On the LS-category and topological complexity of projective product spaces
aa r X i v : . [ m a t h . A T ] D ec ON THE LS-CATEGORY AND TOPOLOGICAL COMPLEXITY OFPROJECTIVE PRODUCT SPACES
SEHER FİŞEKCİ AND LUCILE VANDEMBROUCQ
Abstract.
We determine the Lusternik-Schnirelmann category of the projectiveproduct spaces introduced by D. Davis. We also obtained an upper bound for thetopological complexity of these spaces, which improves the estimate given by J.González, M. Grant, E. Torres-Giese, and M. Xicoténcatl. Introduction
We let ¯ n symbolize an r -tuple ( n , . . . , n r ) of positive integers with n ≤ . . . ≤ n r .We consider the product S ¯ n := S n × · · · × S n r and, given x i ∈ S n i , we write ¯ x =( x , . . . , x r ) for the corresponding element of S ¯ n . The quotient space P ¯ n := S ¯ n / (¯ x ∼ − ¯ x ) = ( S n × · · · × S n r ) / (( x , . . . , x r ) ∼ ( − x , . . . , − x r )) with respect to the diagonal action of Z on S ¯ n has been introduced by D. Davis [5]and is called projective product space . This is a manifold of dimension dim ( P ¯ n ) = dim ( S ¯ n ) = P n i and, when r = 1 , the space P ¯ n coincides with the usual real projec-tive space P n . Recently, the notion of projective product space was used in [2] andgeneralized in [11].In this note, we study the (normalized) Lusternik-Schnirelmann category (cat) andFarber’s topological complexity (TC) of the space P ¯ n . This study has been initiatedby J. González, M. Grant, E. Torres-Giese, M. Xicoténcatl in [10], where the followingresult is established: Theorem 1.1. [10, Theorem 3.8]
Let k represent the number of spheres S n q with n q even and q > . Then T C ( P ¯ n ) < ( T C ( P n ) + 1)( r + k ) . In this paper, we first determine the LS-category of P ¯ n : Theorem 1.2. cat ( P ¯ n ) = cat ( P n ) + r − n + r − . This is obtained through the construction of an explicit categorical cover togetherwith the knowledge of the cohomology of P ¯ n over Z , which has been determined by Date : December 10, 2020.This work has been realized during a stay of the first author at the Centre of Mathematics of theUniversity of Minho during the period October 2019-July 2020. The first author is granted by afellowship by the Scientific and Technological Research Council of Turkey TUBITAK-2211/A andsupported by the Scientific and Technological Research Council of Turkey International DoctoralResearch Fellowship Programme TUBITAK-2214/A (Project Number:1059B141801086). The re-search of the second author was partially supported by Portuguese Funds through FCT – Fundaçãopara a Ciência e a Tecnologia, within the Project UID/MAT/00013/2020.
Davis [5, Theorem 2.1] (see Theorem 2.2 below).For the topological complexity, we establish:
Theorem 1.3. TC ( P ¯ n ) ≤ TC ( P n ) + P rq =2 TC ( S n q ) . In terms of the number k of spheres S n q with n q even and q > , the upperbound of Theorem 1.3 can be written TC ( P n ) + r + k − , which permits us tosee that Theorem 1.3 improves Theorem 1.1. As mentionned in [10], using Davis’description of H ∗ ( P ¯ n ; Z )) , the zero-divisor-cup-length of P ¯ n over Z can be expressedas zcl Z ( P n ) + r − . We then obtain: Corollary 1.4.
If zcl Z ( P n ) = TC ( P n ) and n q is odd for q > then TC ( P ¯ n ) = TC ( P n ) + P rq =2 TC ( S n q ) = TC ( P n ) + r − . The upper bound in Theorem 1.3 is obtained through the construction of an explicitmotion planner which uses the characterization of TC ( P n ) in terms of non-singularmaps due to M. Farber, S. Yuzvinsky and S. Tabachnikov [8]. We note that, usinga strong version of non-singular maps, an explicit motion planner for polyhedralproducts of real projective spaces has recently been constructed in [1].2. Lusternik-Schnirelmann category of P ¯ n As P ¯ n is a finite, path-connected CW complex, we give all the useful definitionsand characterizations for such spaces. Definition 2.1.
