Graduate J. Math. 2 (2017), 29 – 36 The con guration space of the three dimensional Lens space L(7, 2) and its model
Giulio Calimici
Abstract obtained as a suitable quotient of the three sphere S3 (de nitions in the text). This Lens space; which is We study the algebraic topology of the two point con guration one in a family of spaces with relatively space on the lens space L(7, 2), describing an explicit nite dimensional L(p, q) p, q model for the algebra of De Rham dierential forms on the universal prime, was used by Longoni and Salvatore [13] to cover of the two point con gurations space Ω∗ (F˜ (L(7, 2))). provide the only known counterexample to the con- DR 2 jecture that the diagonal complement of a closed ori- MSC 2010. Primary 55R80; Secondary 55S30. ented manifold, or equivalently its two points con- guration space, is a homotopy invariant. In con- structing the counterexample, one takes advantage of the fact that some Lens spaces are known to be 1 Introduction homotopy equivalent but not homeomorphic. Note that the di culty in dealing with this conjecture is Rational (or real) homotopy theory provides a rare that plain homological or homotopy group calcula- instance in topology where spaces, which one takes to tions cannot distinguish between the various diagonal be simply connected, can be completely understood complements associated to L(p, q). There is a need by an algebraic model once we only consider the non- to look for deeper invariants, like Massey products, torsion invariants of the space. The topological dic- and thus the need to go through universal covers to tionary and the algebraic dictionary are completely detect those subtle dierences, as was done in the equivalent in this case. This theory is vast and we Longoni-Salvatore paper. only refer to the main reference on the subject [7]. Denote by ˜ the universal cover of the To better explain what we do in this paper, we F2(L(7, 2)) Lens space F2(L(7, 2)). Our main results are given must say more about these algebraic models which in section 3 where the model is constructed and then always come in the form of commutative graded dif- in section 4 where we furthermore observe that the ferential algebras, or CGDAs for short. The dier- model is equivariant with respect to the action of the ential is denoted by d. For a smooth dierential fundamental group. As an application of our con- manifold , the deRham dierential forms ∗ M ΩDR(M) struction, we recover the non-trivial massey product form a CDGA over the real numbers and this deter- in the cohomology of F2(L(7, 2)) discovered by Lon- mines the real homotopy type of M (in particular, goni and Salvatore. the ranks of the homotopy groups). This CDGA is however generally very large and one seeks to nd a smaller more manageable CDGA quasi-isomorphic to Acknowledgements: This work was part of the master thesis Spazi di con gurazione delle varietá Ω∗ (M), thus yielding the same real homotopy type. “ DRMore precisely, a CDGA is called a model for lenticolari from the academic year 2014/2015 of the A author at the University of Roma Tor Vergata under Ω∗ (M) if there is a third intermediate CDGA B mappingDR homomorphically to both the guidance of Prof. Paolo Salvatore.
∗ A ←− B −→ ΩDR(M) 1.1 Con guration spaces with the arrows inducing an isomorphism in coho- The con guration space of n points in a manifold mology. All algebras in our case are de ned over R. M is the space Our goal in this paper is to exhibit a nite di- mensional model for the De Rham dierential ×n (A, d) Fn(M) := {(x1, . . . , xn) ∈ M : xi 6= xj for i 6= j}. forms on the universal cover of a particular three di- (1.1) mensional manifold which is the Lens space L(7, 2),
