Affine configurations of 4 lines in R 3

Transcription

1 Arch. Math. 00 (2005) X/05/ DOI /s Brkhäuser Verlag, Basel, 2005 Archv der Mathematk Affne confguratons of 4 lnes n R 3 By Jorge L. Arocha, Javer Bracho, Cham Goodman-Strauss and Lus Montejano Abstract. We prove that affne confguratons of 4 lnes n R 3 are topologcally and combnatorally homeomorphc to affne confguratons of 6 ponts n R Introducton. Consder four lnes l 1,l 2,l 3,l 4 n 3-dmensonal space R 3 ; ther affne confguraton s ther equvalence class under the natural (dagonal) acton of the affne group Aff(3). We say that ther drectons are n general poston f ther correspondng four ponts at nfnty are n general poston n the projectve plane P 2. We say that they fx R 3 (or that ther affne confguraton fxes R 3 ) f the only affne somorphsm that fxes them, as sets, s the dentty (.e., they le on a free orbt of the acton) t s easy to see that four lnes wth drectons n general poston fx R 3 f and only f they are not concurrent. The purpose of ths paper s to descrbe the space, whch we denote A 3 4,1, of affne confguratons of four lnes n R 3 that fx and have drectons n general poston. Topologcally, t s the 4 dmensonal projectve space P 4, but furthermore, t has the polyhedral structure of the space of affne confguratons of 6 ponts n R 4. The combnatoral structure, or decomposton, of A 3 4,1 arses naturally from the fact that there s a clear noton of degeneracy: confguratons where some of the lnes meet. So that we can say that two affne confguratons of lnes are equvalent f they have the same meetng pattern (a set of pars of meetng lnes) and one representatve may be contnuously moved to the other wthout ever changng that pattern. We wll prove that these equvalence classes are cells (n fact, the nteror of products of smplces) correspondng to the Radon parttons of the sx possble pars of the (four) ndces. An affne confguraton of ponts s an affne equvalence class of k ponts n R n that affnely generate t; the space of such s denoted A n. Followng the deas of [3] for vector Mathematcs Subject Classfcaton (2000): 52C35, 14N20, 52B70. J. L. Arocha, J. Bracho and L. Montejano, Partally supported by grants CONACYT F and DGAPA-IN C. Goodman-Strauss partally, supported by Catedra Patrmonal CONACYT-IMUNAM and NSF award DMS

2 2 J. L. Arocha, J. Bracho, C. Goodman-Strauss and L. Montejano arch. math. confguratons, these spaces are seen to be grassmannans namely, A n = G(k 1 n, n): the grassmannan of (k n 1) dmensonal subspaces of R k 1. They come wth a natural stratfcaton gven, agan, by the noton of degeneracy, wth cells correspondng to orented matrods. In ths context, the present paper studes one of the frst examples of ther natural generalzaton to spaces of confguratons of flats of dmensons other than 0. It s remarkable that A 3 4,1 s agan a grassmannan because the space of 4 dfferent lnes that fx the plane R 2 (modulo the affne group Aff(2), of course) whch we call A 2 4,1 s the surface of non-orented genus 5 wth combnatoral symmetry group S 5 [1]. 2. The homeomorphsm A 3 4,1 A1 6,0. Let l 1,l 2,l 3,l 4 be four lnes n R 3 wth respectve drectonal vectors d 1,...,d 4 n general poston. We then have a non trval lnear relaton µ d = 0 wth µ = 0 for all (f otherwse, three of the ponts at nfnty would be colnear), whch s unque up to a constant non-zero factor. Then, rescalng the drectons (d := µ d ), we may assume that 4 d = 0 =1 n whch case we say that they are normalzed. Now, we assocate to each par of lnes l,l j a number λ j whch, n a sense, measures the dstance between them. To fx deas, consder the lnes l 1,l 2. It s easy to see that there are unque segments wth drectons d 3 and d 4 and endponts n l 1 and l 2 ; call them σ3 12 and σ4 12 accordngly (Fgure 1). These segments together wth the segments wthn l 1 and l 2 between ther endponts form a quadrlateral, whch, walked around, clearly gves a relaton 4 α d = 0. Snce the drectons d are normalzed, then all the coeffcents (α ) are equal, to λ 12 say. =1 Fgure 1. We then have that, as a vector, σ3 12 = λ 12 d 3, and smlarly σ4 12 = λ 12 d 4. Observe that λ 12 s well defned up to sgn, because walkng around the quadrlateral n the opposte drecton smply changes ts sgn. The two drectons correspond to choosng one of the cyclc orders 1324 or 1423 ndcatng the order of (the drectons of) the segments n the quadrlateral. Observe also that ths procedure analogously gves λ j for the sx pars of ndces n the set

