aa r X i v : . [ m a t h . A T ] A p r MULTIPLICITY OF CONTINUOUS MAPS BETWEENMANIFOLDS
R.N. KARASEV
Abstract.
We consider a continuous map f : M → N betweentwo manifolds and try to estimate its multiplicity from below, i.e.find a q -tuple of pairwise distinct points x , . . . , x q ∈ M such that f ( x ) = f ( x ) = . . . = f ( x q ).We show that there are certain characteristic classes of vectorbundle f ∗ T N − T M that guarantee a bound on the multiplicityof f . In particular, we prove some non-trivial bound on the mul-tiplicity for a continuous map of a real projective space of certaindimension into a Euclidean space. Introduction
In this paper we consider a continuous map f : M → N betweentwo manifolds and try to find some sufficient conditions for existence of multiple points , i.e. the q -tuples of pairwise distinct point x , . . . , x q ∈ M such that f ( x ) = f ( x ) = . . . = f ( x q ) . We call such a q -tuple x , . . . , x q a coincident q -tuple , and call the multiplicity of f the maximum q such that there exists a coincident q -tuple for f .The results of this kind for double points of continuous maps wereobtained in [19, 2, 16, 10]. They have obvious relation to embeddabilityand immersibility of manifolds. Some results about the multiplicity inthe case f is a smooth immersion are also known, see [7, 3] for example.In the recent paper [5] the lower bounds for the multiplicity are givenin the case when the domain space is a polyhedron of high enoughcomplexity (a skeleton of a simplex) and f is a generic piecewise linearor piecewise smooth map. In this paper we investigate the case when f is continuous without any other restrictions. Mathematics Subject Classification.
Key words and phrases. multiplicity, multiple points, singularities, configurationspaces.This research is supported by the Dynasty Foundation, the President’s of Rus-sian Federation grant MK-113.2010.1, the Russian Foundation for Basic Researchgrants 10-01-00096 and 10-01-00139, the Federal Program “Scientific and scientific-pedagogical staff of innovative Russia” 2009–2013.
In Section 6 we show that there exist certain characteristic classesof vector bundle f ∗ T N − T M that guarantee the existence of multiplepoints for f . Then we give some particular applications of these classes.We prove Theorem 3 on the multiplicity for continuous maps from aprojective space to a Euclidean space, calculate some characteristicclasses of coincident 4-tuples in Section 8, calculate the characteristicclasses modulo a prime p of coincident p -tuples in Section 9.In Section 10 we consider another question, having a lot in commonmultiple points of maps, that is the question of estimating from be-low the Krasnosel’skii-Schwarz genus and the Lyusternik-Schnirelmanncategory of configuration spaces of manifolds.The author thanks S.A. Melikhov for pointing out the relation ofthis problem to the singularity theory and detailed discussions of thesubject.2. Local multiplicity of generic smooth maps – theapproach of singularity theory
The theory of singularities for smooth maps gives some approach tomultiplicity. For example, it is known that a generic (in some sense)smooth map f : M → N may have singularities of type Σ k , with thefollowing canonical form [11].Let the local coordinates be ( x , . . . , x m ) in M and ( y , . . . , y n ) in N .Then the map is given by y i = x i , i = 1 , . . . , m − y i = k X l =1 x ( i − m ) k + l x lm , i = m, . . . , n − y n = k − X l =1 x ( n − m ) k + l x lm + x k +1 m , (3)here the inequality k ( n − m + 1) ≤ m must hold. If we select thenumbers x i ( i = 1 , . . . m −
1) so that the polynomials in x m in the rightpart of (2) are zero, and the right part of (3) has k + 1 distinct roots,then we obtain a coincident ( k + 1)-tuple, since the coordinate x m has k + 1 possible choices.Such singularities for generic maps are guaranteed by the appropri-ate characteristic classes of the virtual bundle f ∗ T N − T M . The classes σ k,m − n for singularities of type Σ k in codimension m − n can be ex-pressed in terms of Stiefel-Whitney classes by some recurrent formulas,see [13] for example.Unlike the singularity theory approach, the approach to multiplicityin this paper is valid for arbitrary continuous maps, not only smoothand generic. This approach has some similarities with the singularitytheory, in particular, some characteristic classes of f ∗ T N − T M that
ULTIPLICITY OF CONTINUOUS MAPS. . . 3 guarantee multiple points are introduced. In particular, in Section 6 weintroduce a characteristic class s q,m − n that guarantee a coincident q -tuple for q = 2 l , the author does not known whether the classes s q,m − n are a particular case of the classes σ q,m − n .3. Global multiplicity and configuration spaces
In this section we consider a continuous map f : M → N and try togive sufficient conditions for the existence of a coincident q -tuple. Themost straightforward approach is to consider the configuration space. Definition 1.