29 2 Two point con guration space of lens spaces 30
or equivalently the complement De nition 1.2. Let (p, q) ∈ Z be relatively prime. n The lens space L(p, q) is de ned to be the quotient Fn(M) = M − ∆, 3 S /hζqi. where ∆ denotes the diagonal Observation 1.3. The projection S3 → L(p, q) is for some a covering space projection since the action of is ∆ = {(x1, . . . , xn)| xi = xj i 6= j}. Zp free. Only the identity element has xed points in One thinks of as all -tuples of points of Fn(M) n M 3, and for each point 3, 2πikq/p , for all that are pairwise distinct. S zj ∈ S e zj 6= zj Con guration spaces have been studied in many 0 < k < p. This is a consequence of the assumption that . dierent situations. For example, 3 is the nat- mcd(p, q) = 1 Fn(R ) ural setting for the -body problem (for a history n Because S3 is simply connected, it follows immedi- 3 of this topic see [4]). Con gurations of points in R ately that S3 is the universal covering of L(p, q) and that are required to adapt to a given geometry are that ∼ . An important theorem of Rei- π1(L(p, q)) = Zp used in robotics and the piano mover's problem (for demeister [18] classi es the Lens spaces up to home- more details on this topic, see [5]). In mathematical omorphism and homotopy equivalence. physics, con guration spaces have been considered in relation to gauge theory, gravity or particle physics. Theorem 1.4. The lens spaces L(p, q) and L(p, q0) In topology and geometry, these spaces enter as a are homotopy equivalent if and only if for some m ∈ we have that: fundamental tool in the study of spaces of functions Zp and moduli spaces. 0 2 Exemple 1.1. The best-known and simplest non q ≡ ±qm mod p. trivial examples are the con guration spaces 2 , Fn(R ) They are homeomorphic if and only if whose fundamental groups 2 are the Pn = π1(Fn(R )) so-called pure braid groups. The classical Artin braid q0 ≡ ±q±1 mod p. groups appear as the fundamental groups of the unordered con guration spaces 2 , Bn = π1(UFn(R )) where UF ( 2) is de ned to be the quotient of F ( 2) 2 Two point con guration space of lens n R n R spaces by the natural action of the symmetric group Sn 2 which permutes the points in R . The braid descrip- tion is used to visualize paths in the con guration In this section we will discuss the cohomology of space, thus providing a very intuitive description. the universal cover of two point con guration spaces of lens spaces, as calculated in [13], and we present a list of generators of the relative homology groups 1.2 Lens Spaces 3 3 H∗(S × S , ∆q), where ∆q is de ned below, and de- The term “lens space usually refers to a speci c scribe their respective Lefshetz duals. class of 3-manifolds, albeit these can be de ned in The universal cover of the two point con guration higher dimensions. They were introduced for the space F2(L(p, q)) is given as follows rst time by H. Tietze in 1908. There is more than ˜ 3 3 k one way to construct lens spaces in the 3-dimensional F2(L(p, q)) = {(x, y) ∈ S ×S : x 6= ζq y ∀ k ∈ Zp}. case. One way is to take the quotient of the unit 3- sphere S3 by an action of a cyclic group. Another For each q = 0, . . . , p − 1, de ne way is by gluing two solid tori together via a home- [ k omorphism of their boundary. ∆q := ∆q In the rest of the paper we identify to the group Zp k∈Zp of pth complex roots of unity generated by where ∆k is the image of the following embedding of 2πi/p q ζ = e . S3 3 3 3 Let S3 the unit 3-sphere viewed as a submanifold of S −−→S × S 2; k C ω 7−→ (ω, ζq ω). 