3 Vol. 00, 2005 Affne confguratons of 4 lnes n R :={1, 2, 3, 4}. So that we are left to gve a rule for choosng orentatons to elmnate the ambguty n the sgns of λ j. The natural rule follows from fxng an orentaton of the tetrahedron wth vertces 4 (let us establsh 432 as the postve cyclc orentaton around vertex 1 as n Fgure 2). Then, for the edge j choose the postve orentaton of the quadrlateral of whch t s a dagonal (e.g., for the edge 12 we must choose the cyclc order 1324). Fgure 2. Before proceedng, let us precse our notaton for later use. The segments σ jk are now orented. They go from l j to l k havng drecton d and, abusng notaton that forgets the startng pont n l j, we may wrte σ jk = λ jk d where, moreover, the trangle jk of has orentaton jk. Thus, accordng to our conventons, we should rewrte σ4 21 nstead of the prevously used σ4 12. Observe also that λ j = λ j because our subndces for λ are understood unordered, but not the ndces for σ. We have assocated sx numbers λ j R to a confguraton l 1,l 2,l 3,l 4 dependng on the choce of a normalzed set of drectons. Snce these are defned up to a non-zero constant factor, so are the λ j. Observe that λ j = 0 l l j = so that all the λ j are zero f and only f the four lnes are concurrent (snce ther drectons are n general poston, f three of them meet by pars then they are concurrent). Therefore, f l 1,l 2,l 3,l 4 fx R 3 we have a well defned pont [λ j ] P 5, whch, moreover, only depends on the affne confguraton. To see ths, observe that a translaton does not change the λ j and that a lnear map sends a normalzed set of drectons to a correspondng normalzed set of drectons so that t does not change them ether. Summarzng, we have defned a map A 3 4,1 P5 [l 1,...,l 4 ] [λ j ] whch, as we wll now see, s really a map to a hyperplane (P 4 ). Lemma 1. Let λ j correspond to the lnes l 1,...,l 4 wth normalzed drectons d 1,...,d 4 as above. Then λj = 0 where the sum s taken over the sx pars of 4 ={1, 2, 3, 4}.

4 4 J. L. Arocha, J. Bracho, C. Goodman-Strauss and L. Montejano arch. math. Proof. Accordng to our orentaton conventon, we have defned three orented segments wth drecton d 1, namely σ1 23,σ34 1,σ42 1, whch jon the lnes l 2,l 3,l 4 n that cyclc order. Therefore, ntercalatng segments on the lnes l 3,l 4,l 2 we get an orented hexagon, whch traversed orentedly yelds, for some β 2,β 3,β 4 R, a relaton (1) λ 23 d 1 + β 3 d 3 + λ 34 d 1 + β 4 d 4 + λ 42 d 1 + β 2 d 2 = 0 Because the drectons are normalzed, ths mples that β 2 = β 3 = β 4 = λ 23 + λ 34 + λ 42 =: γ 1 Ths can clearly be done for any 4, yeldng that the segment n l j from the endpont of σ kj to the startng pont of σ jr s precsely γ d j, where γ := λ jk + λ kr + λ rj wth {, j, k, r} = 4. Now we have enough measures between the sx ponts defned as endponts of the segments σ jk n any gven lne. In l 1 for example, we know the dstances between consecutve l 1 -endponts of the segments σ2 31,σ14 2,σ41 3,σ12 3,σ21 4,σ13 4 whch, preservng that cyclc order, yelds the relaton (2) (γ 2 λ 14 + γ 3 λ 12 + γ 4 λ 13 )d 1 = 0 from whch the Lemma follows drectly by the defnton of γ. Fgure 3. The precedng constructon and proof mplctly uses the combnatoral structure of the truncated octahedron. In Fgure 3, the dfferent styles of drected edges correspond to the four drectons; the coeffcents of the vectors used n the proof appear respectvely as labels

