HHomotopy groups and quantitative Sperner–type lemma
Oleg R. Musin
Abstract
We consider a generalization of Sperner’s lemma for triangulations of m –discs whose verticesare colored in at most m colors. A coloring on the boundary ( m − d , then the lower bound is 3 d + 3. Keywords:
Hopf invariant, homotopy group of spheres, Sperner lemma, framed cobordism
Sperner’s lemma is a discrete analog of the Brouwer fixed point theorem. This lemma states:
Every Sperner ( n +1) –coloring of a triangulation T of an n –dimensional simplex ∆ n containsan n –simplex in T colored with a complete set of colors [18].We found several generalizations of Sperner’s lemma [8–15].Let K be a simplicial complex. Denote by Vert( K ) the vertex set of K . Let an ( m + 1)–coloring (labeling) L be a map L : Vert( K ) → { , , . . . , m } . Setting f L ( u ) := v k , where u ∈ Vert( K ) , k = L ( u ) , and { v , ..., v m } = Vert(∆ m ) , we have a simpicial map f L : K → ∆ m . We say that an n –simplex s in K is fully labeled if s is labeled with a complete set of labels { , . . . , m } .Suppose there are no fully labeled simplices in K . Then f L ( p ) lies in the boundary of∆ m . Since the boundary ∂ ∆ m is homeomorphic to the sphere S m − , we have a continuousmap f L : K → S m − . Denote the homotopy class of f L in [ K, S m − ] by [ f L ].Let T be a triangulation of a manifold M with boundary ∂M . Let L : Vert( T ) →{ , . . . , n + 1 } be a labeling of T . Define ∂L : Vert( ∂T ) → { , , . . . , n + 1 } , ∂ f L : ∂T → Vert(∆ n +1 ) . a r X i v : . [ m a t h . A T ] J u l
312 31 3 2 1232 3
Figure 1: An illustration of Theorem A with d = 3Observe that if the dimension of M n +1 is n + 1, then dim( ∂M ) = n and the map ∂ f L : ∂T → ∂ ∆ n +1 is well defined. By the Hopf theorem [7, Ch. 7] we have [ ∂M, S n ] = Z and[ ∂ f L ] = deg( ∂ f L | ) ∈ Z . Theorem A. [12, Theorem 3.4]
Let T be a triangulation of an oriented manifold M n +1 withnonempty boundary ∂M . Let L : Vert( T ) → { , . . . , n + 2 } be a labeling of T . Then T mustcontain at least d = | deg( ∂ f L ) | fully labelled simplices. In Fig.1 is shown an illustration of Theorem A. Here n = 1, M = D and d = [ ∂ f L ] = 3.The theorem yields that there are at least three fully labeled triangles.Observe that for a Sperner labelling we have d = 1. Actually, Theorem A can be consid-ered as a quantitative extension of the Sperner lemma.In [12] with ( n +2)–covers of a space X we associate certain homotopy classes of maps from X to n –spheres. These homotopy invariants can be considered as obstructions for extendingcovers of a subspace A ⊂ X to a cover of all of X . We are using these obstructions toobtain generalizations of the classic KKM (Knaster–Kuratowski–Mazurkiewicz) and Spernerlemmas. In particular, we proved the following theorem: Theorem B. ([12, Corollary 3.1] & [13, Theorem 2.1])
Let T be a triangulation of a disc D n + k +1 . Let L : Vert( T ) → { , . . . , n + 1 } be a labeling of T such that T has no fully labelled n –simplices on the boundary ∂D ∼ = S n + k . Suppose [ ∂f L ] (cid:54) = 0 in π n + k ( S n ) . Then T mustcontain at least one fully labeled n –simplex. We observe that for k = 0 and M = D Theorem A yields Theorem B. However, inthis case Theorem A is stronger than Theorem B. In this paper we are going to prove aquantitative extension of Theorem B. First we consider the case n = 2 and k = 1. In Section2, the following theorem is proved. Theorem 1.1.
