Definition and Examples

Transcription

1 56 Chpter 1 The Fundmentl Group We come now to the second min topic of this chpter, covering spces. We hve in fct lredy encountered one exmple of covering spce in our clcultion of π 1 (S 1 ). This ws the mp R S 1 tht we pictured s the projection of helix onto circle, with the helix lying ove the circle, covering it. A numer of things we proved for this covering spce re vlid for ll covering spces, nd this llows covering spces to serve s useful generl tool for clculting fundmentl groups. But the connection etween the fundmentl group nd covering spces runs much deeper thn this, nd in mny wys they cn e regrded s two viewpoints towrd the sme thing. This mens tht lgeric fetures of the fundmentl group cn often e trnslted into the geometric lnguge of covering spces. This is exemplified in one of the min results in this section, giving n exct correspondence etween the vrious connected covering spces of given spce X nd sugroups of π 1 (X). This is strikingly reminiscent of Glois theory, with its correspondence etween field extensions nd sugroups of the Glois group. Definition nd Exmples Let us egin with the definition. A covering spce of spce X is spce X together with mp p : X X stisfying the following condition: There exists n open cover {U α } of X such tht for ech α, p 1 (U α ) is disjoint union of open sets in X, ech of which is mpped y p homeomorphiclly onto U α. In the helix exmple one hs p : R S 1 given y p(t) = (cos 2πt,sin 2πt), nd the cover {U α } cn e tken to consist of ny two open rcs whose union is S 1. A relted exmple is the helicoid surfce S R 3 consisting of points of the form (s cos 2πt,ssin 2πt,t) for (s, t) (0, ) R. This projects onto R 2 {0} vi the mp (x,y,z) (x, y), nd this projection defines covering spce p : S R 2 {0} since for ech open disk U in R 2 {0}, p 1 (U) consists of countly mny disjoint open disks in S, ech mpped homeomorphiclly onto U y p. Another exmple is the mp p : S 1 S 1, p(z) = z n where we view z s complex numer with z =1 nd n is ny positive integer. The closest one cn come to relizing this covering spce s liner projection in 3 spce nlogous to the projection of the p helix is to drw circle wrpping round cylinder n times nd intersecting itself in n 1 points tht one hs to imgine re not relly intersections. For n lterntive picture without this defect, emed S 1 in the oundry torus of solid torus S 1 D 2 so tht it winds n times monotoniclly round the S 1 fctor without self-intersections, like the strnds of circulr cle, then restrict the projection S 1 D 2 S 1 {0} to this emedded circle.

2 Covering Spces Section As our generl theory will show, these exmples for n 1 together with the helix exmple exhust ll the connected coverings spces of S 1. There re mny other disconnected covering spces of S 1, such s n disjoint circles ech mpped homeomorphiclly onto S 1, ut these disconnected covering spces re just disjoint unions of connected ones. We will usully restrict our ttention to connected covering spces s these contin most of the interesting fetures of covering spces. For covering spce p : X X the crdinlity of the sets p 1 (x) is loclly constnt over X,soifX is connected it is independent of x nd clled the numer of sheets of the covering spce. Thus the covering S 1 S 1, z z n,isnsheeted nd the covering R S 1 is infinite-sheeted. This terminology rises from regrding the disjoint suspces of p 1 (U α ) mpped homeomorphiclly to U α in the definition of covering spce s the individul sheets of the covering spce. When X is disconnected the numer of sheets cn e different over different components of X, nd cn even e zero over some components since p 1 (U α ) is not required to e nonempty. The covering spces of S 1 S 1 form remrkly rich fmily illustrting most of the generl theory very concretely, so let us look t few of these covering spces to get n ide of wht is going on. To revite nottion, set X = S 1 S 1. We view this s grph with one vertex nd two edges. We lel the edges nd nd we choose orienttions for nd. Now let X e ny other grph with four edges meeting t ech vertex, nd suppose the edges of X hve een ssigned lels nd nd orienttions in such wy tht the locl picture ner ech vertex is the sme s in X, so there is n edge oriented towrd the vertex, n edge oriented wy from the vertex, edge oriented towrd the vertex, nd edge oriented wy from the vertex. To give nme to this structure, let us cll X 2oriented grph. The tle on the next pge shows just smll smple of the infinite vriety of possile exmples. Given 2 oriented grph X we cn construct mp p : X X sending ll vertices of X to the vertex of X nd sending ech edge of X to the edge of X with the sme lel y mp tht is homeomorphism on the interior of the edge nd preserves orienttion. It is cler tht the covering spce condition is stisfied for p. The converse is lso true: Every covering spce of X is grph tht inherits 2 orienttion from X. As the reder will discover y experimenttion, it seems tht every grph hving four edges incident t ech vertex cn e 2 oriented. This cn e proved for finite grphs s follows. A very clssicl nd esily shown fct is tht every finite connected grph with n even numer of edges incident t ech vertex hs n Eulerin circuit, loop trversing ech edge exctly once. If there re four edges t ech vertex, then leling the edges of n Eulerin circuit lterntely nd produces leling with two nd two edges t ech vertex. The union of the edges is then collection

