implies that X is a coH-space, and its fundamental group is free.
Proof. The estimates are easily proved by solving the inequalities in Theorem 2.2 and Corollary 2.3 for cat(X). The special case follows from known properties of spaces whose category is at most 2, see [4, Section 1.6].
The last inequality gives an interesting relation between the category of a subcomplex in ∆ n and its (homotopy) codimension n − hdim(|K|).
Corollary 2.7. If K ≤ ∆ n is a combinatorial triangulation of a d-dimensional and c-connected closed manifold M , then The estimate can be improved, if the dimension of X is known, as then In particular, if hdim(X) ≥ n − 3, then cat(X) ≤ 2, which implies that X is a coH-space, and its fundamental group is free.
Proof. The estimates are easily proved by solving the inequalities in Theorem 2.2 and Corollary 2.3 for cat(X). The special case follows from known properties of spaces whose category is at most 2, see [4, Section 1.6].

Cohomological estimates
It is well-known that the Lusternik-Schnirelmann category of a space X is closely related to the structure of the cohomology ringH * (X). Indeed, cat(X) is bounded below by the so-called cup-length of X, which is defined as the maximal number of factors among all non-trivial products inH * (X) (and with any coefficients, see see [4,Proposition 1.5]). However, that estimate does not involve the respective dimensions of the factors in the product. We are going to show that the latter play an important role in the estimate of covering type, which will lead to considerable improvements in our estimates of the covering type of X .
Given an n-tuple of positive integers i 1 , . . . , i n ∈ N we will say that a space X admits an essential (i 1 , . . . , i n )-product if there are cohomology classes x k ∈ H i k (X), such that the product x 1 · x 2 · . . . · x n is non-trivial. For every (i 1 , . . . , i n ) there exist a space X that admits an essential (i 1 , . . . , i n )product, for example we can take X = S i 1 × · · · × S in . Clearly, if X admits an essential (i 1 , . . . , i n )-product then so does every Y ≃ X, since their cohomology rings are isomorphic. We may therefore define the covering type of the n-tuple of positive integers (i 1 , . . . , i n ) as ct(i 1 , . . . , i n ) := min ct(X) | X admits an essential (i 1 , . . . , i n )−product The following proposition follows immediately from the definition. ct(X) ≥ max{ct(|x 1 |, . . . , |x n |) | for all 0 = x 1 · · · x n ∈ H * (X)} Although the covering type of a specific product of cohomology classes may appear as a coarse estimate it will serve very well our purposes. We will base our computations on the following technical lemmas. The first is a standard argument that we give here for the convenience of the reader.
Lemma 3.2. Let X = U ∪V where U, V are open in X, and let x, y ∈ H * (X) be cohomology classes whose product x · y is non-trivial.
Proof. Assume by contradiction that i * V (x) = 0. Exactness of the cohomology sequence implies that there is a classx ∈ H * (X, V ) such that j * V (x) = x. Moreover i * U (y) = 0, because i U : U ֒→ X is null-homotopic, so there is a classȳ ∈ H * (X, U ) such that j * U (ȳ) = y. Then x · y = j * V (x) · j * U (ȳ) is by naturality equal to the image ofx ·ȳ ∈ H * (X, U ∪ V ) = 0, therefore x · y = 0, which contradicts the assumptions of the lemma.
By inductive application of the above lemma we obtain the following: Let x 1 , . . . , x n ∈ H * (X) be cohomology classes whose product x 1 · · · x n is non-trivial, and let  Proof. If ct(X) ≤ hdim(X)+2, then X is homotopy equivalent to a subcomplex of ∆ hdim(X)+1 . The only subcomplex of ∆ hdim(X)+1 that has homotopy dimension equal to hdim(X) is ∂∆ hdim(X)+1 , which has only one non-trivial reduced homology group.
We are ready to prove the main result of this section, an 'arithmetic' estimate for the covering type of a n-tuple: ct(i 1 , . . . i n ) ≥ i 1 + 2 i 2 + · · · + ni n + (n + 1) If i 1 , . . . i n are not all equal, then ct(i 1 , . . . i n ) ≥ i 1 + 2 i 2 + · · · + ni n + (n + 2) Proof. The first statement can be proved by induction. Unfortunately, the same approach is not sufficient to prove the stronger statement, and a modified inductive argument turns out to be quite complicated, and we find it easier to give a direct proof. Although the second proof covers the first statement as well, we believe that it still of some interest to be able compare the two methods.
Toward the proof of the first statement, we begin the induction by observing that if 0 = x 1 ∈ H i 1 (X) then hdim(X) ≥ i 1 , hence ct(i 1 ) ≥ i 1 + 2 by [7, Proposition 3.1].
