Geometric descriptions of polygon and chain spaces. HAUSMANN, Jean-Claude. Abstract

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1 Proceeings Chapter Geometric escriptions of polygon an chain spaces HAUSMANN, Jean-Claue Abstract We give a few simple methos to geometically escribe some polygon an chain-spaces in R^. They are strong enough to give tables of m-gons an m-chains when m Reference HAUSMANN, Jean-Claue. Geometric escriptions of polygon an chain spaces. In: Farber, Michael, Ghrist, R., Burger, M., Koitschek, D. Topology an robotics. Provience, R.I. : American Mathematical Society, p arxiv : math/ Available at: Disclaimer: layout of this ocument may iffer from the publishe version.

2 arxiv:math/ v1 [math.gt] 18 Feb 2007 Geometric escriptions of polygon an chain spaces Jean-Claue HAUSMANN Abstract. We give a few simple methos to geometically escribe some polygon an chain spaces in R. They are strong enough to give tables of m-gons an m-chains when m 6. Introuction For a = (a 1,...,a m ) R m >0 an an integer, efine the subspace Cm (a) of m 1 i=1 S 1 by C m (a) = { z = (z 1,..., z m 1 ) m 1 i=1 S 1 m 1 i=1 a i z i = a m e 1 }, where e 1 = (1, 0,...,0) is the first vector of the stanar basis e 1,...,e of R. An element of C m (a), calle a chain, can be visualize as a configuration of (m 1)- segments in R, of length a 1,...,a m 1, joining the origin to a m e 1. The group O( 1), seen as the subgroup of O() stabilizing the first axis, acts naturally (on the left) on C m (a). The quotient space by SO( 1) coincies with the polygon space N m (a) = SO( 1) C m(a) {ρ SO() = (ρ1,...ρ m ) (R ) m ρ i = a i an m i=1 ρ i = 0 }. The notations are that of [HR04] where it is emphasize how the union N m of N m(a) for all a Rm >0 is relate to the spaces stuie in statistical shape analysis (see, e.g. [KBCL99]). An element a R m >0 is generic if C1 m (a) =, that is to say there is no line chain or polygon configuration. When a is generic, C m (a) is a smooth close manifol of imension (m 2)( 1) 1 (see, e.g. [Ha89]). Mathematical robotics is specially intereste in the chain an polygon spaces for = 2, 3. When a is generic, the action of SO( 1) on C m is then free an therefore N2 m (a) an N3 m (a) are close smooth manifols of imension m 3 an 2(m 3) respectively (in aition, N3 m (a) carries a symplectic structure, see e.g. [KM96]). One has C2 m(a) = N 2 m(a) an Cm 3 (a) N 3 m (a) is a principal circle bunle. In this paper, we present a few geometrical methos permitting us to escribe in some cases the spaces C m(a) an N m (a). From the classification results (see 1991 Mathematics Subject Classification. Primary 55R80, 70G40; Seconary 57R65. 1

3 2 JEAN-CLAUDE HAUSMANN Section 1), this enables us to escribe all the chain or polygon spaces in R when m 6 (tables in Section 3). 1. Review of the classification results The iea of the classification of the polygon an chain spaces goes back to [Wa85]. Details may be foun in [HR04] Short subsets. Let a = (a 1,...,a m ) R m >0. A subset J of {1,...,m} is calle short if i J a i < i/ J a i. Short subsets form, with inclusion, a poset S(a). Define S m (a) = {J S(a) m J}. Lemma 1.2. Let a an a be generic elements in R m >0. Suppose that S m(a) an S m (a ) are poset isomorphic. Then: (i) C m (a) an Cm (a ) are O( 1)-equivariantly iffeomorphic. (ii) N m (a) an N m (a ) are iffeomorphic. Proof: If S m (a) S m (a ), then there is a poset isomorphism ϕ: S(a) S(a ) with ϕ(m) = m (see [HK98, Proposition 2.5]). It is well known that S(a) S(a ) implies (ii) (see, e.g. [HK98, Proposition 2.2] or [HR04, Theorem 1.1]). We give however the variation of the proof to get the less classical (stronger) fact that S(a) S(a ) implies (i). Let K (a) = {z = (z 1,..., z m ) m i=1 S 1 m i=1 a iz i = 0}. The group O() acts on the left on K (a) an N m(a) = SO()\K (a). The function F : K (a) S 1 given by F(z) = z m is a submersion (since F is O()-equivariant). One has C m(a) = F 1 ( e 1 ) with its resiual O( 1)-action. Let σ be the bijection of {1,..., m 1} giving the poset isomorphism S m (a) S m (a ) an then S(a) S(a ). Then (z 1,..., z m 1, z m ) (z σ(1),..., z σ(m 1), z m ) inuces a O( 1)-equivariant iffeomorphism from C m(a 1,..., a m 1, a m ) onto