3 58 Chpter 1 The Fundmentl Group Some Covering Spces of S 1 S 1 ( 1) ( 2) 2 2,, 2 1,, ( 3) ( 4) ,,, ,,, ( 5) ( 6) ,,, 3 3,,, ( 7) ( 8) 4, 4,,, ,,( ),( ) 2, 2 ( 9) ( 10) 2 4,,, 2, n 2n 1, 2n 2n n Z ( 11) ( 12) n n n Z ( 13) ( 14), 1

4 Covering Spces Section of disjoint circles, s is the union of the edges. Choosing orienttions for ll these circles gives 2 orienttion. It is theorem in grph theory tht infinite grphs with four edges incident t ech vertex cn lso e 2 oriented; see Chpter 13 of [König 1990] for proof. There is lso generliztion to n oriented grphs, which re covering spces of the wedge sum of n circles. A simply-connected covering spce of X cn e constructed in the following wy. Strt with the open intervls ( 1, 1) in the coordinte xes of R 2. Next, for fixed numer λ, 0<λ< 1 / 2, for exmple λ = 1 / 3, djoin four open segments of length 2λ, t distnce λ from the ends of the previous segments nd perpendiculr to them, the new shorter segments eing isected y the older ones. For the third stge, dd perpendiculr open segments of length 2λ 2 t distnce λ 2 from the endpoints of ll the previous segments nd isected y them. The process is now repeted indefinitely, t the n th stge dding open segments of length 2λ n 1 t distnce λ n 1 from ll the previous endpoints. The union of ll these open segments is grph, with vertices the intersection points of horizontl nd verticl segments, nd edges the susegments etween djcent vertices. We lel ll the horizontl edges, oriented to the right, nd ll the verticl edges, oriented upwrd. This covering spce is clled the universl cover of X ecuse, s our generl theory will show, it is covering spce of every other connected covering spce of X. The covering spces (1) (14) in the tle re ll nonsimply-connected. Their fundmentl groups re free with ses represented y the loops specified y the listed words in nd, strting t the sepoint x 0 indicted y the hevily shded vertex. This cn e proved in ech cse y pplying vn Kmpen s theorem. One cn ( lso interpret the list of words s genertors of the imge sugroup p π1 ( X, x 0 ) ) in π 1 (X, x 0 ) =,. A generl fct we shll prove out covering spces is tht the induced mp p : π 1 ( X, x 0 ) π 1 (X, x 0 ) is lwys injective. Thus we hve the tfirst-glnce prdoxicl fct tht the free group on two genertors cn contin s sugroup free group on ny finite numer of genertors, or even on countly infinite set of genertors s in exmples (10) nd (11). Another generl fct we shll ( prove is tht the index of the sugroup p π1 ( X, x 0 ) ) in π 1 (X, x 0 ) is equl to the numer of sheets of the covering spce. Chnging the sepoint x 0 to nother point in p 1 (x 0 ) chnges the sugroup ( p π1 ( X, x 0 ) ) to conjugte sugroup in π 1 (X, x 0 ), with the conjugting element of π 1 (X, x 0 ) represented y ny loop tht is the projection of pth in X joining one sepoint to the other. For exmple, the covering spces (3) nd (4) differ only in the choice of sepoints, nd the corresponding sugroups of π 1 (X, x 0 ) differ y