Let T be a triangulation of D with a labeling L : Vert( T ) → { A, B, C, D } such that T has no fully labelled 3–simplices on its boundary ∂T ∼ = S . Let ∂ f L on ∂T be of opf invariant d (cid:54) = 0 . Then T must contain at least 9 fully labeled 3–simplices and for d ≥ this number is at least d + 3 . In Section 3 we consider framed cobordisms Ω frk ( X ) and relative framed cobordismsΩ frk ( X, ∂X ). In particular, we prove the following extension of Pontryagin’s theorem [16].
Theorem 1.2.
For all k ≥ and n ≥ we have Ω frk ( D n + k +1 , S n + k ) ∼ = π n + k +1 ( D n +1 , S n ) ∼ = π n + k ( S n ) ∼ = Ω frk ( S n + k )In Section 4 we prove a simplicial extension of Theorem 1.2 that can be considered as asmooth version of a quantitative Sperner–type lemma. Definition 1.1.
Let T m and T n with m ≥ n be triangulations of spheres S m and S n . Let f : T m → T n be a simplicial map. Let s be an n –simplex of T n and s (cid:48) be a smaller n –simplexthat lies in the the interior of s . Let Π( f, s ) := f − L ( s (cid:48) ). Then Π( f, s ) is an m –dimensionalsubmanifold in S m and by using orientations of S m and S n a natural orientation can beassigned to it. It is clear that under the simplicial homeomorphism Π( f, s ) does not dependon the choice of s (cid:48) .We observe that ∂ Π( f, s ) = f − L ( ∂s (cid:48) ) and t is an interior n –simplex of Π( f, s ) if and onlyif f ( t ) = s (cid:48) . Denote by µ ( f, s ) the number of internal n –simplices in Π( f, s ).Let a ∈ π m ( S n ). Denote by F a the space of all simplicial maps f : S m → S n with [ f ] = a in π m ( S n ). Define µ ( a ) := min f ∈F a ,s µ ( f, s ) . We obviously have µ (0) = 0 and µ ( − a ) = µ ( a ). Theorem 1.3.
Let T be a triangulation of D n + k +1 and L : Vert( T ) → { , . . . , n + 1 } be alabeling of T such that T has no fully labelled n –simplices on its boundary. Suppose [ ∂ f L ] (cid:54) = 0 in π n + k ( S n ) . Then T must contain at least µ ([ ∂ f L ]) fully labeled ( n + 1) –simplices. The
Hopf invariant of a smooth or simplicial map f : S → S is the linking number H ( f ) := link( f − ( x ) , f − ( y )) ∈ Z , (2 . x (cid:54) = y ∈ S are generic points [3]. Actually, f − ( x ) and f − ( y ) are the disjoint inverseimage circles or unions of circles.The projection of the Hopf fibration S (cid:44) → S → S is a map h : S → S with Hopfinvariant 1. The Hopf invariant classifies the homotopy classes of maps from S to S , i.e. H : π ( S ) → Z is an isomorphism.We assume that S and S are triangulated and f : S → S is a simplicial map. Let s bea 2–simplex of S with vertices A, B and C . We have that Π = Π( f, s ) is a simplicial complex3n S (see Definition 1.1). Actually, Π is the disjoint union of k ≥ k closed chains of 3–simplices with a labeling L : Vert(Π) → { A, B, C } . (Herewithout loss of generality we may assume that s (cid:48) has the same labels as s .)We observe that the Hopf invariant of Π is well defined by (2.1) and H (Π) = H ( f ). Usingthis fact in [14] is considered a linear algorithm for computing the Hopf invariant.Since the equality π ( S ) = Z allows us to identify integers with elements of the group π ( S ), in this section we write µ ( d ) bearing in mind that d is an element of π ( S ) . Lemma 2.1. µ (1) = µ (2) = 9 and µ ( d ) ≥ d + 3 for all d ≥ .Proof. Madahar and