Assume that the estimate holds for all sequences of (n − 1) positive integers and consider the classes x 1 ∈ H i 1 (X), . . . , x n ∈ H in (X) such that the product x 1 · · · x n ∈ H i 1 +...+in (X) is non-trivial. The cohomological dimension of X is at least i 1 + . . . + i n , therefore in every good cover U of X one can find i 1 +· · ·+i n +1 elements that intersect non-trivially. Denote their union by U and the union of the remaining elements of U by V . Then U is contractible and by Lemma 3.3 there exists in H * (V ) a non-zero product of elements whose degrees are i 2 , . . . , i n . By induction we obtain ct(X) ≥ (i 1 + · · · + i n + 1) + (i 2 + 2i 3 + . . . + (n − 1)i n + n) = = i 1 + 2i 2 + . . . + ni n + (n + 1), which proves the first statement.
For the second statement, let U be a good cover of X, and assume that the product of classes x 1 ∈ H i 1 (X), . . . , x n ∈ H in (X) is non-trivial. As before, there exists U 1 ⊆ U , such that U 1 contains (i 1 + . . . + i n + 1) sets that intersect non-trivially. If we denote by V 1 the union of sets in U − U 1 , then by Lemma 3.3 the restriction to V 1 of any sub-product of x 1 · · · x n of length (n − 1) is non-trivial. In particular, H i 2 +...+in (V 1 ) = 0, and so there exists U 2 ⊆ U − U 1 , such that U 2 contains (i 2 + . . . + i n + 1) sets that intersect nontrivially. By continuing this procedure we end up with disjoint collections U 1 , . . . , U n−1 ⊆ U , where each U k has (i k + . . . + i n + 1) elements and the union of its elements is contractible.
Let V denote the union of all elements in U n := U − U 1 − . . . − U n−1 . By Lemma 3.3 H * (V ) has non trivial cohomology classes in dimensions i 1 , . . . , i n . Since we assumed that they are not all equal, Lemma 3.4 implies that U n has at least i n + 3 elements. By adding up the cardinalities of all U k we conclude that U has at least i 1 + 2 i 2 + · · · + ni n + (n + 2) elements.
It is worth to emphasize that it is usually not difficult to identify the cup product in H * (X) which provides the best estimate for the covering type. In particular, it clearly makes sense to consider only products whose terms have non-decreasing degrees. The rest of the section is dedicated to computations of specific examples (projective spaces, products of spheres, etc.) based on Theorem 3.5.
The last estimate can be sometimes improved by ad-hoc methods -see Example 3.9.
The LS-category of unitary groups is cat(U (n)) = n and cat(SU (n)) = n − 1 (see [4,Theorem 9.47]), so our of the covering type estimate is a cubical function of the category (as compared with results from Section 2 where we obtained a general quadratic relation between the category and the covering type).
Remark 3.8. The estimates of Corollary 3.6 applied to the number of vertices of triangulation of RP n and CP n or spaces with the same cohomology algebra reproves the result of [1]. The corresponding estimate for HP n was not stated in the literature, up to our knowledge.
The estimate of number of vertices in a triangulation of U (n), or SU (n), that follow from Corollary 3.7 is new.
The computation for unitary groups can be easily extended to all finite Hspaces. In fact the Z p -cohomology of a finite associative H-space is given as (see [12,Theorem III,8.7]).
where k i is a power of 2 if p = 2, while for p odd there are two cases: In particular we can compute lower estimates for the covering type of all classical Lie groups, since their cohomology rings are well-known.
Let us mention that for spaces whose cohomology algebra has several linearly independent generators in low dimensions it is possible to improve the general estimates of the covering type. Since the actual improvements arise only in few cases we do not attempt to develop a theory but instead illustrate this method on an example.
Example 3.9. We are going to estimate ct(S 1 × S 1 × S 1 ), or equivalently ct (1, 1, 1). Let U be a good cover of some X that is homotopy equivalent to S 1 × S 1 × S 1 . Then with respect to any field coefficients we have H * (X) ∼ = Λ(x, y, z), where |x| = |y| = |z| = 1. Since hdim(X) = 3 there are at least 4 open sets, say U 1 , U 2 , U 3 , U 4 ∈ U that intersect non-trivially. The union of the remaining elements of U has category at least 3 and hdim at least 2. By Corollary 2.3 a good cover of it has at least six sets, so there are also sets U 5 , . . . , U 10 ∈ U . Since H 1 (X) is 3-dimensional the kernel of the restriction homomorphism contains a non-trivial element u ∈ H 1 (X). Moreover, the kernel of the restriction homomorphism is at least 2-dimensional so we may find in it a non-trivial element v ∈ H 1 (X) which is linearly independent from u. Finally, we can choose w ∈ H 1 (X) such that the set {u, v, w} is a basis of H 1 (X).