5 60 Chpter 1 The Fundmentl Group conjugtion y. The min clssifiction theorem for covering spces sys tht y ssociting the ( sugroup p π1 ( X, x 0 ) ) to the covering spce p : X X, we otin one-to-one correspondence etween ll the different connected covering spces of X nd the conjugcy clsses of sugroups of π 1 (X, x 0 ). If one keeps trck of the sepoint vertex x 0 X, then this is one-to-one correspondence etween covering spces p : ( X, x 0 ) (X, x 0 ) nd ctul sugroups of π 1 (X, x 0 ), not just conjugcy clsses. Of course, for these sttements to mke sense one hs to hve precise notion of when two covering spces re the sme, or isomorphic. In the cse t hnd, n isomorphism etween covering spces of X is just grph isomorphism tht preserves the leling nd orienttions of edges. Thus the covering spces in (3) nd (4) re isomorphic, ut not y n isomorphism preserving sepoints, so the two sugroups of π 1 (X, x 0 ) corresponding to these covering spces re distinct ut conjugte. On the other hnd, the two covering spces in (5) nd (6) re not isomorphic, though the grphs re homeomorphic, so the corresponding sugroups of π 1 (X, x 0 ) re isomorphic ut not conjugte. Some of the covering spces (1) (14) re more symmetric thn others, where y symmetry we men n utomorphism of the grph preserving the leling nd orienttions. The most symmetric covering spces re those hving symmetries tking ny one vertex onto ny other. The exmples (1), (2), (5) (8), nd (11) re the ones with this property. We shll see tht covering spce of X hs mximl symmetry exctly when the corresponding sugroup of π 1 (X, x 0 ) is norml sugroup, nd in this cse the symmetries form group isomorphic to the quotient group of π 1 (X, x 0 ) y the norml sugroup. Since every group generted y two elements is quotient group of Z Z, this implies tht every two-genertor group is the symmetry group of some covering spce of X. After this extended preview-y-exmples let us return to generl theory y defining nturl generliztion of the symmetry group ide tht we just encountered. Group Actions Given group G nd spce X,nction of G on X is homomorphism ρ from G to the group Homeo(X) of ll homeomorphisms from X to itself. Thus to ech g G is ssocited homeomorphism ρ(g) : X X, which for nottionl simplicity we write s just g : X X. For ρ to e homomorphism mounts to requiring tht (g 1 g 2 )(x) = g 1 (g 2 (x)) for ll g 1,g 2 G nd x X, so the nottion g 1 g 2 (x) is unmiguous. If ρ is injective then it identifies G with sugroup of Homeo(X), nd in prctice not much is lost in ssuming ρ is n inclusion G Homeo(X) since in ny cse the sugroup ρ(g) Homeo(X) contins ll the topologicl informtion out the ction.

6 Covering Spces Section We shll e interested in ctions stisfying the following condition: ( ) Ech x X hs neighorhood U such tht ll the imges g(u) for vrying g G re disjoint. In other words, g 1 (U) g 2 (U) implies g 1 = g 2. Note tht it suffices to tke g 1 to e the identity since g 1 (U) g 2 (U) is equivlent to U g 1 1 g 2 (U). Thus we hve the equivlent condition tht U g(u) only when g is the identity. Given n ction of group G on spce X, we cn form spce X/G, the quotient spce of X in which ech point x is identified with ll its imges g(x) s g rnges over G. The points of X/G re thus the orits Gx ={g(x) g G } in X, nd X/G is clled the orit spce of the ction. If n ction of group G on spce X stisfies ( ), then the quotient mp p : X X/G, p(x) = Gx, is covering spce. For, given n open set U X s in condition ( ), the quotient mp p simply identifies ll the disjoint homeomorphic sets { g(u) g G } to single open set p(u) in X/G. By the definition of the quotient topology on X/G, p restricts to homeomorphism from g(u) onto p(u) for ech g G so we hve covering spce. In view of this fct, we shll cll n ction stisfying ( ) covering spce ction. This is not stndrd terminology, ut there does not seem to e universlly ccepted nme for ctions stisfying ( ). Sometimes these re clled properly discontinuous ctions, ut more often this rther unttrctive term mens something weker: Every point x X hs neighorhood U such tht U g(u) is nonempty for only finitely mny g G. Mny symmetry groups hve this proper discontinuity property without stisfying ( ), for exmple the group of symmetries of the fmilir tiling of R 2 y regulr hexgons. The reson why the ction of this group on R 2 fils to stisfy ( ) is tht there re fixed points: points x for which there is nontrivil element g G with g(x) = x. For exmple, the vertices of the hexgons re fixed y the 120 degree rottions out these points, nd the midpoints of edges re fixed y 180 degree rottions. An ction without fixed points is clled free ction. Thus for free ction of G on X, only the identity element of G fixes ny point of X. This is equivlent to requiring tht ll the imges g(x) of ech x X re distinct, or in other words g 1 (x) = g 2 (x) only when g 1 = g 2, since g 1 (x) = g 2 (x) is equivlent to g 1 1 g 2 (x) = x. Though condition ( ) implies freeness, the converse is not lwys true. An exmple is the ction of Z on S 1 in which genertor of Z cts y rottion through n ngle α tht is n irrtionl multiple of 2π. In this cse ech orit Zy is dense in S 1, so condition ( ) cnnot hold since it implies tht orits re discrete suspces. An exercise t the end of the section is to show tht for ctions on Husdorff spces, freeness plus proper discontinuity implies condition ( ). Note tht proper discontinuity is utomtic for ctions y finite group.