Sarkaria [6] give the minimal simplicial map h : ˜ S → S of Hopfinvariant one (Hopf map) that has µ ( h , s ) = 9, see [6, Fig. 2]. Madahar [5] gives the minimalsimplicial map h : S → S of the Hopf invariant two with µ ( h , ABC ) = 9 [5, Fig. 3]. Itis clear that µ ( d ) ≥ d (cid:54) = 0, then we have µ (1) = µ (2) = 9.Suppose H (Π) = d . Let P be a connected component of Π. Then P is a triangulatedsolid torus in S that is a closed oriented labeled tetrahedral chain. All vertices of P lie onthe boundary ∂P and have labels A , B , and C . Moreover, all internal 2–simplices (triangles)are fully labeled, i.e. have all three labels A, B, C .Take any internal triangle T of P . This triangle is oriented and we assign the order of itsvertices v v v in the positive direction. In accordance with the orientation of the chain thenext vertex v is uniquely determined as well as v and so on. Then we have a closed chainof vertices v , v , ..., v m which uniquely determines the triangulations of ∂P and P . Now wehave a closed chain of internal triangles T = v v v , T = v v v , , ..., T k = v m − − j v m − − j v m − j , k = (cid:98) m/ (cid:99) , j = m − k, that have no common vertices and are fully labeled.Let M := L ( v ) L ( v ) ...L ( v m ). Then M is a sequence (“word”) which contains only threeletters A, B, C . Let deg( M ) := p ∗ − n ∗ , where p ∗ (respectively, n ∗ ) is the number of consecutive pairs AB (respectively, BA ) in M (cid:48) = M L ( v ). (For instance, deg( ABCABCABC ) = 3 and deg(
ABCBACABC ) = 1.)Note that if instead of AB we take BC or CA , then we obtain the same number, see [8,Lemma 2.1].It is easy to see that the minimum length of M of degree n is 3 n . We may assume that L ( T ) = ABC . Then M has the maximum degree k if m = 3 k and M = ABCABC...ABC .In this case every triangle T i has labels ABC .Suppose Π has only one connected component. Then H ( P ) = d , i.e. link( γ A , γ B ) = d ,where γ A = f − ( A ) and γ B = f − ( B ) are curves on the torus ∂P . Observe that the linkingnumber can be computed as the rotation number of curves γ A or γ B on ∂P [19].The labeling L divides Vert( P ) into three groups which of them contains at least k vertices. Since m − k ≤
2, we have that at least one group, say A , contains exactly k vertices. Hence we have a chain of vertices A , ..., A k , where f ( A i ) = A and A i is a vertex4f T i . Note that the rotation angle from A i to A i +1 is less than 2 π . Therefore, the sum ofrotation angles of this chain is less than 2 πk and the rotation number is at most k − k ≥ d + 1 and m ≥ d + 3. (Moreover, we can have the equality only if M = ABCABC...ABC .) Obviously, this inequality gives the minimum for Π that containsonly one connected component. Thus, µ ( d ) ≥ d + 3. Remark 2.1.
It is not clear, is the lower bound µ ( d ) ≥ d + 3 sharp for d >
2? Madahar[5] gives a simplicial map h d : S d → S of Hopf invariant d ≥ µ ( h d , ABC ) = 6 d − µ ( h d , ABD ) = µ ( h d , ACD ) = µ ( h d , BCD ) = (2 d − d − d .) However, µ ( h d , ABC ) > µ ( d ) for d >
3. Indeed, if we take foreven d the connected sum of d/ S with labeling h and ( d − / S andone ˜ S with labeling h for odd d , then we obtain the triangulation and labeling of S with µ = 9 (cid:100) d/ (cid:101) . Hence we have µ ( d ) ≤ (cid:100) d/ (cid:101) . Remark 2.2.