C m(a σ(1),..., a σ(m 1), a m ). We can therefore suppose that S(a) = S(a ) an σ = i. We claim that C m(a) an Cm (a ) are then canonically iffeomorphic. Inee, if S(a) = S(a ), the segment [a, a ] contains only generic elements. Hence, the union X = ( K (b) {b} ) ( m S 1) [a, a ] b [a,a ] is an O()-coborism between K (a) an K(a ) an the projection π : X [a, a ] has no critical point. One still has the map F : X S 1 given by F(z, t) = z m an Y = F 1 ( e 1 ) is an O( 1)-coborism between C m(a) an Cm (a ), with again the projection π over [a, a ] being a submersion. The stanar metric on m 1 i=1 S 1 inuces an O( 1) invariant Riemannian metric on Y. Following the graient lines of π for this metric gives the require O( 1)-equivariant iffeomorphism Ψ : C m(a) C m(a ) Walls an chambers. For J {1,...,m}), let H J be the hyperplane (wall) of R m efine by H J := {(a 1,...,a m ) R m a i = a i }. i J i/ J The union H(R m ) of all these walls etermines a set Ch((R >0 ) m ) of open chambers in (R >0 ) m whose union is the set of generic elements. Two generic elements a an i=1

4 GEOMETRIC DESCRIPTIONS OF POLYGON AND CHAIN SPACES 3 a are in the same chamber if an only if S(a) = S(a ). We call Ch(a) the chamber of a generic element a. If α is a chamber, the poset S(a) is the same for all a α an is enote by S(α) Permutations. Let σ be a permutation of {1,...,m}. The map which sens (z 1,...,z m ) to (z σ(1),..., z σ(m) ) inuces a iffeomorphism from N m(a 1,...,a m ) onto N m(a σ(1),..., a σ(m) ). For the sake of the classification of N m (a), we may as well assume that a R m where ր R m ր := {(a 1,...,a m ) R m 0 < a 1 a m }. Observe that we then o not classify all the chain spaces C m (a) but only those for which a m a i for i < m. Inee, the permutation σ inuces a iffeomorphism from C m(a 1,..., a m ) onto C m(a σ(1),..., a σ(m) ) if an only if σ(m) = m. We enote by Ch(R m ր) the set of chambers etermine in R m ր by the hyperplane arrangement H(R m ) The genetic coe of a chamber. A chamber α Ch(R m ր) is etermine by S(α) which, in turn, is etermine by S m (α). Consier the partial orer on the subsets of {1,...,m} where A B if an only if there exits a non-ecreasing map ϕ : A B such that ϕ(x) x. For instance X Y if X Y since one can take ϕ being the inclusion. The genetic coe of α is the set of elements A 1,...,A k of S m (α) which are maximal with respect to the orer. Thus, the chamber α is etermine by its genetic coe; we write α = A 1,..., A k an call the sets A i the genes of α. As, in this paper m 9, we abbreviate a subset A by the sequence of its igits, e.g. {6, 2, 1} = 621. In [HR04], an algorithm is presente to list by their genetic coes all the elements of Ch(R m ) an then all the chambers ր up to permutation of the components. Tables for m 6 are given in [HR04] (an in Section 3 below); more tables, for m 9, may be foun in [HRWeb]. The algorithm prouces, in each chamber α, a representative a min (α) α; though this is not prove theoretically, a min (α) turne out in all known cases to have integral components a i an minimal a i. See examples in the tables below. 2. Proceures of escription 2.1. Aing a tiny ege. Let a = (a 2,..., a m ) be a generic element of R m 1 ր. If ε > 0 is small enough, the m-tuple a + := (δ, a 2,..., a m ) is a generic element of R m ր for 0 < δ ε. This efines a map Ch(R m 1) + Ch(R m), sening α to ր ր α+, which is injective (see [HR04, Lemma 5.1]). The genetic coe of α + has the same number of genes than that of α an the corresponence goes as follows. If {p 1,...,p r } is a gene of α, then {p + 1,..., p+ r, 1} is a gene of α+, where p + i = p i + 1. For example: 631, 65 + = 7421, 761. The minimal integral representative a min (α + ) of α + is a conventional representative: it starts with a 0 followe by the components of a min (α). Example: as a min ( 3 ) = (1, 1, 1), then a min ( 3 + ) = a min ( 41 ) = (0, 1, 1, 1), a min ( 41 + ) = a min ( 521 ) = (0, 0, 1, 1, 1), etc. It has to be unerstoo that these vanishing components stan for small enough positive real numbers, whose sum is less than 1. Proposition 2.1. There is a O( 1)-equivariant iffeomorphism Φ: C m (α + ) S 1 C m 1 (α), where S 1 C 1 m (α) is equippe with the iagonal O( 1)-action.