For a topological space X denote the configuration space K q ( X ) = { ( x , . . . , x q ) ∈ X q : ∀ i, j x i = x j } . Note that the permutation group Σ q acts freely on K q ( X ). Forany continuous map f : M → N denote its power f q : K q ( M ) → N q its power restricted to K q ( M ). This is an Σ q -equivariant map.A coincident q -tuple is an intersection of f q ( K q ( X )) with the (thin)diagonal ∆( N ) ⊆ N q .Thus the preimage of the diagonal ( f q ) − (∆( N )) ⊆ K q ( M ) can beconsidered as an obstruction to deforming the map f so that it has nocoincident q -tuples. In the case when M and N are smooth manifoldsof dimensions m and n respectively, and M is compact, this preimage ofthe diagonal can be considered as an Σ q -equivariant cohomology class s q ( f ) ∈ H n ( q − q ( K q ( M )) = H n ( q − ( K q ( M ) / Σ q ) , the coefficients of the cohomology being Z , or Z , possibly with the signaction of the group Σ q depending on the orientability of N and parityof its dimension. Certainly, this obstruction can also be consideredas an oriented or non-oriented cobordism class in Ω n ( q − S ) O ( K q ( M ) / Σ q ),but we do not use the cobordism in this paper.The global cohomology class for double points has certain relationwith the local double points in the case N = R n , see [19, 16, 10] forexample. For multiplicity > Local multiplicity of continuous maps – theconfiguration space bundle
In order to prove that the class s q ( f ) is nonzero, it sometimes makessense to restrict it to the intersection of K q ( M ) with a certain neigh-borhood of the thin diagonal ∆( M ) ⊂ M q . This approach would givemultiple points, that are close enough to each other in M . We call suchmultiple points local .Note that this type of local multiplicity is stronger than the localmultiplicity in the smooth generic case (see Section 2), because here wemay guarantee the existence of a coincident q -tuple { x , . . . , x q } with R.N. KARASEV some small but bounded from below diameter, the bound depending on M only. In will be clear from the definition of Q q ( M, . . . ) in Section 6.Similar to what is done in the singularity theory, we are going toreformulate the problem as a problem for bundle maps. Suppose M is a compact Riemannian manifold, and N is a Riemannian manifoldwith the injectivity radius r ( M ).Consider the tangent bundle T M and the exponential map exp :
T M → M × M , induced by the Riemannian metric, and sending atangent vector τ at x to the ends of a geodesic ( x, y ) with length | τ | and starting direction τ . Let us fix some x ∈ M , then the q -th powerexp q : K q ( T x M ) → M q maps q -tuples ( τ , . . . , τ q ) ∈ ( T x M ) q of vectorswith lengths < r ( M ) to q -tuples of distinct points in M . Hence byreplacing T x M with an open disc D x M of radius r ( M ) we obtain awell-defined map exp q : K q ( D x M ) → K q ( M ). Hence there exists Σ q -equivariant map exp q : K qM ( DM ) → K q ( M ) , here we assume the following definition. Definition 2.