Moore spaces
In this section we estimate the covering type of various Moore spaces and use the results to derive estimates for related spaces. Recall that for every abelian group A and positive integer i one can construct a CW complex X with Any such space is called a Moore space of type M (A, i) (cf. [6, Example 2.40]). We will always assume that the fundamental group of X is abelian, because then A and i uniquely determine the homotopy type of X, and we may write M (A, i) instead of X.
The simplest examples are wedges of spheres: r-fold wedge of i-dimensional spheres is a Moore space of type M (Z r , i).
By Theorem 1.2 every space X with ct(X) = n is homotopy equivalent to a subcomplex of ∆ n−1 . Therefore, for any given n there exist only finitely many homotopy types of spaces whose covering type is equal to n, and we may even attempt a classification, at least for small values of n. For each n there is always the trivial example of a space with ct(X) = n, namely the discrete space with n points. These are the only spaces with covering type 1 or 2. The first non-trivial example is the circle, whose covering type is 3, and belongs to the family of spheres S n whose covering is ct(S n ) = n + 2. Apart from the discrete space and the sphere, there are two other spaces with covering type 4, namely the wedges of 2 and of 3 circles. Similarly, the spaces of covering type 5 are wedges of spheres of various dimensions. The number of homotopy distinct complexes increases rapidly with the covering type, but there is a reasonably complete classification for manifolds whose covering type is at most 11 (cf. [5, Section 5]).
The following theorem gives the covering type of Moore spaces with free homology. n−1 is obtained by truncating the simplicial chain complex for ∆ n−1 at degree i: The homology of ∆ n−1 is trivial, so the above chain complex is exact, except at the beginning. The rank of each C k (∆ n−1 ) is n k+1 , and the rank of H i (∆ (i) n−1 ) = ker ∂ i can be computed by exploiting the exactness: We conclude that ∆ (i) n−1 is a Moore space of type M (Z ( n−1 i+1 ) , i) (it is even simply-connected, but we will not need that fact).
It is obvious from the definition of simplicial homology that the rank of H i (∆ To show the converse, note that im(∂ i ) is n−2 i+1 -dimensional, so we may find up to n i+1 − n−2 i+1 = n−1 i+1 i-simplices in ∆ (i) n−1 whose removal does not alter the image of ∂ i . In particular, if r ≤ n−1 i+1 then we may remove n−1 i+1 − r simplices of dimension i, so that the remaining simplices form a Moore space of type M (Z r , i). We conclude that M (Z r , i) can be represented by a subcomplex of ∆ n−1 if, and only if n−1 i+1 ≥ r, which proves our claim.
The theorem that we have just proved allows to improve some of our previous estimates. Let M be a (n − 1)-connected closed 2n-dimensional manifold. Up to homotopy type it can be built by attaching a 2n-dimensional sphere to a wedge of n-dimensional spheres. Its LS-category is 3, so by Corollary 2.4 every combinatorial triangulation of M has at least 1 + 2n + (n − 1) + 1 2 · 3 · 2 = 3n + 3 vertices. Similarly, Poincaré duality implies that there are cohomology classes in H n (M ) whose product is non zero, so by Proposition 3.1 and Theorem 3.5 the covering type of M is bounded by ct(M ) ≥ ct(n, n) = 3n + 3.
We can obtain better estimates by taking into account the rank of H n (M ). If n > 1 then the exactness of the homology sequence of the pair (M, V ) immediately implies that V is a Moore space of type M (Z r , n) where r = rank(H n (M )). If n = 1, then we observe that V can be deformed to a surface with boundary, and these are well-known to be homotopy equivalent to wedges of circles. In that case V is a Moore space of type M (Z r , 1) where r = rank Z 2 (H 1 (M ; Z 2 )) (we use Z 2 -coefficients to obtain a statement that is valid for both orientable and non-orientable surfaces).
By Theorem 4.1 V is the union of at least n + k + 2 open stars of vertices in K, where k is the minimal integer for which n+k+1 k = n+k+1 n+1 ≥ r. We conclude that K has at least (2n + 1) + (n + k + 2) = 3n + 3 + k vertices, where k is defined as above. Proof. For m < n the estimate follows by Theorem 3.5, and for m = n by the previous Corollary and the observation that k = n + 2 is the minimal integer for which k n+1 ≥ rank H n (S n × S n ) = 2.
We must add that the corresponding estimate of number of vertices of a triangulation of a combinatorial manifold which is homotopy equivalent to S m × S n which follows from Corollary 4.3 was shown in [2] (see also [3]).