3 ∼ 2 2 2 S = {(z1, z2) ∈ C | |z1| + |z2| = 1}. If we write ω = (x1, x2) in complex coordinates, then k k kq De ne for each relatively prime to de ne (ω, ζ ω) = ((x1, x2), (ζ x1, ζ x2)), and we can view q ∈ Zp p q the action of on 3 such that the generator acts ˜ as the subset of 2 2 given as follows Zp S F2(L(p, q)) C × C by the map ζq with k qk q {((z1, z2), (z3, z4)) | (z1, z2) 6= (ζ z3, ζ z4), k ∈ Zp}. ζq(z1, z2) = (ζz1, ζ z2) where we denote by z1 and z2 the complex coordi- nates of S3. 2 Two point con guration space of lens spaces 31
¯ 2.1 Representatives in homology and duality closure of the manifold Ak denoted by Ak. The long exact sequence restricts to From now on we denote I(k) the open interval (k − 1, k) and x the positive orientation of I(k) as 3 3 the canonical one.We also choose the orientation of H4(S × S ) = 0 S3 to correspond to the orientation at the point (0, 1) given by the basis tangent vectors (i, 0), (1, 0) and ∂ (0, i). In [13], that authors computed the following H (S3 × S3, ∆ ) / H (∆ ) / H (S3 × S3) ··· cohomology groups 4 q 3 q 3 ¯ k k−1 6 ∂ :[Ak] 7−→ [∆q ] − [∆q ]. R ∗ = 2 ∗ ˜ the algebraic boundary is deduced from the geo- H (F2(L(p, q))) = R ∗ = 3 ∂ 6 metric boundary of ¯ which is exactly k k−1. R ∗ = 5. Ak ∆q ∪ ∆q The exact sequence takes then the form where the cup product of the two dimensional classes φ with the three dimensional class give the ve dimen- 0−−−→H (S3 × S3, ∆ ) −−−−→∂ p −−−−→ ⊕ sional classes. By Lefshetz duality we have the iso- 4 q Z Z Z morphisms where the map φ is given by
3 3 [ k ∗ ˜ p p H6−∗(S × S , ∆q ) = H (F2(L(p, q))), X X (v1, . . . , vp) 7→ ( vi, vi). k∈Zp i=1 i=1 for each . The explicit generators of this ∗ = 1,..., 6 p−1 intersection homology ring are listed and labeled as The kernel of φ is isomorphic to Z and by the injectivity of we have that 3 3 ∼ p−1. follows: ∂ H4(S ×S , ∆q) = Z 3 3 ¯ The generators of H4(S × S , ∆q) are exactly [Ak] ∗ H∗ with the relation P [A¯ ] = 0. To represent the k∈Zp k 4 [Ak] generator [S] ∈ H (S3 × S3, ∆ ) we de ne S to be 3 [S] 3 q the image of the embedding S3−−→S3 × S3 constant 1 [S ∩ Ak] on the rst coordinate sending ω 7→ ((1, 0), ω). For 3 3 where the indices k have to be read modulo p. These the generators of H1(S × S , ∆q) we must consider generators are represented by submanifolds and their the intersection S ∩ A¯ given by (transversal) intersections. We need make this de- k scription explicit, and to that end we recall a de ni- S ∩ A¯ = {(1, 0); (ζs, 0) : s ∈ I(¯k)}. tion from M.S. Miller [15]. k We notice that the intersection is always transversal, De nition 2.1. Given a map between topological indeed taking ˜ we have spaces , the track of , denoted , w = (w1, w2) ∈ F2(L(p, q)) f : X × Y → X f Γ(f) that the following vectors are linearly independent is the image of 2 where π1 × f :(X × Y ) → X × X π1 and generate the tangent space at respectively of is the canonical projection on the rst factor. w the manifolds Ak and S: De nition 2.2. The submanifold Ak is the track of s qs the following map Tw(Ak) = span{((0, 0), (iζ , 0)); ((0, 1), (0, ζ )); ((0, i), (0, iζqs)); ((i, 0), (iζs, 0))}, 3 3 αk : S × I(k) −→ S s qs Tw(S) = span{((0, 0), (iζ , 0)); ((0, 0), (0, ζ )); (ω, s) 7−→ ζs ω, q ((0, 0), (0, iζqs))}. With prior notation Ak = Γ(αk). The space Tw(Ak)+Tw(S) has dimension six and this proves the