5 4 JEAN-CLAUDE HAUSMANN Proof: Let a = (a 2,..., a m ) α an a + = (ε, a 2,..., a m ) α +. The map Φ is of the form (Φ 1, Φ 2 ), where Φ 1 : C m(a+ ) S 1 an Φ 2 : C m(a+ ) C m 1 (a) are O( 1)-equivariant maps. The map Φ 1 is just given by Φ 1 (z 1,..., z m ) = z 1. It remains to efine Φ 2. If p R satisfies p p e 1, there is a unique R p SO() such that R p (p) = p e 1 an R p (q) = q if q EV(p, e 1 ), the orthogonal complement to the vector space EV(p, e 1 ) generate by p an e 1. In particular, R e1 = i. The map p R p is smooth. We shall apply that to p = p(z), where m p(z) = a i z i = a m e 1 εz 1. i=2 We may suppose that ε < a m, so p(z) p(z) e 1. The corresponence (z 1,...,z m ) (R p(z) z 2,..., R p(z) z m ) gives a smooth map Φ 2: C m (a + ) C m 1 (a 2,..., a m 1, p(z) ). The fact that (δ, a 2,...,a m ) is generic when 0 < δ ε implies that Ch(a 2,...,a m 1, p(z) ) = Ch(a). We can then use the canonical O( 1)-equivariant iffeomorphism Ψ: C m 1 (a 2,...,a m 1, p(z) ) C m 1 (a 2,..., a m ) constructe in the proof of Lemma 1.2 an efine Φ 2 = Ψ Φ 2. If A O( 1), the formula AR p = R A(p) A hols in O( 1), as easily seen on EV(p, e 1 ) an on EV(p, e 1 ). This implies that φ 2 is O( 1)-equivariant. We have thus constructe an O( 1)-equivariant smooth map Φ: C m(α+ ) S 1 C m 1 (α). The reaer will easily figure out what the inverse Φ 1 of Φ is like, proving that Φ is a iffeomorphism. We now turn our interest to N3 m (α + ). Let D(α) be the total space of the D 2 - isk bunle associate to C m 1 3 (α) N m 1 3 (α). We call ouble of D(α) the union of two copies of D(α), with opposite orientations, along their common bounary C m 3 (α). Proposition 2.2. (a) N3 m(α+ ) is iffeomorphic to S 2 S 1 C3 m 1 (α). (b) N3 m (α + ) is iffeomorphic to the ouble of D(α). In Part (a), S 2 S 1 C m 1 3 (α) enotes the quotient of S 2 C m 1 3 (α) by the 3 (α) is 3 (α). iagonal action of S 1 = SO(2). The projection S 2 S 1 C m 1 3 (α) N m 1 then the S 2 -associate bunle to the SO(2)-principal bunle C m 1 3 (α) N m 1 A irect proof of Part (b) may be foun in [HR04, Prop.6.4]. Proof: For Part (a), we check that the iffeomorphism Φ: C m 3 (α+ ) S 2 C m 1 3 (α) of Proposition 2.1 escens to a iffeomorphism from N3 m (α + ) to S 2 S 1 C3 m 1 (α). For Part(b), we observe that D(α) is the mapping cyliner of the projection C3 m 1 (α) N3 m 1 (α). The ouble of D(α) is then iffeomorphic to M = [ 1, 1] C3 m 1 (α) /, where is the equivalence relation generate by ( 1, z) ( 1, Az) an (1, z) (1, Az) for all z C3 m 1 (α) an all A S 1. Each S 1 -orbit of S 2 has an unique point of the form (u 1, 0, u 3 ). To (u, z) S 2 C m 1 (α) with u = (u 1, 0, u 3 ), we