Let X be a finite, path-connected CW complex. A subset A ⊂ X is called categorical if the inclusion i : A ֒ → X is nullhomotopic. The Lusternik-Schnirelman category cat ( X ) of X is defined as the least integer k that admits acover of X by k + 1 open categorical subsets U , . . . , U k ⊂ X .Let R be a commutative unitary ring. Recall that the cup-length over R ofa path-connected space X , cuplength( X ) = cuplength R ( X ) , is the longest length k of a nonzero product c ⌣ · · · ⌣ c k = 0 of cohomology classes c , · · · , c k ∈ H + ( X ; R ) and provides a lower bound for the Lusternik-Schnirelmann category of X , cuplength( X ) ≤ cat ( X ) .As is well-known, the cup-length over Z of the m -dimensional projective space P m is equal to m . The Z cohomology of P ¯ n has been determined by Davis: Theorem 2.2. ( [5, Theorem 2.1] ) Let ¯ n = ( n , . . . , n r ) such that n ≤ . . . ≤ n r . If n < n or n is odd, the mod 2 cohomology ring of P ¯ n is given by H ∗ ( P ¯ n ; Z ) = H ∗ ( P n ; Z ) ⊗ Λ[ a , . . . , a r ] where dim ( a ) = 1 , dim ( a i ) = n i for i > , and Λ denotes an exterior algebra. If n is even and n = · · · = n k < n k +1 for some k > , H ∗ ( P ¯ n ; Z ) is the same as abovewith the extra relation given by a i = a n a i for ≤ i ≤ k . N THE LS-CATEGORY AND TOPOLOGICAL COMPLEXITY OF PROJECTIVE PRODUCT SPACES3
By considering the longest nonzero product a n ⌣ a · · · ⌣ a r = 0 in H ∗ ( P ¯ n ; Z ) we obtain: Proposition 2.3. cuplength Z ( P ¯ n ) = cuplength Z ( P n ) + r − n + r − . In order to prove Theorem 1.2, we will construct an explicit cover of P ¯ n withcontractible subsets using the following characterization of the category: Proposition 2.4. [4, Lemma 1.35]
Suppose that X is a path-connected finite CW-complex. We have cat ( X ) ≤ k if and only if there exists an increasing sequence ofopen sets ∅ = U − ⊂ U ⊂ U ⊂ . . . ⊂ U k = X such that, for any i ∈ { , · · · , k } , U i − U i − is contractible in X . Remark 2.5.
The sets F i = U i − U i − provide a cover of X by k + 1 disjoint subsetswhich are contractible in X . We note that, in [4, Lemma 1.35], X is not supposed tobe a finite CW-complex but it is required that each F i = U i − U i − is contained in anopen set of X which is contractible in X . Assuming that X is a finite CW-complex(and therefore an ENR space) and adapting the proof of [7, Prop. 4.12], we can justask that F i is contractible in X . Actually, when X is a metric ANR space, we knowby [12] (see also [9]), that cat( X ) ≤ k if and only there exists a cover by k + 1 subsetscontractible in X without further condition on the subsets. Proof of Theorem 1.2.