Let ξ : E ( ξ ) → M be a vector or disc bundle over M .The subspace of K q ( E ( ξ )), consisting of configurations lying in thesame fiber of ξ , is denoted K qM ( ξ ) and called the configuration spacebundle .Now let us consider a continuous map f : M → N . Let us takesmall enough tangent disc bundle DN so that exp (of N ) is invertibleon it. Then take small enough tangent disc bundle DM so that theinclusion for the map of pairs f ◦ exp M DM ⊂ exp N DN holds. Thena fiberwise map φ = exp − N ◦ f ◦ exp M : DM → DN is defined. To find a local coincident q -tuple it is sufficient to finda coincident q -tuple of the fiberwise map φ q : K qM ( DM ) → DN ⊕ q ,the latter is the q -fold Whitney sum of bundles over N . Consider thepullback f ∗ ( DN ) with the natural fiberwise map f ∗ : f ∗ ( DN ) → DN .Note, that there exists a natural fiberwise map ψ : DM → f ∗ ( DN ),and the corresponding map ψ q : K qM ( DM ) → ( f ∗ ( DN )) ⊕ q such that φ q = f ∗ ◦ ψ q . So the local multiplicity is bounded from below if webound from below a multiplicity in the fiberwise map ψ : DM → f ∗ ( DN )over the same space M .In the sequel we do not distinguish between a vector bundle and itsdisc bundle, since they and their configuration spaces are diffeomorphic.Let us generalize a problem to finding multiple points for a fiberwisemap ψ : ξ → η of some vector bundles ξ and η over the same space M .Let A q be the q − q , arising fromthe natural permutation representation on R q by taking the quotient ULTIPLICITY OF CONTINUOUS MAPS. . . 5 R q / (1 , , . . . , ξ q : K qM ( ξ ) → M . Now the map ψ q gives an Σ q -equivariant section of the vector bundle ( ξ q ) ∗ ( η ⊕ q ). Composed withthe natural projection( ξ q ) ∗ ( η ⊕ q ) → A q ⊗ ( ξ q ) ∗ ( η ) , it also gives a section ψ q of the vector bundle A q ⊗ ( ξ q ) ∗ ( η ) over K qM ( ξ ).Note that ψ q is an equivariant section w.r.t the natural action of Σ q on K qM ( ξ ) and on A q . Now the coincident q -tuples correspond to thezero set of the section ψ q , and we have reduced the problem of findinglocal multiple points to proving that the zero set of the Σ q -equivariantvector bundle A q ⊗ ( ξ q ) ∗ ( η ) over K qM ( ξ ) is nonempty.5. The Euler class of local multiple points
Now let us consider the zero set Z ( ψ q , ξ, η ) of some Σ q -equivariantsection of the bundle A q ⊗ ( ξ q ) ∗ ( η ) over K qM ( ξ ). We shall denote thebundle simply A q ⊗ η since it does not lead to a confusion.Certainly, the set Z ( ψ q , ξ, η ) may be considered a manifold for genericsections, and it is Poincare dual to the Euler class e ( A q ⊗ η ), taken inthe equivariatn cohomology or some bordism theory. It can be cal-culated directly sometimes, but we are going to “stabilize” it in somesense to simplify the computation, though some information can belost.Consider some other vector bundle ζ and two fiberwise maps ψ : ξ → η and ι : ζ → ζ , the latter being the identity. Now it is readily seenthat(4) Z (( ψ ⊕ ι ) q , ξ ⊕ ζ , η ⊕ ζ ) = Z ( ψ q , ξ, η ) × M ζ . Hence if the Euler class e ( A q ⊗ ( η ⊕ ζ )) is nonzero over K qM ( ξ ⊕ ζ ),then the set Z ( ψ q , ξ, η ) cannot be empty. Lemma 1.
The coincident q -tuples in a continuous fiberwise map ψ : ξ → η over M are guaranteed by a nonzero Σ q -equivariant Euler class e ( A q ⊗ ( ξ ⊥ ⊕ η )) in the cohomology (or cobordism) of K q ( R µ ) × M ,where µ = dim ξ + dim ξ ⊥ .Proof. In the above reasoning take ζ to be ξ ⊥ such that ξ ⊕ ξ ⊥ = ε µ , ε denotes a trivial vector bundle. Then it suffices to note that K qM ( ε µ ) = K q ( R µ ) × M. (cid:3) If we consider the cohomology with coefficients in a field Z p , then bythe K¨unneth formula the algebra H ∗ Σ q ( K q ( R µ ) × M ) is a free H ∗ ( M )-module, spanned by the linear basis of H ∗ ( K q ( R µ ) / Σ q ). Denote thelatter basis by ( h , . . . , h N ( q,µ ) ), it is known (see [4, 18, 14]) that in the R.N. KARASEV case p = 2 the elements h i can be selected to be part of the basis of H ∗ ( B Σ q , Z ).Now we can decompose the Euler class e ( A q ⊗ ( ξ ⊥ ⊕ η )) = N ( q,µ ) X i =1 s i h i and obtain the elements s i ∈ H ∗ ( M, Z p ), that depend naturally on thebundle ξ ⊥ ⊕ η . Hence s i only depend on the Stiefel-Whitney (in case p = 2) or the Pontryagin and Euler (in case p odd) classes of the virtualbundle η − ξ , since it is sufficient to consider the situation over someGrassmann variety and then use the naturality of construction. Thisis in accordance with the similar result for smooth map singularities,where the characteristic classes depend on the virtual bundle f ∗ T N − T M .6.