transversality of the intersection. Equivalently using complex coordinates ω = (z1, z2) we have: We summarize the calculation in the following ta- ble, where the generators in H6−∗ are the Lefshetz s qs ∗ Ak = {(z1, z2); (ζ z1, ζ z2): s ∈ I(k)} dual to the generators in H : 3 3 2 2 ⊂ S × S ⊂ × . ∗ C C ∗ H H6−∗ 6 − ∗ 3 3 2 [ak] [Ak] 4 To prove that the classes [Ak] generate H4(S ×S , ∆q), it is enough to consider a portion of the long exact 3 [η] [S] 3 3 3 5 [ηa ] [S ∩ A ] 1 sequence of the pair (S × S , ∆q). To de ne the k k homological relative classes we have to consider the where the indices k have to be read modulo p. 3 The models 32
2.2 Intersection and transversality of the De nition 3.1. Let A, B be two CDGA. We will say that they are quasi-isomorphic if there exists a manifolds Ak CDGA C and a zigzag of homomorphisms f and g In this section we will discuss the intersections of the manifolds . The intersections of more than f g Ak A ←−−−− C −−−−→ B three manifolds Ak is empty. From now on we denote by the projection of the interval (a, b)S1 (a, b) ⊂ R such that the induced maps in cohomology are iso- onto the quotient R/pZ. morphisms. In this case and as discussed in the in- De nition 2.3. Let j be an integer modulo p. De ne troduction, we say that A is a model for B, and vice- j to be a q-covering if versa. −1 for De nition 3.2. Let A be the free CDGA on the j ≡ q m (mod p) |m| ∈ (0, q)S1 . following generators De nition 2.4. Let be a -covering. Then j q k ∈ Zp is an interloper of j if 6 0 5 ηak, xijak 4 k ∈ [j, p]S1 for m > 0 aiaj 3 η, x k ∈ [0, j]S1 for m < 0. ij 2 ak If j is not a q-covering then there are no interlopers 1 of j. In [15] section III.2, it was proved that if j is not a q-covering then the submanifold A ∩ A is empty truncated in degree six, with the indices i, j, k all in k k+j . We de ne the dierential of the generators in and that Z7 this way: Proposition 2.5. If j is a q-covering and , then the intersection Iτ = (qI(k))S1 ∩ (qI(k + j))S1 d(ak) = 0, Ak ∩ Ak+j is non-empty, transversal and homeomor- 1 d(η) = 0, phic to S × Iτ . d(x ) = a a . In order to calculate the triple intersections we con- ij i j sider the submanifold Xij ∩Ak where Xij is the man- ifold such that ∂Xij = Ai ∩ Aj. We see that Ak ∩ Now we set the following relations: the elements . xij ∂Xij = Ak ∩ Ai ∩ Aj are de ned to be non-zero for all the indices i < j such that . By the proposition 2.5 we Proposition 2.6. [15] Let j and j0 be q-covering, so Ai ∩ Aj 6= ∅ −1 0 0 −1 0 have that the only non-empty intersections are j ≡ mq , and j ≡ m q with |m|, |m | ∈ (0, q)S1 . Then the triple intersection A1 ∩ A1+j ∩ A1+j0 is non such that 0 Ai ∩ Ai+3 i ∈ Z7, empty if and only if |m − m | is in (0, q)S1 . so the only indices , for which is de ned are in We conclude with this i j xij the set Proposition 2.7. Let be -covering and let . j q l ∈ Zp If l is also a q-covering assume that |m−ql| ∈/ (0, q)S1 . G = {(1, 4); (2, 5); (3, 6); (4, 7); (5, 1); (6, 2); (7, 3)} Then ⊂ Z7 × Z7. ( ∅ if l is not an interloper of j The relations for the product are: A1+l ∩ Xi,j+1 = A1+l ∩ S if l is an interloper of j X For the proof see again [15]. By the action of ak = 0, (3.1) it is su cient to consider the intersection k∈ 7 Zp = hζi Z with . A1 xijak = 0 for k not interloper, (3.2)