Observe that Madahar’s triangulation S in [5] is not geometric. Indeed, inthis case H (Π( h , ABC )) = link( h − ( A ) , h − ( B )) = 2. However, for geometric triangulations h − ( A ) and h − ( B ) are triangles, therefore their linking number cannot be 2. Lemma 2.2.
Let T be a triangulation of D . Let L : Vert( T ) → { A, B, C, D } be a labelingsuch that T has no fully labelled 3–simplices on the boundary ∂T ∼ = S . If the Hopf invariantof ∂f L on ∂T is d , then T must contain at least µ ( d ) fully labeled 3–simplices.Proof. This lemma is a particular case of Theorem 1.3. We have d = [ ∂f L ] ∈ π ( S ) = Z .Then there are at least µ ( d ) fully labeled 3–simplices.It is easy to see that Lemmas 2.1 and 2.2 yield Theorem 1.1. A framing of an k –dimensional smooth submanifold M k (cid:44) → X n + k is a smooth map whichfor any x ∈ M assigns a a basis of the normal vectors to M in X at x : v ( x ) = { v ( x ) , ..., v n ( x ) } , where vectors { v i ( x ) } form a basis of T ⊥ x ( M ) ⊂ T x ( X ).A framed cobordism between framed k –manifolds M k and N k in X n + k is a ( k + 1)–dimensional submanifold C k +1 of X × [0 ,
1] such that ∂C = C ∩ ( X × [0 , M × { } ) ∪ ( N × { } ) (3 . C that restricts to the given framings on M × { } and N × { } .This defines an equivalence relation on the set of framed k –manifolds in X . Let Ω frk ( X )denote the set of equivalence classes. 5he main result concerning Ω frk ( X ) is the theorem of Pontryagin [16]: Ω frk ( X n + k ) with n ≥ and k ≥ corresponds bijectively to the set [ X, S n ] of homotopy classes of maps X → S n . In particular, Ω frk ( S n + k ) ∼ = π n + k ( S n ) . Let f : X n + k → S n be a smooth map and y ∈ S n be a regular image of f . Let v = { v , ..., v n } be a positively oriented basis for the tangent space T y S n . Note that for every x ∈ f − ( y ), f induces the isomorphism between T y S n and T ⊥ x f − ( y ). Then v induces aframing of the submanifold M = f − ( y ) in X . This submanifold together with a framing iscalled the Pontryagin manifold associated to f at y . We denote it by Π( f, y ).Actually, the Pontryagin theorem states that1. Under the framed cobordism Π( f, y ) does not depend on the choice of y ∈ S n .2. Under the framed cobordism Π( f, y ) depends only on homotopy classes of [ f ].3. Π : [ X, S n ] → Ω frk ( X ) is a bijection.Let A (cid:96) + k be a submanifold of X m + k . It is not hard to define relative framed cobordisms and the set of equivalence classes Ω frk ( X, A ).Let us describe the case A = ∂X , dim X = n + k + 1, in more details. Let M k be asubmanifolds of X \ ∂X with a framing { v ( x ) , v ( x ) , ..., v n ( x ) } . Let N k be a submanifolds of ∂X with a framing { u ( x ) , ..., u n ( x ) } . We say that ( M, N ) is a framed relative pair if there aresubmanifold W in X and n –framing ω = { w ( x ) , ..., w n ( x ) } of W such that ∂W = M (cid:116) N , ω | M = { v , ..., v n } and ω | N = { u , ..., u n } . Then the framed cobordisms of framed relativepairs define the set of equivalence classes Ω frk ( X, ∂X ). Theorem 3.1.