We first fix some general notation. Writing S m = { ( u , . . . , u k , . . . , u m ) ∈ S m ⊂ R m +1 | X u k = 1 } , we denote by p j : S m → R the obvious projection ( j = 0 , · · · , m ) and by a j the uniquepoint of S m such that p j ( a j ) = 1 . Note that, for any x ∈ S m , p j ( − x ) = − p j ( x ) . When j = m , we will use the special notation A m := a m = (0 , · · · , , and we fix a merid-ian path µ ( A m , − A m ) : I → S m such that µ (0) = A m and µ (1) = − A m . Wewill denote by µ ( − A m , A m ) the path given by µ ( − A m , A m )( t ) = − µ ( A m , − A m )( t ) ,that is µ ( − A m , A m ) = − µ ( A m , − A m ) . For non-antipodal points A, B ( A = − B ) of S m , let λ ( A, B ) : I → S m be the geodesic path from A to B . Note that λ ( − A, − B ) = − λ ( A, B ) and that λ ( A, A ) is the constant path.We now define a cover of S n which induces a cover of P n by categorical subsets.In our constructions we use a formalism which was inspired by [3].For a subset L ⊂ { , , . . . , n } , let | L | denote the cardinality of L and consider S L = { x ∈ S n : p l ( x ) = 0 if l ∈ L } . By setting U i = [ | L | =( n +1) − i S L SEHER FİŞEKCİ AND LUCILE VANDEMBROUCQ for ≤ i ≤ n and U − = ∅ , we have an increasing sequence of open subsets of S n : ∅ = U − ⊂ U ⊂ . . . ⊂ U n = S n Note that U n = S n since the projections p ,..., p n cannot vanish all at the sametime.Considering, for L ⊂ { , , . . . , n } , Q L = { x ∈ S n : p l ( x ) = 0 if l ∈ L and p l ( x ) = 0 if l / ∈ L } we have, for ≤ i ≤ n , F i := U i − U i − = [ | L | =( n +1) − i Q L . Note that all the sets S L , Q L , U i , F i are saturated with respect to the antipodalrelation x ∼ − x on S n . Note also that, for L = L with same cardinality | L | = | L | the sets Q L and Q L are topologically disjoint and that S n i =0 F i is a cover of S n .For L ⊂ { , , . . . , n } , we consider the map ψ L : Q L → ( P n ) I ψ L ( x ) = [ λ ( x, a l )] , if p l ( x ) > , [ λ ( x, − a l )] , if p l ( x ) < . where l = min L . The map is continuous on Q L and satisfies ψ L ( x )(0) = [ x ] , ψ L ( x )(1) = [ a l ] , and ψ L ( x ) = ψ L ( − x ) . This shows that the subset Q L / ∼ is con-tractible in P n . As the sets Q L with | L | constant are topologically disjoints, weobtain that, for ≤ i ≤ n , F i / ∼ is a union of topologically disjoint subsets whichare contractible in P n . Since P n is path-connected, we can conclude that F i / ∼ isitself contractible in P n .For ≤ q ≤ r , we consider the increasing sequence of open sets V − = ∅ V = { x ∈ S n q | x = ± A n q } V = S n q as well as the sets G := V − V − = V and G := V − V = {± A n q } . These sets aresaturated with respect to the antipodal relation.Let L ⊂ { , , . . . , n } with l = min L and, for ≤ q ≤ r , let j q ∈ { , } . Wedefine ψ ( L,j ,...,j r ) : Q L × Π rq =2 G j q → P I ¯ n by ψ ( L,j ,...,j r ) ( x , x , . . . , x r ) = [ λ ( x , a l ) , y , . . . , y r ] , if p l ( x ) > λ ( x , − a l ) , z , . . . , z r ] , if p l ( x ) < where y q = µ ( − A n q , A n q ) , if x q = − A n q λ ( x q , A n q ) , if x q = − A n q and N THE LS-CATEGORY AND TOPOLOGICAL COMPLEXITY OF PROJECTIVE PRODUCT SPACES5 z q = µ ( A n q , − A n q ) , if x q = A n q λ ( x q , − A n q ) , if x q = A n q This map is continuous, well-defined on Q L × Π rq =2 G j q and satisfies ψ ( L,j ,...,j r ) (¯ x )(0) = [¯ x ] ψ ( L,j ,...,j r ) (¯ x )(0) = [ a l , A n , . . . , A n r ] Moreover, we can check that the map is compatible with the diagonal antipodalrelation and hence permits us to see that ( Q L × Π rq =2 G j q ) / ∼ is a subset contractiblein P ¯ n .For i ∈ { , , . . . , n } and j q ∈ { , } for each ≤ q ≤ r , we built W s = S i + P rq =2 j q = s U i × Q rq =2 V j q ⊂ S n × S n . . . × S n r where s = 0 , . . . , n + r − . Therefore, there exists a tower of open subsets ∅ = W − ⊂ W ⊂ . . . ⊂ W n + r − = S n × S n . . . × S n r We have W s − W s − = [ i + P rq =2 j q = s ( U i − U i − ) × r Y q =2 ( V j q − V j q − ) = [ i + P rq =2 j q = s F i × r Y q =2 G j q which is a topologically disjoint union and each F i × Q rq =2 G j q is itself a topologicallydisjoint union of ( Q L × Π rq =2 G j q ) / ∼ with | L | constant. As already mentioned, allthe subsets are saturated with respect to the diagonal antipodal relation. Passing tothe quotient we get a tower of open subsets of P ¯ n ∅ = ˜ W − ⊂ ˜ W ⊂ . . . ⊂ ˜ W n + r − = P ¯ n and ˜ W s − ˜ W s − = [ i + P rq =2 j q = s F i × Q rq =2 G j q ∼ is a topologically disjoint union. As ( F i × Q rq =2 G j q ) / ∼ is itself a topologically disjointunion of ( Q L × Π rq =2 G j q ) / ∼ and as these spaces are contractible in P ¯ n , which is path-connected, we can conclude that each ˜ W s − ˜ W s − is contractible in P ¯ n . Therefore byProposition 2.4, we obtain that cat ( P ¯ n ) ≤ n + r − . By Proposition 2.3, we concludethat cat ( P ¯ n ) = n + r − . (cid:3) Topological complexity of P ¯ n Definition 3.1.