xijai = 0 ∀ (i, j) ∈ G, (3.3) 3 The models xijak = ηak for (i, j) ∈ G and k = i + 1 or k = i + 2, (3.4) In this section we will discuss the explicit con- (3.5) struction of the model A(p, q). From now on we xijaj = −η(ai+1 + ai+2) ∀ (i, j) ∈ G, will x the parameters and , there- p = 7 q = 2 Now we are able to give the following fore the associated lens space is L(7, 2). The model A(7, 2) is a commutative dierential graded algebra De nition 3.3. Let A(7, 2) be the algebra A modulo (CDGA) that is quasi-isomorphic to the universal the relations above. cover of the two point con guration space of the lens space ∗ ˜ . ΩDR(F2(7, 2)) 3 The models 33
By standard calculations we will verify that the co- By standard computations (which are omitted), homology with real coe cients of A(7, 2) is the same we nd that the algebra B(7, 2) has the required co- as that of F˜ (L(7, 2)). Indeed we have homology, which is the same as that of A(7, 2), up to 2 degree ve. In order to kill the cohomology in degree 6 six we add cycles in degree ve, so that R ∗ = 2 w1, . . . , wm ∗ the cohomology in lower degrees does not change and H (A(7, 2)) = R ∗ = 3 simultaneously we annihilate the cohomology in de- 6 R ∗ = 5. gree six. Moreover these generators do not produce cohomology in degree six because for dierent values To prove that the CDGA is quasi-isomorphic A(7, 2) of i the images of zw under the dierential to ∗ ˜ we will construct a CDGA i ΩDR(F2(7, 2)) B(7, 2) and a zigzag of quasi-isomorphisms and between |z| λ µ d(zwi) = d(z)wi + (−1) z d(wi) them X = d(z)wi + ··· = ( ak)wi + ... λ µ ∗ ˜ k∈ A(7, 2) ←−−−− B(7, 2) −−−−→ ΩDR(F2(7, 2)). Z7 De nition 3.4. Let B the free graded dierential are linearly independent. This procedure is formal algebra generated in degree one by z, in degree two because the cohomology is nitely generated in ev- ery degree. Since 1 we can iterate by ak, in degree three by η, xij, and in degree four by H (B(7, 2)) = 0 , for . De ne the dierential to be this process for each degree introducing generators tijl i, j, k ∈ Z7 in degree l to annihilate the cohomology in degree X l + 1 for each l ≥ 5. Therefore the three algebras d(z) = ak, (3.6) ∗ ˜ A(7, 2),B(7, 2), Ω (F2(7, 2)) have the same coho- k∈Z7 mology. DR (3.7) d(xij) = aiaj, Next we analyze the algebra of dierential forms d(t ) = (x − η)a , (3.8) ∗ ˜ ijk ij k Ω (F2(7, 2)). As discussed in proposition 2.7 the intersectionDR for and zero on other generators. Ak ∩Xij = Ak ∩S k ∈ {i, i+1, i+2, j} and (i, j) ∈ G, so for these indices there exists a Now we de ne the algebra B(7, 2) to be the alge- Thom form τijk in degree four with support small bra modulo the following relations: B enough such that Supp(τijk) ⊇ Xij ∩ Ak and at the same time for each . Its (3.9) Supp(τijk) ∩ Al = ∅ k 6= l xij = 0 ∀ (i, j) ∈/ G, dierential is such that: a a = 0 (i, j) ∈/ G, (3.10) i j ∗ ˜ ∗ ˜ d :ΩDR(F2(7, 2)) −−−→ ΩDR(F2(7, 2)) tijk 6= 0 (i, j) ∈ G and k ∈ {i + 1, i + 2}, (3.11) τijk 7−→ (χij − σ)ωk, and xijak = 0 (i, j) ∈ G k ∈ {i, i + 1, i + 2, j}, where the forms ωk, χij and σ are respectively Thom (3.12) forms whose support is in a tubular neighborhood of the submanifolds Ak, Xij and S. tijkal = 0 ∀ l 6= k. (3.13) To prove that the algebras and ∗ ˜ A(7, 2) ΩDR(F2(7, 2)) For the sake of clarity, we list the generators of B(7, 2) as claimed, we have to map B to both of them via up to degree six maps inducing isomorphism in cohomology. We start by de ning the homomorphism 7 ··· 6 , , , , 2 , 3 zakxij ηzak ηxij tijkak ai aj ak λ : B(7, 2) −→ A(7, 2), 5 , , 2, , ηak xijak zak zaiaj ztijk 4 2, , , , ak aiaj ηz zxij tijk λ(z) = 0, 3 η, xij, zak λ(ak) = ak, 2 ak 1 z λ(xij) = xij, λ(η) = η, for appropriate indices i, j, k. λ(tijk) = 0. Observation 3.5. The algebra B(7, 2) unlike