5 Vol. 00, 2005 Affne confguratons of 4 lnes n R 3 5 of the quadrlaterals or the edges. The hexagons wth only one type of edge le wthn the lnes; they gve equatons of type (2) at the end of the proof. The edges between these hexagons correspond to the segments σ kj and they group n the quadrlaterals that defned the λ s. The other type of hexagons were used to defne the γ s and gve equatons of type (1). We must fnally remark that the map A 3 4,1 P4 we have defned s a homeomorphsm. To see ths, observe that f the sx λ j, addng zero, are gven, one can construct the lnes. Fx one of them wth an arbtrary base pont. Then, enough geometrc nformaton s gven by the coeffcents to know where on the lne precsely defned segments should go to the other three lnes. The lnear condton, and Fgure 3 mply that the result s ndependent of the choces. 3. Dualty of affne confguratons of ponts. Two affne confguratons of ponts are dual f they are the orthogonal projectons of (the vertces of) the standard regular smplex to a par of orthogonal complementary subspaces; where the standard regular smplex has all vertces equdstant. We wll see that dualty gves a homeomorphsm A n Ak n 1 and characterze t completely for the case n = 1. The basc deas come from the classc dualty of matrods and vector confguratons [4], see also [3]. Consder an affne confguraton of k ponts n dmenson n, p A n. It s represented by ponts p 1,...,p k R n (wrtten, p = [p 1,...,p k ]) such that they affnely generate R n. Snce ther barycenter (1/k)p s a well defned affne nvarant, we may translate t to the orgn and assume that p 1,...,p k s centered, that s, that k p = 0. =1 Observe that choosng centered confguratons leaves our ambguty n the general lnear group Gl(n), that s, { A n = p 1,...,p k R n p 1,...,p k lnearly generater n }, / Gl(n). p = 0 Gven p = [p 1,...,p k ] A n as above, we have a lnear map ϕ p : R k R n defned by ϕ(e ) = p, where e 1,...,e k s the canoncal bass of R k. It s unto by hypothess, so that ξ p := Ker(ϕ p) s a subspace of dmenson k n (ths s the subspace assocated to the vector confguraton). Observe that ξ p does not depend on our choce of the centered representatve p 1,...,p k of p, because ϕ p followed by a lnear somorphsm has the same kernel. Observe also that from ξ p one can obtan p, because the mage of the canoncal bass n R k /ξ p (somorphc va ϕ p to R n ) s lnearly equvalent to p 1,...,p k.

6 6 J. L. Arocha, J. Bracho, C. Goodman-Strauss and L. Montejano arch. math. Let 1 = (1,...,1) = e, and let be ts normal hyperplane defned by 1 x = 0. Let v 1,...,v k be the orthogonal projecton of e 1,...,e k to (namely, v = e (1/k)1), so that v 1,...,v k are the vertces of a standard regular smplex n. Because p 1,...,p k s centered, then 1 ξ p, so that ξ p := ξ p, whch s a subspace of dmenson k n 1 of the (k 1)-dmensonal space, has all the nformaton to recover p. Indeed, p s equvalent to the mage of v 1,...,v k n /ξ p R n. Let q 1,...,q k be, respectvely, the (orthogonal) projectons of v 1,...,v k (or e 1,...,e k ) to ξ p, and let q A k n 1 be the correspondng affne confguraton. By constructon, ξ p and ξ q (defned analogously for q) are orthogonal complementary subspaces of ( R k 1 ) and the projecton of the standard regular smplex v 1,...,v k to them gves, respectvely, the affne confguratons q and p (because we can dentfy /ξ p = ξ q ). So they are dual. We can summarze by sayng that both A n and Ak n 1 are naturally homeomorphc to the grassmannans G(k n 1,n) and G(n, k n 1) wth dualty correspondng to orthogonal complementaton. In the case that nterests us (n = 1), dualty can be characterzed more explctly. Theorem 2. Let λ 1,...,λ k be a centered affne confguraton n R 1 and p = [p 1,...,p k ] an affne confguraton n R k 2. Then they are dual f and only f (3) k λ p = 0. =1 Proof. Because multplcaton by non zero constant factors does not affect the affne confguraton or the equaton, we may assume that λ 1,...,λ k s normalzed,.e., that λ 2 = 1; so that λ := (λ 1,...,λ k ) R k s well defned up to sgn. Let ξ [λ] be the (k 2)-dmensonal subspace of defned as above, and q 1,...,q k the (orthogonal) projectons to ξ [λ] of v 1,...,v k respectvely; so that [λ] A 1 and q := [q 1,...,q k ] A k 2 are dual. Observe that λ (by the centered hypothess) and that the defnng map for ξ [λ] (e λ ) s precsely x x λ, so that ξ [λ] s the orthogonal hyperplane to λ n. Then, t s easy to see that (4) q = e (1/k)1 λ λ where one uses that λ s normalzed. Therefore, ( k k ) ( k λ q = λ (1/k) λ 1 =1 =1 =1 λ 2 ) λ = 0 because λ 1,...,λ k s centered and normalzed; provng the only f sde. Suppose now that p = [p 1,...p k ] A k 2 satsfes the relaton (3). Then λ ξ p and moreover, ξ p s the lne generated by λ. By equaton (4) λ λ s the orthogonal projecton of v to ξ p. So that λ 1,...,λ k represent the dual confguraton to p.