Let X be a finite, path-connected CW complex. The (normalized)topolological complexity of X , T C ( X ) , is the least integer k such that there exists acover of X × X by k + 1 open subsets U , U , ..., U k ⊂ X × X on each of which thefibration ev , : X I → X × X , γ ( γ (0) , γ (1)) ,admits a continuous section. SEHER FİŞEKCİ AND LUCILE VANDEMBROUCQ
In order to establish Theorem 1.3, we will use the following characterization:
Proposition 3.2. [7, Proposition 4.12]
Let X be a finite, path-connected CW com-plex. We have TC ( X ) ≤ k if and only if there exists s : X × X → X I and anincreasing sequence of open sets ∅ = U − ⊂ U ⊂ U ⊂ . . . ⊂ U k = X × X such that ev , ◦ s = id and, for any i ∈ { , · · · , k } , s is continuous on U i − U i − . Remark 3.3.
The sets F i = U i − U i − provide a cover of X × X by k + 1 disjointsubsets on each of which there is a continuous section of ev , . This defines a motionplanner for X . Proof of Theorem 1.3.
Through the obvious homeomorphism we think of P ¯ n × P ¯ n = S ¯ n × S ¯ n / ∼ as the quotient of ( S n × S n ) × · · · × ( S n r × S n r ) with respect to therelation(3.1) ( x , y , . . . , x r , y r ) ∼ ( x ′ , y ′ , . . . , x ′ r , y ′ r ) ⇔ ∀ i x i = x ′ i and y i = y ′ i or ∀ i x i = − x ′ i and y i = y ′ i or ∀ i x i = x ′ i and y i = − y ′ i or ∀ i x i = − x ′ i and y i = − y ′ i We first recall from [8] and [6] the construction of motion planners for the realprojective space P n and for a sphere S n q . We will next see how to assemble them inorder to obtain a motion planner for P ¯ n .Suppose that TC ( P n ) = k . As proven in [8], there exists a non-singular map f = ( f , · · · , f k ) : R n +1 × R n +1 → R k +1 . The k + 1 scalar maps f , f , . . . , f k : R n +1 × R n +1 → R satisfy f i ( ax , by ) = abf i ( x , y ) for ( x , y ) ∈ S n × S n and a, b ∈ R and do not vanish all at the same time (except in (0 , ). Moreover, accordingto [8, Lemma 11], we can assume that f ( x , x ) > for any x ∈ S n .For a subset L ⊂ { , , . . . , k } , let S L = { ( x , y ) ∈ S n × S n : f l ( x , y ) = 0 if l ∈ L } . By setting U i = [ | L | =( k +1) − i S L for ≤ i ≤ k and U − = ∅ , we have an increasing sequence of open subsets of S n × S n : ∅ = U − ⊂ U ⊂ . . . ⊂ U k = S n × S n Note that U k = S n × S n since the k + 1 scalar functions f ,..., f k cannot vanish allat the same time.Considering, for L ⊂ { , , . . . , k } , Q L = { ( x , y ) ∈ S n × S n : f l ( x , y ) = 0 if l ∈ L and f l ( x , y ) = 0 if l / ∈ L } N THE LS-CATEGORY AND TOPOLOGICAL COMPLEXITY OF PROJECTIVE PRODUCT SPACES7 we have, for ≤ i ≤ k , F i := U i − U i − = [ | L | =( k +1) − i Q L . Note that all the sets S L , Q L , U i , F i are saturated with respect to the equivalencerelation on S n × S n induced by the antipodal relation x ∼ − x on S n . Notealso that, for L = L with same cardinality | L | = | L | the sets S L and S L aretopologically disjoint