A(7, 2) The induced map in cohomology is an isomorphism. is not truncated. In fact it has generators in each de- On the other hand, we de ne the map gree, but only a nite number of them. In degree six the triple product has been omitted because ∗ ˜ aiajak µ : B(7, 2) → ΩDR(F2(7, 2)), for each i 6= j 6= k at least one couple between (i, j), as follows: since P ω represents a trivial cocy- (i, k), (j, k) is not in G, and so the corresponding k∈Z7 k product is zero. cle, there exists a dierential form ∗ ˜ ξ ∈ ΩDR(F2(7, 2)) 4 Equivariance 34
P such that ωk = dξ. Therefore we set: trivially on z and η, and by translation of indices on k∈Z7 the other generators: µ(z) = ξ, ζ · ak := ak+1, µ(xij) = χij, ζ · xij := xi+1,j+1, µ(η) = σ, ζ · tijk := ti+1,j+1,k+1. µ(tijk) = τijk. We still need to de ne the action on the other gener- By de nition of µ the induced map in cohomology is ators in degree higher than ve. To this aim we pro- ceed as follows: since a -action on a vector space an isomorphism as well. This implies the existence Z7 is equivalent to a real -representation, we consider of a zigzag of quasi-isomorphism between A(7, 2) and Z7 ∗ ˜ the short exact sequence for (i > 5) Ω (F2(7, 2)). DRWe observe that in there exists a non- A(7, 2) i ˜ i ˜ π i ˜ trivial Massey product, and so using the quasi- iso- 0 → B (B(7, 2))−−→Z (B(7, 2))−−→H (B(7, 2)) → 0, morphism we have the same non-trivial product in µ where B˜(7, 2) is the subalgebra of B(7, 2) generated ∗ ˜ found in Longoni and Salvatore [13]. ΩDR(F2(7, 2)) by the generators up to degree i − 2, and we indicate Proposition 3.6. The Massey product respectively with Bi and Zi the ith co-boundaries and the ith co-cycles. The vector space Hi(B˜(7, 2)) is iso- ha4, a1, a2 + a6i morphic to the direct sum of one-dimensional vector spaces i−1 (equal to the number of added elements is non-trivial. Wj in degree i − 1): Proof. We have that i ˜ ∼ i−1 H (B(7, 2)) = ⊕jWj . d(x14) = a1a4, For each i, the space W i−1 is the real vector space and also j generated by the elements added in degree i−1 to kill a1(a2 + a6) = 0, the cohomology in degree . The irreducible real - i Z7 so by de nition of the Massey product: representations are only four: the trivial one which is one-dimensional and three others of dimension two ha4, a1, a2 + a6i = x14(a2 + a6) (up to isomorphism) induced by multiplication by the third roots of unity. Using Schur's lemma we deduce = x14 a2 + x14 a6 that each -representation on i ˜ can be Z7 H (B(7, 2)) = x14 a2 = η a2, decomposed as a direct sum of irreducible real sub- that is non-trivial modulo hη a , η (a + a )i. representations of dimension one and two. So the 4 2 6 problem to de ne the action is reduced to nding an equivariant section 4 Equivariance s : Hi(B˜(7, 2)) → Zi(B˜(7, 2)) In this nal section we suitably modify the maps which we know exists by [10]. The existence of this λ and µ so that they become equivariant with respect to the action of the second factor of the fundamental section allows us to chose equivariant representatives in i ˜ . The -action on ∗ ˜ is group of π (F (L(7, 2))) ∼= × . Z (B(7, 2)) Z7 ΩDR(F2L(7, 2)) 1 2 Z7 Z7 induced by the action of the fundamental group on De nition 4.1. (Equivariant map) Let G a group, the universal cover. We must appropriately modify X1 and X2 two G-sets and ϕ : X1 → X2 a map the map µ in order to make it equivariant and so that between sets. Then ϕ is G-equivariant if it commutes the following diagram commutes: with the G-action: µ ∗ ˜ B(7, 2) −−−−−→ ΩDR(F2L(7, 2)) ϕ(g · x) = g · ϕ(x), ζ·y ζ·y for all x ∈ X1 and g ∈ G. µ ∗ ˜ B(7, 2) −−−−−→ ΩDR(F2L(7, 2)). 