Let X n + k +1 with n ≥ and k ≥ be a compact orientable smooth manifoldwith boundary ∂X . Then Ω frk ( X, ∂X ) corresponds bijectively to the set [( X, ∂X ) , ( D n +1 , S n )] of relative homotopy classes of maps ( X, ∂X ) to ( D n +1 , ∂D n +1 ) .Proof. The proof of Pontryagin’s theorem is cogently described in many textbooks, for in-stance, Milnor’s book [7], Hirsch’s and Ranicki’s books [2, 17]. Actually, this theorem can beproved by very similar arguments as the Pontryagin theorem.Let f : ( X, ∂X ) → ( D n +1 , S n ) be a smooth map, y ∈ S n be a regular value of ∂f , z ∈ D n +1 \ S n be a regular value of f , v = { v , ..., v n } be a positively oriented basis forthe tangent space T y S n and v be a vector in R n such that { v , v , ..., v n } is its basis. Let γ be a smooth non-singular path in D n +1 framed with v , connecting z and y such that thetangent vector to γ at z is v . Then Π( f, y, z, γ ) can be defined as a framed relative pair( f − ( z ) , f − ( y )) with W = f − ( γ ).To prove the theorem we can use the same steps 1, 2, 3 as above. It can be shown thatΠ : [( X, ∂X ) , ( D n +1 , S n )] → Ω frk ( X, ∂X ) is well–defined and is a bijection. In the next sectionwe consider details of this construction for simplicial maps.6 roof of Theorem 1.2.
Pontryagin’s theorem and Theorem 3.1 yield bijective correspon-dences Ω frk ( S n + k ) ∼ = π n + k ( S n ) and Ω frk ( D n + k +1 , S n + k ) ∼ = π n + k +1 ( D n +1 , S n ). The well–knownisomorphism π n + k +1 ( D n +1 , S n ) ∼ = π n + k ( S n ) follows from the long exact sequence of relativehomotopy groups: ... → π n + k +1 ( D n +1 ) → π n + k +1 ( D n +1 , S n ) → π n + k ( S n ) → π n + k ( D n +1 ) = 0 → ... This completes the proof. (cid:3)
Theorem 1.2 can be considered as a smooth version of a quantitative Sperner–type lemma.In this section we consider the bijective correspondence Ω frk ( D n + k +1 , S n + k ) ∼ = Ω frk ( S n + k ) forlabelings (simplicial maps).Let T be a triangulation of a smooth manifold X n + k . An S – framing of a k –dimensionalsubmanifold M k (cid:44) → X is a simplicial embedding h : P → T , where P ∼ = M × D n withVert( P ) ⊂ ∂P , and a labelling L : Vert( P ) → { , ..., n + 1 } such that (i) an n –simplex of P is internal iff it is fully labeled, (ii) M lies in the interior of h ( P ) and (iii) h − ( M ) ∼ = M .An S – framed cobordism between two S –framed manifolds M k and N k can be defined bythe same way as the framed cobordism in (3.1). If between M and N there is an S –framedcobordism then we write [ M ] = [ N ]. Let Ω Sfrk ( X ) denote the set of equivalence classes under S –framed cobordisms.Let f : T → Y be a simplicial map, where Y is a triangulation of S n . For any simplex s in Y can be defined a simplicial complex Π = Π( f, s ) in X , see Definition 1.1. Let s (cid:48) ⊂ s bean n –simplex with vertices v , ..., v n +1 . If Π is not empty, then it is an ( n + k )–submanifoldof X , all vertices of Π lie on its boundary and f : Vert(Π) → { v , ..., v n +1 } . Moreover, if y ∈ int( s (cid:48) ) then M = f − ( y ) is a k –dimensional submanifold of Π ⊂ X . ( Here int( S ) denotethe interior of a set S . ) Thus Π is an S –framing of M .There is a natural framing of M . Let u = { u , ..., u n } , where u i is a vector yv i . Then u induces a framing of M in X . Hence we have a correspondence between Π( f, s ) and Π( f, y ).It is not hard to see that this correspondence yield a bijection. Lemma 4.1. Ω Sfrk ( X ) ∼ = Ω frk ( X ) . We observe that relative S –framining, relative S –framed cobordisms and a correspon-dence between relative S –framed and relative framed manifolds can be defined by a similarway. It can be shown that Ω Sfrk ( X, ∂X ) ∼ = Ω frk ( X, ∂X ) . Let us take a closer look at the bijectionΩ