Local coincident p k -tuples and their characteristicclasses Let us consider the case when q is a power of a prime p . In this casethe cohomology H ∗ ( K q ( R µ ) / Σ q , A ) is zero in dimensions > ( q − µ − q − µ − e ( A q ) n − , see [17, 12, 14, 9] for different cases of these results.Here the coefficients A are Z p for p = 2 or odd µ , and Z p with signaction of Σ q in other cases.Thus similar to the above definition, we can put e ( A q ⊗ ( ξ ⊥ ⊕ η )) = s q,d ( ξ ⊥ ⊕ η ) e ( A q ) µ − + . . . , where d = dim η − dim ξ = dim( ξ ⊥ ⊕ η ) − µ and . . . denotes the termswith the dimension of the corresponding class in H ∗ ( M, Z p ) larger, thenthat of s q,d . Note that the dimension of s q,d equals ( q − d + 1). Definition 3.
We call s q,d ( η − ξ ) the leading characteristic class oflocal coincident q -tuples for prime powers q at codimension d .In some cases this class (and possibly some higher classes) can becalculated.We are going to prove the following result, showing that s q,d arenontrivial in the case of q = 2 k . Theorem 1.
Let q be a power of two. Denote w i the Stiefel-Whitneyclasses of η − ξ . Then as a polynomial in the Stiefel-Whitney classesof η − ξ s q,d ( η − ξ ) ≡ w q − d +1 mod w d +2 , w d +3 , . . . . In order to prove Theorem 1 we are going to consider some subspaceof the configuration space K q ( R n ). Such subspaces were introducedin [8] and proved to be very useful in describing the cohomology of thesymmetric group modulo 2. ULTIPLICITY OF CONTINUOUS MAPS. . . 7
Definition 4.
Let q = 2 k and consider a sequence δ , . . . , δ k of positiveintegers such that for any l < kδ l > k X i = l +1 δ i . Let Q ( R n ) be the configuration, consisting of one point at the origin.Let by induction Q q ( R n , δ , . . . , δ k ) be the set of all q -point configura-tions, such that the first q/ Q q/ ( R n , δ , . . . , δ k ),shifted by a vector u of length δ , and the other q/ Q q/ ( R n , δ , . . . , δ k ), shifted by a vector − u . Definition 5.
A configuration in Q q ( R n , δ , . . . , δ k ) can also be de-scribed inductively as x , . . . , x q ∈ R n such that all the distancesdist( x i − , x i ) = 2 δ k and the midpoints of [ x i − , x i ] form a config-uration of Q q/ ( R n , δ , . . . , δ k − ).Note that Q q ( R n , δ , . . . , δ k ) is always a product of q − n −
1, and we shall omit δ i in the notation since it doesnot change the diffeomorphism type of Q q ( R n ). Then we can naturallydefine the space Q qM ( ξ ) ⊂ K qM ( ξ ) for any vector bundle ξ : E ( ξ ) → M as a bundle of corresponding Q q ( ξ − ( x )) for x ∈ M .Note that the Definition 5 (distance and midpoint characterization)can be applied to any Riemannian manifold M , if we allow the lastcenter point (configuration Q ) be any x ∈ M . The distances shouldbe chosen small enough in order for the midpoint to be unique. Definition 6.
Let M be a Riemannian manifold. Define Q q ( M, δ , . . . , δ k ) ⊂ K q ( M ) for q = 2 k inductively as follows. Q ( M ) = M .For q ≥ Q q ( M, δ , . . . , δ k ) be the set of q -tuples x , . . . , x q ∈ M such that all the distances dist( x i − , x i ) = 2 δ k and the midpoints of[ x i − , x i ] form a configuration of Q q/ ( M, δ , . . . , δ k − ).The following lemma allows to guarantee not only local singularities,but singularities of some finite size from considering Q qM ( T M ) as theconfiguration space.
Lemma 2.
Let the injectivity radius of M be r and for all i = 1 , . . . , k δ i < r. Then Q q ( M, δ , . . . , δ k ) is a fiber bundle (the bundle map is the laststage midpoint) over M , and is naturally homeomorphic to Q qM ( T M ) Proof.