6 GEOMETRIC DESCRIPTIONS OF POLYGON AND CHAIN SPACES 5 associate the class [u 1, z] in M an check that this corresponence gives rise to a iffeomorphism from S 2 S 1 C m 1 3 (α) to the ouble of D(α). Example 2.3. When m = 3, there is only one chamber α = 3, with a min (α) = (1, 1, 1), for which C 3 (α) is not empty. Its image uner aing tiny eges gives a chamber {m, m 3, m 2,...,1} Ch(R m ր) with a min = (0,...,0, 1, 1, 1) (conventional representative, 2.1). As C 3 3 = S 2 with the stanar O( 1)- action, Propositions 2.1 an 2.2 give the following The chamber α = {m, m 3, m 2,...,1} a min (α) N m 2 (α) N m 3 (α) Cm (α) (0,...,0, 1, 1, 1) T m 3 T m 3 (S 2 ) m 3 (S 1 ) m 3 S 2 Remark 2.4. Let A = {m, m 3, m 2,...,1}. We claim that α = A as above is the only chamber in R m having J S ր m(α) with J = m 3. Inee, let β Ch(R m ր) having J S m (β) with J A an J = m 3. Then A = {m, m 2, m 4,...,1} woul satisfy A J. Then, Ā = {m 3, m 1} woul be long, which contraicts {m 3, m 1} {m 2, m} S m (β). Now, if A S m (β), then {m, m 2} is long, since {m, m 2} A. Therefore, A S m (β) implies β = A. For an application of this remark, see Propositions 2.7 an The manifol V (a). Let a R m >0. Define V (a) = {z = (z 1,..., z m 1 ) m 1 i=1 S 1 m 1 i=1 a i z i = te 1 with t a m }. Let f : V (a) R efine by f(z) = m 1 i=1 a iz i. The group O( 1) acts on V (a). The following proposition is proven in [Ha89, Th. 3.2]. Proposition 2.5. Suppose that a R m >0 is generic. Then (i) V (a) is a smooth O( 1)-submanifol of m 1 i=1 S 1, of imension (m 2)( 1), with bounary C m (a). (ii) f is a O( 1)-equivariant Morse function, with one critical point p J for each J S m (a), where p J = (z 1,...,z m 1 ) with z i equal to e 1 if i J an e 1 otherwise (aligne configuration). The inex of p J is ( 1)( J 1). This permits us to get some information on C m (a). Example 2.6. The chamber m. If S m = {m}, f : V (a) R has only one critical point, of inex 0. Hence, C m (a) S(m 2)( 1) 1 an the O( 1) action is conjugate to that obtaine by the embeing S (m 2)( 1) 1 (R ) m 2 with the stanar iagonal action [Ha89, Prop. 4.2]. The chamber of a has here genetic coe m, with minimal representative (1,..., 1, m 2). One then has: The chamber α = m a min (α) N m 2 (α) N m 3 (α) Cm (α) (1,...,1, m 2) S m 3 CP m 3 S (m 2)( 1) 1 Another consequence of Proposition 2.5 is the connectivity of C m (α). We saw in Example 2.3 that C m(β) = (S 1 ) m 3 S 2 if β = {m, m 3, m 2,...,1}.