Moore spaces for an arbitrary abelian group are usually constructed as wedges of Moore spaces of cyclic groups. It is therefore important to have estimates of the covering type of a wedges of spaces but one should expect some irregular behaviour. For example, by [7, Proposition 4.1] we have the following relations We will first show that the covering type of a wedge exceeds the covering type of its summands. Proof. Denote by i : A ֒→ X be the inclusion of A in X, and by r : X → A the retraction of X to A. If H : U × I → U is a contraction of some U ⊆ X, then it is easy to check that defines a contraction of i −1 (U ).
As a consequence, if U is a good cover of X, then {i −1 (U ) | U ∈ U } is a good cover of A, which means that the covering type of A does not exceed the covering type of X.
It is easy to extend the above estimate to homotopy retracts. Proof. Let A be a homotopy retract of X, i.e., there exist a map r : X → A and a homotopy H : ri ≃ 1 A . Then A ≡ A × 1 is a strict retract of the mapping cylinder M i := (A×I)+X (a,0)∼i(a) . Explicit retraction r : M i → A is given by r(a, t) := H(a, t) and r(x) := r(x).
Since a wedge retracts to any of its summands, we obtain the following monotonicity property of the covering type with respect to wedges. Corollary 4.6. ct(X ∨ Y ) ≥ max{ct(X), ct(Y )} On the other hand, we can derive an upper estimate for the covering type of a wedge as follows. Let m = ct(X) and n = ct(Y ). Then there are simplicial complexes K ≤ ∆ m−1 and L ≤ ∆ n−1 , such that X ≃ |K| and Y ≃ |L|. Clearly, K ∨ L can be realized as a one-point union of K and L and is thus a subcomplex of ∆ m+n−1 . That estimate can be improved by gluing K and L along bigger subcomplexes. Indeed, if hdim(X) = k and hdim(Y ) = l, then K and L contain respectively a k-dimensional simplex σ ≤ K and a l-dimensional simplex τ ≤ L. If we assume that k ≤ l and we glue together K and L so that σ is identified with a face of τ , then the resulting complex is a subcomplex of ∆ m+n−k−1 , while its geometric realization is homotopy equivalent to X ∨ Y . Thus we have proved the following estimate. ct(X ∨ Y ) ≤ ct(X) + ct(Y ) − min{hdimX, hdimY } − 1 .
A similar argument yields an estimate for the covering type of a connected sum of manifolds. Proof. As above, we find simplicial complexes K with ct(M ) vertices and |K| ≃ M , and L with ct(N ) vertices and |L| ≃ N . If we form the union of K and L along a common d-dimensional simplex and excise the interior of the common simplex, we obtain a model for the connected sum M ♯N . The number of vertices is as stated in the Corollary, because we have to subtract the vertices of the common simplex that are counted twice.
Karoubi and Weibel [7,Theorem 7.1] have shown that the suspension of a space can increase its covering type at most by one. However, it happens frequently that the covering type drops after suspension. Example 4.9. We have shown in Corollary 4.3 that ct(S m × S n ) ≥ m + 2n + 4. On the other hand, after suspension a product of spheres splits as a wedge of spheres Σ(S m × S n ) ≃ S m+1 ∨ S n+1 ∨ S m+n+1 . Therefore, by Proposition 4.7 ct(Σ(S n × S n ) ≤ m + n + 5, so the covering type of the suspension of S m × S n is smaller at least by n − 1 than the covering type of S m × S n . Indeed, the result is not surprising at all if we have in mind our estimates based on the LS-category and the cohomology products, and recall that the category of a suspension is always equal to 2, and that the cohomology products in a suspension are always trivial.
We may now estimate the covering type of Moore spaces whose homology is a finite cyclic group. i + 3 ≤ ct(M (Z k , i)) ≤ i + 3k .
Proof. The lower estimate follows immediately from the observation that hdimM (Z k , i) = i + 1. For the upper estimate observe that M (Z k , 1) is the mapping cone of the k-sheeted covering map between circles. By [7,Theorem 7.2] and the fact that ct(S 1 ) = 3 we obtain ct(M (Z k , 1)) ≤ 1 + 3k. Since M (Z k , i) can be obtained as a (i − 1)-fold suspension of M (Z k , 1) [7, Theorem 7.1] implies ct(M (Z k , 1)) ≤ i + 3k.
By combining Theorem 4.1, Proposition 4.7 and Proposition 4.10 we obtain an upper bound for the covering type of Moore spaces with finitely generated homology. In the next result we will assume that there is at least some torsion in homology, since the torsion-free case is settled by Theorem 4.1. If we add more finite cyclic summands both terms in the wedge are (i + 1)dimensional, and so the covering type increases at most by 3k j − 2 at each step. Observe that the formula is valid even if r = 0, because in that case k 0 = 0.