Let us prove by induction. For any configuration ( x , . . . , x q ) ∈ Q q ( M, δ , . . . , δ k ) the midpoints of pairs [ x , x ] , [ x , x ] , . . . , [ x q − , x q ]form a configuration in Q q/ ( M, δ , . . . , δ k − ). Since 2 δ k < r , thenknowing the midpoint of [ x , x ], the possible positions of the points x , x form a sphere. R.N. KARASEV So Q q ( M, . . . ) is a product-of-spheres bundle over Q q/ ( M, . . . ). More-over, these spheres are spheres of the vector bundles π ∗ i ( T M ), where π i : Q q/ ( M, . . . ) → M is the map, assigning to a configuration its i -thpoint. Note that the maps π i are all homotopic to the centerpoint map π : Q q/ ( M, . . . ) → M (the homotopy can be obtained by deforminga point x i − or x i to the midpoint of [ x i − , x i ], and then repeatinginductively), hence all the vector bundles are equivalent to π ∗ ( T M ).Now the proof is completed by applying the inductive assumption. (cid:3)
The space Q q (or Q q ) is not invariant under the natural Σ q -action,but it is invariant under the action of a certain Sylow subgroup. Definition 7.
Let q = 2 k . Denote Σ (2) q the Sylow subgroup of Σ q ,generated by all permutations of two consecutive blocks [ a l + 1 , a l +2 l − ] and [ a l +2 l − +1 , ( a +1)2 l ], where 2 ≤ l ≤ k and 0 ≤ a ≤ k − l − Lemma 3.
The manifold Q q ( R n ) is Σ (2) q -invariant. The cohomology H ( q − n − (2) q ( Q q ( R n ) , Z ) is generated by the Euler class e ( A q ) n − .Proof. The first claim is obvious by definition.Consider the natural projection ψ : R n → R n − , defined by ψ ( x , . . . , x n − , x n ) = ( x , . . . , x n − ) . For this projection the only configurations in Q q ( R n ) that give co-incident q -tuples are those with all coordinates zero except x n . Butsuch configurations form exactly one orbit of Σ (2) q . Since this orbit isPoincare dual to the Euler class e ( A q ) n − , which is responsible for co-incident q -tuples in this case, we see that e ( A q ) n − coincides with thefundamental class of the manifold Q q ( R n ) / Σ (2) q . (cid:3) In is well-known [1], that if we consider the Σ q -equivariant cohomol-ogy with coefficients Z p , then the cohomology does not change whenpassing to p -Sylow subgroup. Here we do not only pass to a Sylowsubgroup, but also refine the configuration space K q to a manifold, toallow some direct geometric reasoning as in the proof of Lemma 3.Using Lemma 3 the leading characteristic class of coincident 2 k -tuples can be defined as follows. Definition 8.
Denote π : Q qM ( ξ ) / Σ (2) q → M the natural projection.Then s q,d ( η − ξ ) = π ! ( e ( A q ⊗ π ∗ η )) , i.e. geometrically it is a projection of the set of coincident q -tuples in Q qM ( ξ ) / Σ (2) q to M .This definition is the same because for the restricted set of coincident q -tuples Z ( ψ, ξ, eta ) ⊆ Q q of a fiberwise map ψ : ξ → η the stabilityholds in the following exact form(5) Z (( ψ ⊕ ι ) q , ξ ⊕ ζ , η ⊕ ζ ) = Z ( ψ q , ξ, η ) . ULTIPLICITY OF CONTINUOUS MAPS. . . 9
Then passing to the case of trivial ξ we see that the topmost cohomol-ogy of Q q ( R µ ) is the same as in K q ( R µ ). The map π ! “divides” by thefundamental class of Q q ( R µ ) / Σ (2) q in H ∗ ( Q q ( R µ ) / Σ (2) q × M ), similar tothe first definition of s q,d .7. Proof of Theorem 1 and some corollaries