7 6 JEAN-CLAUDE HAUSMANN Thus, π 2 (C m(β)) Z if 3 an π 0(C2 m (β)) has 2 elements. But this is an exceptional case: Proposition 2.7. Let α be a chamber of R m with α {m, m 3, m ր 2,...,1}. Then, C m(α) is ( 2)-connecte, i.e. π i(c m (α)) = 0 for i 2. Proof: Let a R m ր be a representative of α. By Proposition 2.5, one has that π i (V (a)) = 0 if i 2. If α {m, m 3, m 2,...,1}, then J m 3 for all J S m (a) by Remark 2.4. Then V (a) has a hanle ecomposition, starting from C m (a), with hanles of inex (m 2)( 1) (m 4)( 1) = 2( 1). Therefore, π i (C m(a)) π i(v (a)) for i 2 2 > 2. Remark 2.8. When = 2, Proposition 2.7 says that {m, m 3, m 2,..., 1} is the only chamber β of R m ր for which N2 m (β) is not connecte. This was prove by Kapovich an Millson [KM95] (see also [FS06, Ex.2 in 1]) Crossing walls an surgeries. Proposition 2.9. Let J {1,...,m}, efining the wall H J in R m. Let α an β be two chambers of R m, with S m (β) = S m (α) {J}. Then C m (β) is obtaine from C m (α) by an O( 1)-equivariant surgery of inex A = ( 1)( J 1) 1: C m (β) ( C m (α)\ (SA D B ) ) S A S B 1 ( D A+1 S B 1), with B = (m 1 J )( 1). The O( 1)-action on D A+1 an D B comes from their natural embeing into a prouct of copies of R 1 with the iagonal action. Proof. Let a α an b β. As S m (β) = S m (α) {J}, the segment [a, b] in R m crosses the wall H J an has no intersection with any other wall. There exists a vector orthogonal to H J with coorinates equal to ±1. Therefore, e 1 is transverse to H J an, by changing a an b if necessary, we assume that a = b + λe 1. By Proposition 2.5, the manifol V (b)\intv (a) is a O( 1)-equivariant coborism W from C m(a) to Cm (b). The map f : W R efine by f(ρ) = m 1 j=1 a jρ j is an invariant Morse function having a single critical point ρ 0 of inex ( 1)( J 1); the components (ρ 0 1,...,ρ0 m 1 ) of ρ0 satisfy ρ 0 i = e 1 if i J an ρ 0 i = e 1 if i / J. By relabeling the ρ i if necessary, we assume that J = {1, 2,..., k, m} an J = {k + 1,...,m 1}. The inex of ρ 0 is then equal to ( 1)k. Therefore, W is obtaine by aing to a collar neighborhoo of C m (a) an O( 1)-equivariant hanle of inex ( 1)k, whence the surgery assertion. For a reference about equivariant Morse theory, see [Wn69, 4]. By [Wn69, Lemma 4.5], the O( 1)- action is etermine by the linear isotropy action on T ρ 0W, which we shall now escribe. Let K ρ = k j=1 a jρ j an L ρ = K ρ a m e 1. Let p 1 : R R 1 an P : R R 1 be the maps p 1 (x 1,..., x ) = x 1 an P(x 1,..., x ) = (x 2,..., x ) For ε > 0, we consier the following open neighborhoo N ε of ρ 0 in W N ε = {ρ W p 1 (K ρ 0) p 1 (K ρ ) < ε an L ρ L ρ 0 < ε}. Consier the unique rotation R ρ SO() such that R ρ (L ρ ) = L ρ e 1 an R ρ (q) = q if q EV(e 1, K ρ ) (if ε is small enough, L ρ is not a positive multiple of e 1 when ρ N ε, thus R ρ is well efine). If ε is small enough, we check, as in [Ha89, Proof of Theorem 3.2] that the smooth maps φ : N ε (R 1 ) k an φ + : N ε (R 1 ) m k 2 given by φ (ρ) = (P(ρ 1 ),..., P(ρ k )) an φ + (ρ) = ( P(R ρ ( ρ k+1 )),..., P(R ρ ( ρ m 1 )) )