7 62 Chpter 1 The Fundmentl Group Exmple Let X e the closed orientle surfce of genus 11, n 11 hole torus s shown in the figure. This hs 5 fold rottionl symmetry, generted y rottion of ngle 2π/5. C 4 C 3 C 5 C 2 C 1 p C Thus we hve the cyclic group Z 5 cting on X, nd the condition ( ) is oviously stisfied. The quotient spce X/Z 5 is surfce of genus 3, otined from one of the five susurfces of X cut off y the circles C 1,,C 5 y identifying its two oundry circles C i nd C i+1 to form the circle C s shown. Thus we hve covering spce M 11 M 3 where M g denotes the closed orientle surfce of genus g. In prticulr, we see tht π 1 (M 3 ) contins the lrger group π 1 (M 11 ) s norml sugroup of index 5, with quotient Z 5. This exmple oviously generlizes y replcing the two holes in ech rm of M 11 y m holes nd the 5 fold symmetry y n fold symmetry. This gives covering spce M mn+1 M m+1. An exercise in 2.2 is to show y n Euler chrcteristic rgument tht if there is covering spce M g M h then g = mn + 1 nd h = m + 1 for some m nd n. Exmple If the closed orientle surfce M g of genus g is emedded in R 3 in the stndrd very symmetric wy centered t the origin then the mp x x restricts to homeomorphism τ of M g generting covering spce ction of Z 2 on M g. The figure t the right shows the cses g = 2, 3, with sphericl ul inserted in the middle of the genus 2 picture to mke the two cses look more like. When g = 0 the homeomorphism τ is the ntipodl mp of S 2 with orit spce RP 2. When g = 1 the mp τ rottes the longitudinl fctor of the torus S 1 S 1 nd reflects the meridionl fctor, so the orit spce is Klein ottle. For higher vlues of g one cn regrd M g s eing otined from sphere or torus y dding symmetric strings of n tori on either side, so the orit spce is projective plne or Klein ottle with n tori dded on. All closed nonorientle surfces rise this wy, so we see tht every closed nonorientle surfce hs two-sheeted covering spce tht is closed orientle surfce. Exmple Consider the grid in R 2 formed y the horizontl nd verticl lines through points in Z 2. Let us decorte this grid with rrows in either of the two wys shown in the figure, the difference etween the two cses eing tht in the second cse the horizontl rrows in djcent lines point in opposition directions. The group G consisting of ll symmetries of the first decorted grid is isomorphic to Z Z

8 Covering Spces Section since it consists of ll trnsltions (x, y) (x + m, y + n) for m, n Z. For the second grid the symmetry group G contins sugroup of trnsltions of the form (x, y) (x + m, y + 2n) for m, n Z, ut there re lso glide-reflection symmetries consisting of verticl trnsltion y n odd integer distnce followed y reflection cross verticl line, either verticl line of the grid or verticl line hlfwy etween two djcent grid lines. For oth decorted grids there re elements of G tking ny squre to ny other, ut only the identity element of G tkes squre to itself. The minimum distnce ny point is moved y nontrivil element of G is 1, which esily implies the covering spce condition ( ). The orit spce R 2 /G is the quotient spce of squre in the grid with opposite edges identified ccording to the rrows. Thus we see tht the fundmentl groups of the torus nd the Klein ottle re the symmetry groups G in the two cses. In the second cse the sugroup of G formed y the trnsltions hs index two, nd the orit spce for this sugroup is torus forming two-sheeted covering spce of the Klein ottle. Theorem For covering spce ction of group G on simply-connected spce X the fundmentl group π 1 (X/G) is isomorphic to G. The proof of the theorem depends on sic lifting property of ll covering spces. Recll from the proof of Theorem 1.7 tht for covering spce p : X X, lift of mp f : Y X is mp f : Y X such tht p f = f. The property we need is the homotopy lifting property, orcovering homotopy property, s it is sometimes clled: Proposition Given covering spce p : X X, homotopy f t : Y X, nd mp f 0 : Y X lifting f 0, then there exists unique homotopy f t : Y X of f 0 tht lifts f t. Proof: For the covering spce p : R S 1 this is property (c) in the proof of Theorem 1.7, nd the proof there pplies to ny covering spce. Tking Y to e point gives the pth lifting property for covering spce p : X X, which sys tht for ech pth f : I X nd ech lift x 0 of the strting point f(0) = x 0 there is unique pth f : I X lifting f strting t x 0. In prticulr, the uniqueness of lifts implies tht every lift of constnt pth is constnt, ut this could e deduced more simply from the fct tht p 1 (x 0 ) hs the discrete topology, y the definition of covering spce. Tking Y to e I, we see tht every homotopy f t of pth f 0 in X lifts to homotopy f t of ech lift f 0 of f 0. The lifted homotopy f t is homotopy of pths, fixing the endpoints, since s t vries ech endpoint of f t trces out pth lifting constnt pth, which must therefore e constnt. Proof of Theorem 1.34: We will construct n explicit isomorphism Φ : G π 1 (X/G), defined in the following wy. Choose sepoint x 0 X. Since X is simply-connected