Now we are ready to prove Theorem 1 using the geometric definitionof s q,d .Let us find s q,d for a certain fiberwise map over the Grassmannian M = G n,d +1 of linear n -subspaces in R n + d +1 . Denote the canonical n -dimensional bundle γ : E ( γ ) → G n,d +1 . Now consider the map f : R n + d +1 → R n + d given by f ( x , . . . , x n + d +1 ) = ( x − x n + d +1 , x − x n + d +1 , . . . , x n + d − x n + d +1 n + d +1 ) . Each fiber of γ is mapped with this map to R n + d , so f can be consideredas a fiberwise map of γ to ε n + d .Let us describe the coincident q -tuples of f in Q qM ( γ ). First, notethat there is a natural inclusion Q qM ( γ ) → Q q ( R n + d +1 ). A configurationof q points ( p , . . . , p q ) ∈ Q q ( R n + d +1 ) is mapped to one point y if theylie on a single curve C ( c , . . . , c n + d ), given by the parameterization x = c + t , x = c + t , . . . , x n + d = c n + d + t n + d +1 , x n + d +1 = t. Consider a single curve C ( c , . . . , c n + m ) and the set Z ( c , . . . , c n + m )of all configurations in Q q ( R n ) (not Q q !), lying entirely on C ( c , . . . , c n + d ).Consider a configuration ( p , . . . , p q ) ∈ Z ( c , . . . , c n + m ) and assumethat the points are ordered w.r.t. the coordinate x n + d +1 , which cor-responds with the parameter on the curve. From Definition 5 it isclear that if δ i are small enough in the definition (so that the cur-vature of C ( c , . . . , c n + m ) becomes negligible), then the configuration( p , . . . , p q ) ∈ Z ( c , . . . , c n + m ) is determined uniquely by any one point p i , which can be chosen arbitrarily. In other words, p i is a smoothparameter on Z ( c , . . . , c n + m ) / Σ (2) q .Denote Z = [ c ,...,c n + d ∈ R Z ( c , . . . , c n + d ) ⊂ Q q ( R n + d +1 ) . We have already noted that the map g i : Z/ Σ (2) q → R n + d +1 takingany configuration to its i -th point w.r.t. the coordinate x n + d +1 is adiffeomorphism. For any configuration ( p , . . . , p q ) ∈ Z put h ( p , . . . , p q ) = q X i =1 g i ( p , . . . , p q ) , i.e. the center point of the configuration. The map h is smooth on Z/ Σ (2) q and for small enough δ i it is a diffeomorphism onto R n + d +1 .Hence h − (0) is the only configuration in Q q ( R n + d +1 ) that is mappedinto single point by f .Since the configuration h − (0) lies on a translate of the momentcurve, which is a convex curve, then its points span some q − L ⊂ R n + d +1 . Now any linear space V ∈ G n,d +1 , that is supposed to have coincident q -tuples in the fiber Q q ( V ),must contain L . Moreover, it can be easily seen that the map f istransversal to zero and the condition V ⊇ L defines the Poincare dualto s q,d homology class. From the well-known description of the (dual)Stiefel-Whitney classes it follows that(6) s q,d = w q − d +1 ( γ ⊥ )in the cohomology H ∗ ( G n,d +1 , Z ). Now it suffices to note that thiscohomology algebra has generators w ( γ ) , . . . , w n ( γ ) and relations(7) w d +2 ( γ ⊥ ) = w d +3 ( γ ⊥ ) = · · · = 0 , hence (6) holds over arbitrary space modulo higher Stiefel-Whitneyclasses of η − ξ , because the number n can be taken arbitrarily large,and the relations in (7) are the only essential relations. Corollary 2.
For double points we have: s ,d = w d +1 . The corollary follows from Theorem 1 because in the right part therecannot be anything, depending on w d +2 , w d +3 , . . . from the dimensionconsiderations.Corollary 2 was actually proved in [2] for fiberwise maps to trivialbundle. Moreover, it is known that the local double points for maps M → R n have a relation to global double points for maps M → R n +1 (informally, they have the same characteristic class), see [16] for alge-braic description, or [10] for some geometric reasoning.Let us prove a theorem that gives coincident q -tuples for maps ofcertain projective spaces to R n by the class s q,d . Theorem 3.
Suppose that q is a power of two, q ( d + 1) < l − . Thenany continuous map f : R P l − − d → R l − has multiplicity ≥ q . Note that this theorem gives a coincident q -tuple on a configurationfrom any subspace Q q ( R P l − − d , δ , . . . , δ k ) ⊂ K q ( R P l − − d )for any sequence of δ i , satisfying δ < π/ , ∀ i δ i > δ i +1 + · · · + δ k . ULTIPLICITY OF CONTINUOUS MAPS. . . 11
Here we measure the distance on R P n as the angle between lines. Proof.