4.1 Equivariance of µ Next we study the action ζ· on the submanifold Ak As we stated in section 1.2 consider as the of the second factor of the fundamental group. For Z7 each we have that: cyclic group of 7th complex roots of unity generated k ∈ Z7 by 2πi/7. It is enough to de ne the -action s+1 2s+2 ζ = e Z7 ζ · A = {(z , z ); (ζ z , ζ z ): s ∈ I(k)} on the elements , , , , . The group acts k 1 2 1 2 z ak xij η tijk Z7 t 2t = {(z1, z2); (ζ z1, ζ z2): t ∈ I(k + 1)} = Ak+1. 4 Equivariance 35
We de ne µ(ak) = ωk where ωk is the Thom form action de ned on the algebra B(7, 2). Let us proceed de ned on a tubular neighborhood of Ak as in section to check this in degree one: 3. So since we have that ζ · Ak = Ak+1 ζ · ωk = ωk+1 X X by de nition and so we have that the action in degree d(ζ · z) = d(z) = ak = ζ · ak = ζ · d(z). 2 is equivariant because k∈Z7 k∈Z7
ζ · µ(a ) = ζ · ω = ω = µ(a ) = µ(ζ · a ), For the elements ak in degree two and for η in degree k k k+1 k+1 k three we have that the equivariance holds because for all k ∈ . Since P ω is a trivial cocycle in the dierential on them is trivial. For the elements Z7 k∈Z7 k cohomology, there exists an element of degree one ξ x we proceed as follows: P ij such that dξ = ωk. An arbitrary choice of such k∈Z7 d(ζ·x ) = d(x ) = a a = ζ·(a a ) = ζ·d(x ). ξ does not assure the equivariance of this map be- ij i+1,j+1 i+1 j+1 i j ij cause the action de ned on B(7, 2) xes the element Finally we check the equivariance also for the ele- z but this is not true for any choice of ξ. Concerning ments in degree four: the one degree element z we rede ne µ in this way: ζ · d(tijk) = ζ · ((xij − η)ak) 1 X µ(z) := ζk · ξ. = (xi+1,j+1 − η)ak+1 7 = d(t ) = d(ζ · t ), k∈Z7 i+1,j+1,k+1 ijk because η is xed by the action. We have that ζ · µ(z) = 1 P ζk+1 · ξ = µ(z) = 7 k∈ 7 for , that givesZ us the equivariance µ(ζ · z) k ∈ Z7 in degree one. Regarding degree three we proceed as 4.3 Equivariance of λ follows: since the action on B(7, 2) xes η we rede ne We have also to de ne an action on the algebra . Let us consider the Thom form de ned on a µ σ A(7, 2) that we still call ζ·, such that λ is equivariant. particular tubular neighborhood of a submanifold S, Equivalently the following diagram: as in section 3, we de ne λ 1 X B(7, 2) −−−−−→ A(7, 2) µ(η) := ζk · σ, 7 ζ· ζ· k∈Z7 y y so due to the cyclicity we have that B(7, 2) −−−−−→λ A(7, 2) commutes. 1 X ζ · µ(η) = ζk+1 · σ = µ(η) = µ(ζ · η), Also in this subsection we will use the same action 7 k∈Z7 that we de ne on B(7, 2) and we will de ne another one on A(7, 2) as follow. For the element z in degree and thus the related diagram is commutative. Also one we have: for the other elements in degree three xij with (i, j) ∈ G we modify the map. By de nition of ζ· we have λ(ζ · z) = λ(z) = 0, so we de ne , where ζ · Xij = Xi+1,j+1 µ(xij) = χij and at the same time we have ζ · λ(z) = 0. For all χij is the Thom form de ned on a particular tubu- , we de ne the action for the all k ∈ Z7 ζ · ak := ak+1 lar neighborhood of submanifold Xij, as in section elements ak that in degree 2 of A(7, 2), so we have: 3. Therefore ζ acts on the forms χij by translation of (i, j): ζ · χij = χi+1,j+1, and thus ζ · µ(xij) = λ(ζ · ak) = λ(ak+1) = ak+1 = ζ · ak = ζ · (λ(ak)) ζ · χij = χi+1,j+1 = µ(xi+1,j+1) = µ(ζ · xij) that gives (care must be taken as we previously de ned the el- us the equivariance also in degree three. Finally in ement ak