9 64 Chpter 1 The Fundmentl Group there is unique homotopy clss of pths γ connecting x 0 to g(x 0 ) for ech g G. The composition of γ with the projection p : X X/G is then loop in X/G, nd we let Φ(g) e the homotopy clss of this loop. To see tht Φ is homomorphism, let γ 1 nd γ 2 e pths from x 0 to g 1 (x 0 ) nd g 2 (x 0 ). The composed pth γ 1 (g 1 γ 2 ) then goes from x 0 to g 1 g 2 (x 0 ). This pth projects to (pγ 1 ) (pγ 2 ),soφ(g 1 g 2 ) = Φ(g 1 )Φ(g 2 ). Using the sme nottion we cn lso see tht Φ is injective. For suppose Φ(g 1 ) = Φ(g 2 ), so the loops pγ 1 nd pγ 2 re homotopic. The homotopy lifting property then gives homotopy of γ 1 to pth which must e γ 2 since it strts t the sme point s γ 2 nd hs the sme projection to X/G, dt which determine γ 2 uniquely y the uniqueness prt of the pth lifting property. Thus γ 1 nd γ 2 re homotopic, nd in prticulr they hve the sme endpoint g 1 (x 0 ) = g 2 (x 0 ), which implies g 1 = g 2 since we hve covering spce ction. Surjectivity of Φ follows from the pth lifting property since ny loop in X/G t the sepoint p(x 0 ) lifts to pth γ in X strting t x 0 nd ending t point x 1 which must equl g(x 0 ) for some g G since the projection pγ is loop. Cyley Complexes Covering spces cn e used to descrie very clssicl method for viewing groups geometriclly s grphs. Recll from Corollry 1.28 how we ssocited to ech group presenttion G = g α r β 2dimensionl cell complex XG with π 1 (X G ) G y tking wedge-sum of circles, one for ech genertor g α, nd then ttching 2 cell for ech reltor r β. We cn construct cell complex X G with covering spce ction of G such tht X G /G = X G in the following wy. Let the vertices of X G e the elements of G themselves. Then, t ech vertex g G, insert n edge joining g to the vertex gg α for ech of the chosen genertors g α. The resulting grph is known s the Cyley grph of G with respect to the genertors g α. Ech reltion r β determines loop in the grph strting t ny vertex g nd pssing cross the edges corresponding to the successive letters of r β, returning in the end to the vertex g since gr β = g in G. After we ttch 2 cell for ech such loop, we hve cell complex X G clled the Cyley complex of G. The group G cts on X G y multipliction on the left. Thus, n element g G sends vertex g G to the vertex gg, nd the edge from g to g g α is sent to the edge from gg to gg g α. The ction extends to 2 cells in the ovious wy. This is clerly covering spce ction, nd the orit spce is just X G. The Cyley complex X G is in fct simply-connected. It is pth-connected since every element of G is product of g α s, so there is sequence of edges joining ech vertex to the identity vertex e. To see tht π 1 ( X G ) = 0, strt with loop t the sepoint vertex e. This loop is homotopic to loop in the 1 skeleton consisting of finite sequence of edges, corresponding to word w in the genertors g α nd their inverses. Since this sequence of edges is loop, the word w, viewed s n element