and that S ki =0 F i is a cover of S n × S n .As before, for non-antipodal points A, B ( A = − B ) of a sphere S m , we denote by λ ( A, B ) : I → S m the geodesic path from A, B . Notice that λ ( − A, − B ) = − λ ( A, B ) and that λ ( A, A ) is the constant path.For L ⊂ { , , . . . , k } , we consider the map ψ L : Q L → ( P n ) I ψ L ( x , y ) = [ λ ( x , y )] , if f l ( x , y ) > , [ λ ( − x , y )] , if f l ( x , y ) < . where l = min L . Recall that, for any x ∈ S n , f ( x , x ) = f ( − x , − x > and, consequently f ( x , − x ) = f ( − x , x ) < . Therefore, if ( ± x , ± x ) ∈ Q L ,then l = min L = 0 . This ensures that ψ L is well-defined on pairs of antipodalpoints. The map is continuous on Q L and satisfies ψ L ( x , y ) = ψ L ( ± x , ± y ) . Asthe sets Q L with | L | constant are topologically disjoints, we obtain, for ≤ i ≤ k ,a continuous map ψ i : F i → ( P n ) I by setting ψ i | Q L = ψ L . This map satisfies ψ i ( x , y ) = ψ i ( ± x , ± y ) and the induced map ¯ ψ i : F i / ∼ → ( P n ) I gives us anexplicit motion planner on P n which essentially corresponds to the one described in[8].For ≤ q ≤ r , we will use the following increasing sequence of open subsets of S n q × S n q together with the associated complements:When n q is odd, we consider: V − = ∅ V = { ( x q , y q ) ∈ S n q × S n q : y q = ± x q } V = S n q × S n q . and G = V − V − = { ( x q , y q ) ∈ S n q × S n q : y q = ± x q } G = V − V = { ( x q , y q ) ∈ S n q × S n q : y q = ± x q } . When n q is even, we consider: V − = ∅ V = { ( x q , y q ) ∈ S n q × S n q : y q = ± x q } V = S n q × S n q \ (cid:8) ( x q , y q ) ∈ S n q × S n q : y q = ± x q and x q = ± A n q (cid:9) SEHER FİŞEKCİ AND LUCILE VANDEMBROUCQ V = S n q × S n q . and G = V − V − = { ( x q , y q ) ∈ S n q × S n q : y q = ± x q } G = V − V = (cid:8) ( x q , y q ) ∈ S n q × S n q : y q = ± x q , x q = ± A n q (cid:9) .G = V − V = (cid:8) ( x q , y q ) ∈ S n q × S n q : y q = ± x q , x q = ± A n q (cid:9) . Here, as before, A n q = (0 , · · · , , ∈ S n q .Recall that the classical motion planner for a sphere can be given as follows: • For ( x q , y q ) ∈ G and for ( x q , x q ) ∈ G ∪ G , we consider the geodesic path λ ( x q , y q ) . • For ( x q , − x q ) ∈ G , we consider the geodesic meridian µ ( x q , − x q ) from x q to − x q in the direction of χ ( x q ) . Here χ is the symmetric tangent vector field on S n q ⊂ R n q +1 given by χ ( u , v , · · · , u m , v m ) = ( − v , u , · · · , − v m , u m ) if n q =2 m − is odd and by χ ( u , v , · · · , u m , v m , u m +1 ) = ( − v , u , · · · , − v m , u m , if n q = 2 m is even. Note that µ ( − x q , x q ) = − µ ( x q , − x q ) . • For ( A n q , − A n q ) , we fix a meridian µ ( A n q , − A n q ) from A n q to − A n q and weset µ ( − A n q , A n q ) = − µ ( A n q , − A n q ) .We first assemble these motion planners in the following way.Let L ⊂ { , , . . . , k } with l = min