both in A(7, 2) and B(7, 2)). In degree 3 degree four we de ne the map µ on the generators we de ne the action on the element x in this way by for and ij µ(tijk) = τijk (i, j) ∈ G k ∈ {i + 1, i + 2} ζ · xij := xi+1,j+1, and with this de nition we have where τijk is the Thom form de ned on a particular that: tubular neighborhood of the submanifold Xij ∩Ak, as in section 3. We consider the action of ζ on this in- λ(ζ · xij) = λ(xi+1,j+1) = xi+1,j+1 = ζ · xij = ζ · λ(xij) tersection and we have . Thus we ζ ·τijk = τi+1,j+1,k+1 Still in degree three, for the element η, we have that: have the commutativity also in degree four, because λ(ζ · η) = λ(η) = η = ζ · η = ζ · λ(η), ζ · µ(t ) = ζ · τ = τ = µ(τ ) ijk ijk i+1,j+1,k+1 i+1,j+1,k+1 where, as usual we use the same name η for elements = µ(ζ · τijk). de ned in the two dierent algebras. Finally in de- gree four for the elements tijk are sent to 0, and the 4.2 Commutativity with the dierential equivariance is respected. The nal statement that we have to check is the equivariance of the dieren- We have also to check that the dierential is equiv- tial of A(7, 2) . By similarity to what has been done ariant. In the following we will check this only for the in subsection 4.2 the calculation is omitted. 4 Equivariance 36
References [18] K. Reidemeister, Complexes and homotopy chains, Bull. Amer. Math. Soc:56, pp. 297-307, [1] M. Artin, Algebra, Prentice Hall, New Jersey, 1950. 1991. [19] H. Seifert e W. Threlfall, A Textbook of Topol- [2] R. Bott, L.W. Tu, Dierrential forms in alge- ogy, Academic Press, 1980. braic topology, Graduate Texts in Mathematics, vol. 82, Springer-Verlag, New York, 1982. [20] S. Warner, Foundations of Dierentiable Man- ifolds and Lie Groups, Springer-Verlag, New [3] G. E. Bredon, Topology and Geometry, York, 1983. Springer-Verlag, New York, 1993.
[4] F.Diacu, The solution of the n-body problem, Giulio Calimici The Mathematical Intelligence 18, pp. 66-70, 1996. Laboratoire Paul Painlevé Université Lille 1, 59655 Villeneuve d'Ascq, [5] R. Ghrist, Con guration spaces, braids, and France robotics, World Scienti c Publishing, Hacken- E-mail address: [email protected] sack, NJ, 2010, pp.263-304. [6] V. Guillemin, A. Pollack, Dierential Topology, Prentice Hall, Englewood Clis, New Jersey, 1974. [7] Y. Felix, S.Halperin, J-C Thomas, Rational ho- motopy theory, Graduate Text in Mathematics 205, Springer-Verlag, New York, 2001. [8] A. Hatcher, Algebraic Topology, Cambridge University Press, 2001. [9] M. Jankins, W. D. Neumann Lectures in Seifert manifolds, lecture notes, 1983. [10] T. Lambre, Modèle minimal équivariant et for- malité, Transactions of the American Mathe- matical Society, vol. 327 no. 2, pp. 621-639, 1991. [11] J. Lee, Introduction to Smooth Manifolds, Springer-Verlag, New York 2003. [12] N. Levitt, Spaces of arcs and con guration spaces of manifolds, Topology 34, pp. 217-230, 1995. [13] R. Longoni, P. Salvatore, Con guration spaces are not homotopy invariant, Topology 44 no.2, pp. 375-380, 2005. [14] W. S. Massey, Journal of Knot Theory and Its Rami cations, Higher order linking numbers, World Scienti c Publishing Company Vol. 7, No. 3, pag. 393-414, 1997. [15] M. S. Miller, PhD Thesis: The rational ho- motopy types of con guration space of three- dimensional lens spaces, September 2007. [16] M. S.Miller, Rational homotopy models for two- points con guration spaces of lens spaces, Inter- national Press, vol. 13(2), pp. 43-62, 2011. [17] J. R. Munkres, Elements of algebraic topology, Addison-Wesley, 1984.