8 GEOMETRIC DESCRIPTIONS OF POLYGON AND CHAIN SPACES 7 are O( 1)-equivariant an give rise to a O( 1)-equivariant chart φ = (φ, φ + ) : N ε (R 1 ) k (R 1 ) m 2 k = (R 1 ) m 2, where (R 1 ) n is enowe with the iagonal action of O( 1). One has φ(ρ 0 ) = 0. The subspaces D + = {ρ N ε L ρ = L ρ 0 } an D = {ρ N ε K ρ = K ρ 0} are submanifols of imensions k( 1) an (m k 2)( 1) respectively, satisfying φ(d + ) (R 1 ) k 0 an φ(d + ) 0 (R 1 ) m 2 k. As in [Ha89, Proof of Theorem 3.2], we prove that f restricts to Morse functions on D ±. The single critical point ρ 0 is a minimum on D + an a maximum on D. Therefore, the Hessian form of f φ 1 is positive efinite on T 0 (R 1 ) k an negative efinite on T 0 (R 1 ) m 2 k. We have seen above that the O( 1)-action on these subspaces is the stanar iagonal action. By [Wn69, Lemma 4.5], this implies the last assertion of Proposition 2.9. Proposition Suppose, in Proposition 2.9, that J = 2. Then (1) C m (β) = Cm (α) (S 1 S (m 3)( 1) 1 ) (2) N m 3 (β) = N m 3 (α) CP m 3 Proof: Since S m (β) = S m (α) {J}, the chamber α is not {m, m 3, m 2,...,1} by Remark 2.4. By Proposition 2.7, C m (α) is ( 2)-connecte. Hence, the sphere S 2 C m (α) on which the surgery of Proposition 2.9 is performe is null-homotopic. We may assume that m 4 an 2 since Proposition 2.10 is empty for m = 3 an C1 m(α) = Cm 1 (β) = because of genericity. Therefore 2( 2) < im C m(α), from which we euce that S 2 C m (α) is isotopic to a sphere containe in a isk. Observe that we are ealing with stably parallelizable manifols (for instance, C m ( ) is the pre-image of a regular value of a map from a prouct of spheres to R ). Part 1 then follows from stanar results in surgery, see e.g. [Ko93, Proposition 11.2 an p. 188]. As for Part 2, we have C m 3 (β) ( C m 3 (α)\ (S 1 D 2(m 3) ) ) S 1 S 2(m 3) 1 ( D 2 S 2(m 3) 1). The quotient space S 1 D 2(m 3) by the action of SO(2) is a isk D 2(m 3). On the other han, consier the tautological line bunle E CP m 4, where E = {(v, l) C m 3 CP m 4 v l}. Seeing D 2 as the unit isk in C, the map g : D 2 S 2(m 3) 1 E given by g(z, w) = (zw, Cw) escens to an embeing from SO(2) (D 2 S 2(m 3) 1 ) to a neighborhoo of the zero section of E. It follows that C3 m(β) is iffeomorphic to Cm 3 (α) blown up at one point, which implies Assertion 2 (see e.g. [MDS95, pp ]). Example The chamber {m, m 3, m 2,...,2}. Let α = {m, m 3, m 2,..., 2} an β = {m, m 3, m 2,...,1}. Then S m (β) = S m (α) {m, m 3, m 2,...,1}. By Proposition 2.9, C m(β) is obtaine from Cm (α) by an O( 1)-equivariant surgery of inex ( 1)(m 3) 1. Then, conversely, C m(α) is obtaine from Cm (β) by an O( 1)-equivariant surgery of inex 2. By Example 2.3 an Proposition 2.7, C m(β) (S 1 ) m 3 S 2 while C m (α) is ( 2)-connecte. This implies that the surgery on C m (β) is performe on a tubular