7 Vol. 00, 2005 Affne confguratons of 4 lnes n R The Radon Complex. We have proved that A 3 4,1 s homeomorphc to P4 whch s naturally dentfed wth A 1 6,0 and then to A4 6,0 by dualty. Now we see that ts combnatoral structure corresponds to the latter, whch s ntmately related to the classc Radon s Theorem. Let p = [p 1,...,p k ] A k 2 and λ = [λ 1,...,λ k ] A 1 be dual (related as n Theorem 2 wth λ 1,...,λ k centered). Then we have a partton of the ndex set k :={1,...,k} nto three components A ={ λ > 0}; B ={ λ < 0}; C ={ λ = 0} wth A and B non-vod, whch s called the Radon partton of the confguraton p. Radon s Theorem states that the nterors of the smplces p A and p B ntersect. It s obtaned by changng the relaton (3) nto an equalty of barycentrc (convex) combnatons n the obvous way. All the confguratons wth the same Radon partton A; B; C can then be parametrzed by the product of the nterors of the two abstract smplces A and B usng the barycentrc coordnates of the ntersecton pont; gvng the natural cell decomposton of A k 2 whch we call the Radon complex. Two confguratons n the same cell (wth the same Radon partton) can be joned by a path of confguratons of the same type (the geodesc n P k 2 ). Confguratons n general poston are exactly those for whch C =, thus, the Radon complex s obtaned by choppng P k 2 by the k hyperplanes λ = 0, n whch the confguratons (p) have some degeneracy. See [2]. Returnng to the affne confguratons of four lnes n R 3, A 3 4,1, we assocated to such a confguraton l 1,...,l 4 a centered confguraton of sx ponts n the lne [λ j ] A 1 6,0 n such a way that the degeneracy l meets l j corresponds to λ j = 0. Thus, the combnatoral structure of A 3 4,1 corresponds by dualty to that of the Radon complex A 4 6,0 ; wth open cells the product of open smplces and wth two confguratons of lnes beng combnatorally equvalent f they can be moved from one to the other wthout changng the ntersecton pattern of the lnes. For each Radon partton of the sx pars j, consstng of the pars that have postve, negatve and zero dstance, there s one cell. We have proved the followng theorem. Theorem 3. There s a stratfed homeomorphsm (preservng degeneraces) between the space A 3 4,1 of affne confguratons of 4 lnes n R3 and the space A 4 6,0 of affne confguratons of 6 ponts n R 4. References [1] J. Arocha, J. Bracho and L. Montejano, On confguratons of flats I; Manfolds of ponts n the projectve lne. To appear n Dscr. Comp. Geom. [2] J. Bracho, L. Montejano and D. Olveros, The topology of the space of transversals through the space of confguratons. Topology Appl. 120(1 2), (2002).

8 8 J. L. Arocha, J. Bracho, C. Goodman-Strauss and L. Montejano arch. math. [3] I. M. Gelfand, R. M. Goresky, R. D. MacPherson and V. V. Serganova, Combnatoral geometres, convex polyhedra and Schubert cells. Adv. n Math. 63(3), (1987). [4] H. Whtney, On the abstract propertes of lnear dependence. Amer. J. Math. 57, (1935). Receved: 14 July 2004; revsed: 18 February 2005 J. L. Arocha J. Bracho Insttuto de Matematcas Insttuto de Matematcas UNAM UNAM Mexco Mexco arocha@math.unam.mx rol@math.unam.mx C. Goodman-Strauss L. Montejano Unversty of Arkansas Insttuto de Matematcas USA UNAM strauss@uark.edu Mexco lus@math.unam.mx