Any map f : R P m → R n induces a fiberwise map between ξ = T R P m and ε n . Suppose we have some normal bundle ξ ⊥ of dimension k . The Stiefel-Whitney class of ε n − ξ is w ( ε n − ξ ) = (1 + u ) − m − = (1 + u ) l − m − = (1 + u ) d +1 , and by Theorem 1 we have s q,d ( ε n − T R P m ) = u ( q − d +1) , which isnonzero since ( q − d + 1) ≤ m = 2 l − − d . (cid:3) Characteristic classes for local coincident -tuples –calculations Let us give some general schema of calculating s q,d for any particular q and d . The class e ( A q ⊗ ( ξ ⊥ ⊕ η )) can be calculated in the assumptionthat the bundle ξ ⊥ ⊕ η is decomposed into one-dimensional bundles τ , . . . , τ ν with respective Stiefel-Whitney classes 1 + t , . . . , t ν . Letthe Stiefel-Whitney class of the representation A q in the cohomology H ∗ ( B Σ (2) q , Z ) be e ( A q ) = 1 + a + · · · + a q − . Now the Euler class equals(8) e ( A q ⊗ ( τ ⊕ · · · ⊕ τ ν )) = ν Y i =1 ( t q − i + t q − i a + · · · + a q − )in the cohomology H ∗ ( B Σ (2) q × M, Z ). Then we have to map thisclass to H ∗ ( K q ( R µ ) × M, Z ) or H ∗ ( Q q ( R µ ) × M, Z ), by the naturalmap K q ( R µ ) / Σ q → B Σ q , or Q q ( R µ ) / Σ (2) q → B Σ (2) q , find the coefficient at a µ − q − , and express it in the Stiefel-Whitney classesof η − ξ . Of course, the knowledge of the cohomology of the symmetricgroup modulo 2 and the relations on these cohomology that describe H ∗ ( Q q ( R µ ) / Σ (2) q ) should be known.Passing to the particular case q = 4 note that Σ (2)4 is the squaregroup D , and its cohomology is multiplicatively generated by threeelements a, b, c such thatdim a = dim c = 1 , dim b = 2 , and the relation ac = 0. The Stiefel-Whitney class of A is w ( A ) = (1 + ( a + c ) + b )(1 + c ) . The space Q q ( R n ) is a product of three n − Q q ( R n ) / Σ (2) q we have relations c n = b n = 0 . Now (8) has the form e ( A q ⊗ ( τ ⊕ · · · ⊕ τ ν )) = ν Y i =1 ( t i + ( a + c ) t i + b )( t i + c ) . to find the leading characteristic class of coincident 4-tuples in codi-mension d we have to find the coefficient at ( bc ) ν − d − after applyingall the relations. We can also add the artificial relation a = 0 to sim-plify the situation, since it does not affect anything. Then some directcalculations give, for example s , = w + w w ,s , = w + w + w w w + w w . Further cases can be considered too, it even seems plausible to havesome explicit formula with summation for q = 4.9. Local coincident p -tuples for prime p Let us consider local coincident p -tuples for odd prime p . In thiscase we consider the Euler class e ( A p ⊗ ( ξ ⊥ ⊕ η )) modulo p , and there-fore we may consider, instead of Σ p its p -Sylow subgroup Z p of cyclicpermutations. This group acts on A p without change of orientation, sowe do not have to worry about the twisted cohomology coefficients.Similar to (8), by the splitting principle we decompose ξ ⊥ ⊕ η ofdimension ν into the sum of k two-dimensional oriented (because wedo everything mod p ) bundles σ ⊕ · · · ⊕ σ k , when ν = 2 k , or into thesum σ ⊕ · · · ⊕ σ k ⊕ τ with dim τ = 1, when ν = 2 k + 1. In either casewe have e ( A p ⊗ ( ξ ⊥ ⊕ η )) = k Y i =1 e ( A p ⊗ σ i ) = k Y i =1 ( u − e p − i ) , for even ν , and e ( A p ⊗ ( ξ ⊥ ⊕ η )) = u k Y i =1 ( u − e p − i )for odd ν . In the last two formulas u = e ( A p ), and e i = e ( σ i ) are theEuler classes of summands. The formula follows from the formula ofPontryagin classes of a tensor product along with the fact that the onlynonzero characteristic classes of A p in H ∗ ( BZ p , Z p ) are its Euler class u and its topmost Pontryagin class u .Let us define the following characteristic classes by the splitting prin-ciple: if the Pontryagin classes are expressed through symmetric func-tions p i = σ i ( t , . . . , t k )Then put α p,i = σ i ( t p − , . . . , t p − k ) , ULTIPLICITY OF CONTINUOUS MAPS. . . 13 in the case p = 3 these are the Pontryagin classes again, in the generalcase these are some classes of dimension 2( p − i . Now we can rewrite e ( A p ⊗ ( ξ ⊥ ⊕ η )) = k X i =0 ( − i u k − i ) α p,i ( ξ ⊥ ⊕ η )for even ν and e ( A p ⊗ ( ξ ⊥ ⊕ η )) = k X i =0 ( − i u k − i )+1 α p,i ( ξ ⊥ ⊕ η )for odd ν .The images of u i in H ( p − i ( K p ( R µ ) , Z p ) are nonzero if i ≤ µ − Theorem 4.