9 8 JEAN-CLAUDE HAUSMANN neighborhoo of pt S 2. Thus, Part 1 of Proposition 2.10 is not true, but one has (1) C m (α) [( (S 1 ) m 3 \ B ) S 2] B S 2 ( B D 1 ), where B is a ((m 3)( 1))-isk in (S 1 ) m 3. This is not a very simple expression, except when = 2 where N2 m(α) = N 2 m (α) becomes (2) N m 2 (α) (S 1 ) m 3 (S 1 ) m 3. On the other han, Part 2 of Proposition 2.10 is vali an we get (3) N m 3 (α) (S2 ) m 3 CP m 3. It was observe by D. Schütz that there are only three chambers α Ch(R m ր ) such that S m (α) contains A = {m, m 3, m 2,...,2}. These are (4) α = A α = A, {m, m 2} α = A, {m, m 1}. Inee, if A is short, then {m 1, m 2, 1} is long an one cannot a to A a gene containing 3 elements. As S m (α ) = S m (α) {m, m 2} an S m (α ) = S m (α ) {m, m 1}, the chain an polygon spaces for α an α may be obtain from the above using Proposition Example The chamber {m, p}. For p 2, one has S m ( {m, p} ) = S m ( {m, p 1} ) {m, p} an S m ( {m, 1} ) = S m ( m ) {m, 1}. Using Proposition 2.10 an Example 2.6, one sees that The chamber α = {m, p} N m 2 (α) N m 3 (α) C m (α) p (S 1 S m 4 ) CP m 3 pcp m 3 p (S 1 S (m 3)( 1) 1 ) Here, p times a manifol V means the connecte sum of p copies of V (hence, a sphere if p = 0). A representative of {m, p} is given by (1,...,1, 2,...,2, 2m p 5). }{{}}{{} p m p 1 The tables of [HRWeb] show that, for m 9 (See Section 3 below for m 6), this representative is a min ( {m, p} ), except for p = 0, 1. As {m, 1} = m 1 +, Proposition 2.2 gives the iffeomorphism CP m 3 CP m 3 N m 3 ( {m, 1} ) N m 3 ( m + ) S 2 S 1 S 2(m 3) 1. In the case m = 5, we get the two topological escriptions of the Hirzebruch surface (see, e.g. [MDS95, Ex. 6.4]). 3. Tables for m = 4, 5, 6 For any m, there is the trivial chamber, where a m is so long that the corresponing chain or polygon spaces are empty. When m = 4, Examples 2.6 an 2.3 give the remaining two chambers:

10 GEOMETRIC DESCRIPTIONS OF POLYGON AND CHAIN SPACES 9 Table A: m = 4 α a min (α) N 4 2 (α) N 4 3 (α) C4 (α) 1 (0, 0, 0, 1) 2 4 (1, 1, 1, 2) S 1 CP 1 S 2( 1) (0, 1, 1, 1) S 1 S 1 S 2 (S 1 ) S 2 Recall that the column C 4 (α) oes not contain all the 4-chains, only those for which a 4 a i for I = 1, 2, 3. For example, C 4(1, 1, 1ε) T 1 S 1, the unit tangent bunle to S 1. For a complete classification of 4-chains, see [Ha89]. When m = 5, there are seven chambers. Lines 2 an 7 come from Examples 2.6 an 2.3. The symbols Σ g enotes the orientable surface of genus g an T r is the torus (S 1 ) r. Within the central block, each line is obtaine from the previous one by Proposition Table B: m = 5 α a min (α) N 5 2 (α) N 5 3 (α) C5 (α) 1 (0, 0, 0, 0, 1) 2 5 (1, 1, 1, 1, 3) S 2 CP 2 S 3( 1) (0, 1, 1, 1, 2) T 2 CP 2 CP 2 S 1 S 2( 1) (1, 1, 2, 2, 3) Σ 2 CP 2 2CP 2 2 [S 1 S 2( 1) 1 ] 5 53 (1, 1, 1, 2, 2) Σ 3 CP 2 3CP 2 3 [S 1 S 2( 1) 1 ] 6 54 (1, 1, 1, 1, 1) Σ 4 CP 2 4CP 2 4 [S 1 S 2( 1) 1 ] (0, 0, 1, 1, 1) T 2 T 2 S 2 S 2 (S 1 ) 2 S 2 Line 3 in Table B together with Equation (3), re-proves the classical fact that (S 2 S 2 ) CP 2 is iffeomorphic to CP 2 2CP 2. When m = 6, there are 21 chambers. In orer to save space, we i not give a min (α) (they can be foun in [HR04, Table 6]). The first line of each block is obtaine from Then, each line is obtaine from the previous one by Proposition 2.10.