Sfrk ( D n + k +1 , S n + k ) ∼ = Ω Sfrk ( S n + k ) ∼ = π n + k ( S n ) . T be a triangulation of D n + k +1 and L : Vert( T ) → { , . . . , n + 1 } be a labeling of T such that T has no fully labelled n –simplices on the boundary ∂T ∼ = S n + k . Then we havesimplicial maps: f L : T ∼ = D n + k +1 → ∆ n +1 ∼ = D n +1 , ∂ f L : ∂T ∼ = S n + k → ∂ ∆ n +1 ∼ = S n , where ∆ = ∆ n +1 denote the ( n + 1)–simplex with vertices { v , v , ..., v n +1 } . Hence the ho-motopy class [ ∂ f L ] ∈ π n + k ( S n ).Let s denote the n –simplex of ∆ with vertices { v , ..., v n +1 } . Define M := f − L ( z ) , z ∈ int(∆ (cid:48) ) , N := ∂ f L − ( y ) , y ∈ int( s (cid:48) ) , W := f − L ([ z, y ]) . Lemma 4.2.
We have that ( M , N ) is an S –framed relative pair in ( D n + k +1 , S n + k ) and F ([( M , N )]) = [ N ] defines a bijection F : Ω Sfrk ( D n + k +1 , S n + k ) → Ω Sfrk ( S n + k ) . Proof.
Since z and y are regular values of f L and ∂ f L , we have that M and N are manifoldsof k dimensions with a cobordism W . In fact, Π( f L , ∆) and Π( ∂ f L , s ) define S –framings of M and N . Lemma 4.3.
Let C be a connected component of W such that N C := ∂C ∩ N (cid:54) = ∅ . Then Π( f L , s ) induces an S –framing of M C := ∂C ∩ M in S n + k and [ M C ] = [ N C ] in Ω Sfrk ( S n ) .Proof. Note that ∂C = M C ∪ N C . Actually, C is a cobordism between M C and N C in D n + k +1 .We obviously have that if M C is empty then N C is null–cobordant, i.e. [ N C ] = 0 in Ω Sfrk ( S n ).Let Γ be the closure of f − L (int(∆)) and K C := C ∩ Γ ⊂ Π( f L , s ). Note that Π( f L , s )induces an S –framing of K C with ( n + 1)–labels. Let t := [ z, y ) in ∆ and C t := f − L ( t ).Since f L is linear on C t we have C t ∼ = M × [0 , S –framing of M C with( n + 1)–labels.The last of the proof to show that this S –framing of M C is in S n + k . We have that S –framing of N C is in S n + k . It can be proved that using shelling along C of fully labeled n -ssimplices we can contract M C to N C such that at each step the boundary lies in S n + k .That completes the proof. Proof of Theorem 1.3.
Lemma 4.1 and Pontryagin theorem yieldΩ
Sfrk ( S n ) ∼ = Ω Srk ( S n ) ∼ = π n + k ( S n ) . Let [ ∂ f L ] = a in π n + k ( S n ) . Then [ N ] = a in Ω Sfrk ( S n ). If { C , ..., C k } are connectedcomponents of W then Lemma 4.3 yields the equality[ M C ] + ... + [ M C k ] = [ N C ] + ... + [ N C k ] = [ N ] = a. Therefore, Π( f L , ∆) contains at least µ ( a ) n –simplices with labels 1 , ..., n + 1. The same wehave for every ( n + 1)-labeling. Since Π( f L , ∆) contains all fully labeled ( n + 1)–simplices, itis not hard to see that this number is not less than µ ( a ). (cid:3) eferences [1] J. A. De Loera, E. Peterson, and F. E. Su, A Polytopal Generalization of Sperner’sLemma, J. of Combin. Theory Ser. A, (2002), 1-26.[2] M. W. Hirsch, Differential topology,
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