L and, for ≤ q ≤ r , let j q ∈ { , } when n q is odd or j q ∈ { , , } when n q is even. We define ψ ( L,j ,...,j r ) : Q L × Π rq =2 G j q → P I ¯ n by ψ ( L,j ,...,j r ) ( x , y , x , y , . . . , x r , y r ) = [ λ ( x , y ) , z , . . . , z r ] , f l ( x , y ) > λ ( − x , y ) , z ′ , . . . , z ′ r ] , f l ( x , y ) < where, for n q odd, z q = µ ( x q , y q ) , if y q = − x q λ ( x q , y q ) , otherwise ( y q = ± x q or y q = x q ) z ′ q = µ ( − x q , y q ) , if y q = x q λ ( − x q , y q ) , otherwise ( y q = ± x q or y q = − x q ) and, for n q even, z q = µ ( x q , y q ) , if y q = − x q , x q = ± A n q µ ( x q , y q ) , if y q = − x q , x q = ± A n q λ ( x q , y q ) , otherwise ( y q = ± x q or y q = x q ) N THE LS-CATEGORY AND TOPOLOGICAL COMPLEXITY OF PROJECTIVE PRODUCT SPACES9 z ′ q = µ ( − x q , y q ) , if y q = x q , x q = ± A n q µ ( − x q , y q ) , if y q = x q , x q = ± A n q λ ( − x q , y q ) , otherwise ( y q = ± x q or y q = − x q ) This map is continuous and well-defined on Q L × Π rq =2 G j q . Moreover, this spaceis saturated with respect to the relation (3.1) and we can check that the map iscompatible with this relation.For i ∈ { , ..., k } , we now define ψ ( i,j ,...,j r ) : F i × Π rq =2 G j q → P I ¯ n by setting ψ ( i,j ,...,j r ) (cid:12)(cid:12) Q L × Π rq =2 G jq = ψ ( L,j ,...,j r ) . Passing to the quotient, we get a con-tinuous map ˜ ψ ( i,j ,...,j r ) : ( F i × Π rq =2 G j q ) / ∼→ P I ¯ n . . For i ∈ { , , . . . , k } , j q ∈ { , } when n q is odd or j q ∈ { , , } when n q is even( ≤ q ≤ r ) we built W s = S i + P rq =2 j q = s U i × Q rq =2 V j q ⊂ ( S n × S n ) × ( S n × S n ) × . . . × ( S n r × S n r ) where s = 0 , . . . , k + P rq =2 T C ( S n q ) = T C ( P n ) + P rq =2 T C ( S n q ) . Therefore, thereexists a tower of open subsets ∅ = W − ⊂ W ⊂ . . . ⊂ W k + P rq =2 T C ( S nq ) = ( S n × S n ) × ( S n × S n ) × . . . × ( S n r × S n r ) . We have W s − W s − = [ i + P rq =2 j q = s ( U i − U i − ) × r Y q =2 ( V j q − V j q − ) = [ i + P rq =2 j q = s F i × r Y q =2 G j q which is a topologically disjoint union. As already mentioned, all the subsets aresaturated subsets of S n × S n × S n × S n × . . . × S n r × S n r ∼ = S ¯ n × S ¯ n with respectto the relation (3.1). In particular, the tower of W s provides the following tower ofopen subsets by passing to the quotient space, ∅ = ˜ W − ⊂ ˜ W ⊂ . . . ⊂ ˜ W k + P rq =2 T C ( S nq ) ∼ = P ¯ n × P ¯ n and ˜ W s − ˜ W s − = [ i + P rq =2 j q = s F i × Q rq =2 G j q ∼ is a topologically disjoint union where s = 0 , . . . , k + P rq =2 T C ( S n q ) . Assembling themaps ˜ ψ ( i,j ,...,j r ) we get a continuous motion planner on ˜ W s − ˜ W s − and, by Proposition3.2, we can conclude thatTC ( P ¯ n ) ≤ k + r X q =2 TC ( S n q ) = TC ( P n ) + r X q =2 TC ( S n q ) . (cid:3) References [1] J. Aguilar-Guzmán, J. González, J. 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