11 10 JEAN-CLAUDE HAUSMANN Table C: m = 6 α N 6 2 (α) N6 3 (α) C6 (α) S 3 CP 3 S 4( 1) S 1 S 2 CP 3 CP 3 R() ( 1 ) (S 1 S 2 ) CP 3 2CP 3 2R() (S 1 S 2 ) CP 3 3CP 3 3R() (S 1 S 2 ) CP 3 4CP 3 4R() (S 1 S 2 ) CP 3 5CP 3 5R() T 3 S 2 S 1(S 2 S 3 ) S 1 C 5 ( 51 ) (2 ) 9 621, 63 T 3 (S 1 S 2 ) [S 2 S 1(S 2 S 3 )] CP 3 [S 1 C 5 ( 51 )] R() , 64 T 3 2(S 1 S 2 ) [S 2 S 1(S 2 S 3 )] 2CP 3 [S 1 C 5 ( 51 )] 2R() , 65 T 3 3(S 1 S 2 ) [S 2 S 1(S 2 S 3 )] 3CP 3 [S 1 C 5 ( 51 )] 3R() Σ 2 S 1 S 2 S 12(S 2 S 3 ) S 1 C 5 ( 52 ) (2 ) , 64 (Σ 2 S 1 ) (S 1 S 2 ) S 2 S 12(S 2 S 3 ) CP 3 [S 1 C 5 ( 52 )] R() , 65 (Σ 2 S 1 ) 2(S 1 S 2 ) S 2 S 12(S 2 S 3 ) 2 CP 3 [S 1 C 5 ( 52 )] 2R() Σ 3 S 1 S 2 S 13(S 2 S 3 ) S 1 C 5 ( 53 ) (2 ) , 65 (Σ 3 S 1 ) (S 1 S 2 ) S 2 S 13(S 2 S 3 ) CP 3 [S 1 C 5 ( 53 )] R() Σ 4 S 1 S 2 S 14(S 2 S 3 ) S 1 C 5 ( 54 ) (2 ) T 3 T 3 (S 2 ) 3 (S 1 ) 3 S T 3 (S 2 ) 3 CP 3 C 6 ( 632 ) (3 ) , 64 2T 3 (S 1 S 2 ) (S 2 ) 3 2CP 3 C 6 ( 632 ) R() , 65 2T 3 2(S 1 S 2 ) (S 2 ) 3 3CP 3 C 6 ( 632 ) 2R() ( 1 ) R() = S 1 S 3( 1) 1. ( 2 ) see Table B. ( 3 ) See Example 2.11 The list for N2 6 (α) is present in [Wa85] with some short-han justification. A version of the column for N3 6 (α) is in [HR04]. For m 7, the above proceure fails to give all the chambers, since surgeries of higher inex are neee. For example, for m = 7, only 49 chambers out of 135 are reache. References [FS06] M. Farber an D. Schütz, Homology of planar polygon spaces, Preprint (2006), math.at/ [Ha89] J.-C. Hausmann, Sur la topologie es bras articulés, in Algebraic topology Poznań 1989, Springer Lectures Notes 1474 (1989), [HK98] J.-C. Hausmann an A. Knutson, The cohomology rings of polygon spaces, Ann. Inst. Fourier (Grenoble) 48 (1998), [HR04] J.-C. Hausmann an E. Roriguez, The space of clous in an Eucliean space Experimental Mathematics. 13 (2004), [HRWeb] WEB site organise by J-Cl. Hausmann an E. Roriguez.

12 GEOMETRIC DESCRIPTIONS OF POLYGON AND CHAIN SPACES 11 [KM95] M. Kapovich an J. Millson, On the mouli space of polygons in the Eucliean plane, J. Differential Geom. 42 (1995), [KM96] M. Kapovich an J. Millson, The symplectic geometry of polygons in Eucliean space, J. Differential Geom. 44 (1996), [KBCL99] D.G. Kenall, D. Baren, T.K. Carne an H. Le, Shape an Shape Theory, John Wiley & Sons Lt., Chichester [Ko93] A. Kosinski. Differentiable manifols. Acaemic Press [MDS95] D. McDuff an D. Salamon Introuction to symplectic topology, Oxfor Science Publ [Wa85] K. Walker Configuration spaces of linkages, Bachelor s thesis, Princeton (1985). [Wn69] A. Wasserman Equivariant ifferential topology, Topology 8 (1969) Section e Mathématiques, Université e Genève,, B.P. 240, CH-1211 Geneva 24, Switzerlan aress: hausmann@math.unige.ch