The characteristic classes of coincident p -tuples in afiberwise map ξ → η are the classes α p,i ( η − ξ ) with i ≥ dim η − dim ξ + 1 . Estimating the equivariant category of configurationspaces
In the previous sections some cohomology classes of the configurationspace K q ( M ) were established to be nonzero, thus bounding from belowthe multiplicity of a map.Note that the same classes give new lower bounds for the Lyusternik-Schnirelmann category of K q ( M ) / Σ q and the Krasnosel’skii-Schwarzgenus of K q ( M ), thus improving the results of [17, 12, 14, 9], wherethe estimate was made depending on the dimension of M only.Let us remind some definitions and lemmas, mainly from [16]. Definition 9.
Let X be a free G -space, the genus of X is the minimalsize of G -invariant open cover (i.e. cover by G -invariant open subsets) { X , . . . , X n } of X such that every X i can be G -mapped to G . Denotethe genus of X by g ( X ).It is also well-known that the genus g ( X ) estimates the Lyusternik-Schnirelmann category cat X/G from below. We need the followinglemma:
Lemma 4. If X is a paracompact free G -space, and for some G -module α the natural cohomology map π ∗ X : H n ( BG, α ) → H nG ( X, α ) is nontrivial, then cat X/G ≥ g ( X ) ≥ n + 1 . Now we can state a special case of the previous lemma.
Lemma 5.
If the natural image of e ( A q ) n is the cohomology of K q ( M ) (or K qM ( T M ) ) is nontrivial, then cat K q ( M ) / Σ q ≥ g ( K q ( M )) ≥ ( q − n + 1 . Proof.
Note that we have an Σ q -equivariant mapexp : K qM ( T M ) → K q ( M ) . If e ( A q ) n = 0 ∈ K qM ( T M ), then e ( A q ) n = 0 ∈ K q ( M ), and then weapply Lemma 4. (cid:3) Now we can state some corollaries of Theorems 1, 3, 4.
Corollary 5.
Suppose that the dual Stiefel-Whitney class of M with dim M = m has the form w ( T M ⊥ ) = 1 + ¯ w + · · · + ¯ w d +1 ,q is a power of two, and ¯ w q − d +1 = 0 . Then g ( K q ( M )) ≥ ( m + d )( q −
1) + 1 . In particular if q ( d + 1) < l − then g ( K q ( RP l − − d )) ≥ (2 l − q −
1) + 1 . Proof.
Denote the dimension of normal bundle k = dim T M ⊥ . Theo-rem 1 claims that the Euler class e ( A q ⊗ ( T M ⊥ ⊕ ε m + d )) ∈ H ( q − k + m + d ) ( K q ( R m + k ) × M )is nonzero under the assumptions.Lemma 1 now claims that the Euler class e ( A q ⊗ ε m + d ) = e ( A q ) m + d isalso nonzero in K qM ( T M ). Now the result follows from Lemma 5. (cid:3)
The results of Section 8 give a similar corollary.
Corollary 6.
Denote ¯ w i the Stiefel-whitney classes of the normal bun-dle T M ⊥ . If the class ¯ w + ¯ w ¯ w is nonzero on M then g ( K ( M )) ≥ m + 1 . If the class ¯ w + ¯ w + ¯ w ¯ w ¯ w + ¯ w ¯ w is nonzero on M then g ( K ( M )) ≥ m + 4 . And here is the corresponding corollary of Theorem 4.
Corollary 7.
Let p be a prime. Consider the classes ¯ α p,i of the normalbundle T M ⊥ , introduced in Section 9. If ¯ α p,i = 0 for some i , then g ( K p ( M )) ≥ ( m + 2 i − p −
1) + 1 . ULTIPLICITY OF CONTINUOUS MAPS. . . 15
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