10 Covering Spces Section of G, is the identity, so s n element of the free group generted y the g α s the word w cn e written s product of conjugtes of the r β s nd their inverses. This mens tht the loop is homotopic in the 1 skeleton to product of loops ech of which consists of three prts: pth from the sepoint to vertex in the oundry of some 2 cell, followed y the oundry loop of this 2 cell, nd finishing with the inverse of the pth from the sepoint. Such loops re evidently nullhomotopic in X G, so the originl loop ws lso nullhomotopic. Let us look t some exmples of Cyley complexes. Exmple When G is the free group on two genertors nd, X G is S 1 S 1 nd X G is the Cyley grph of Z Z pictured t the right. The ction of on this grph is rightwrd shift long the centrl horizontl xis, while cts y n upwrd shift long the centrl verticl xis. The composition of these two shifts then tkes the vertex e to the vertex. Similrly, the ction of ny w Z Z tkes e to the vertex w e Exmple For G = Z 2 = 2, X G is RP 2 nd X G = S 2. More generlly, for Z n = n, X G is S 1 with disk ttched y the mp z z n nd X G consists of n disks D 1,,D n with their oundry circles identified. A genertor of Z n cts on this union of disks y sending D i to D i+1 vi 2π/n rottion, the suscript i eing tken mod n. The common oundry circle of the disks is rotted y 2π/n. Exmple The group G = Z Z with presenttion, 1 1 hs X G the torus S 1 S 1, nd X G is R 2 with vertices the integer lttice Z 2 R 2 nd edges the horizontl nd verticl segments etween these lttice points. This is the first figure in Exmple 1.33, with the ddition of lels on the horizontl edges, on the verticl edges, nd with the integer lttice point (m, n) leled y the element m n G. Exmple The Klein ottle is X G for G =, 1. Here X G is shown in the second figure of Exmple 1.33 with exctly the sme leling of vertices nd edges s in the preceding exmple of the torus. In prticulr, elements of G re gin uniquely representle s products m n. But with rrows in lternte horizontl rows going in opposite directions, the rule for multipliction of such products ecomes m n p q = m±p n+q, the ± eing + when n is even nd when n is odd. This formul cn e red off directly from the Cyley grph. If we modify this group y dding the new reltor 2 we otin the infinite dihedrl group D. The Cyley grph in this cse cn e drwn on n infinite cylinder, the quotient of the previous Cyley complex R 2 y verticl trnsltion y even integer

11 66 Chpter 1 The Fundmentl Group distnces. The Cyley complex for D is otined from this cylinder y inserting n infinite sequence of inscried spheres formed from pirs of 2 cells ttched long ech reltor cycle e 2 2 As further vrint, if we dd the reltor n s well s 2 we otin the finite dihedrl group D 2n of order 2n. The Cyley grph lies on torus, the quotient of the previous infinite cylinder y horizontl trnsltion 2 y n units. The Cyley complex hs the inscried 3 spheres nd lso n 2 cells ttched long ech of the two n cycles. The figure t the right shows 3 2 the cse n = 5. The usul ction of D 2n on regulr n gon is not free, nd the Cyley grph cn e 4 4 regrded s n exploded version of the n gon tht mkes the ction free. Vertices of the n gon re replced y circles in the Cyley grph, nd ech edge e of the n gon is replced y two prllel edges. Exmple If G = Z 2 Z 2 =, 2, 2 then the Cyley grph is union of n infinite sequence of circles ech tngent to its two neighors. e We otin X G from this grph y mking ech circle the equtor of 2 sphere, yielding n infinite sequence of tngent 2 spheres. Elements of the index-two norml sugroup Z Z 2 Z 2 generted y ct on X G s trnsltions y n even numer of units, while ech of the remining elements of Z 2 Z 2 cts s the ntipodl mp on one of the spheres nd flips the whole chin of spheres end-for-end out this sphere. The orit spce X G is RP 2 RP 2. The Cyley grph for Z 2 Z 2 my look little like the erlier Cyley grph for the infinite dihedrl group D, nd in fct these two groups re isomorphic, with the elements nd in D corresponding to nd in Z 2 Z 2. Geometriclly, it is ovious tht the symmetry groups of the two Cyley grphs re isomorphic. Thus we see tht two different presenttions for the sme group cn hve different Cyley grphs, ut not so different tht their symmetry groups re different. As nother exmple, the Cyley grph for the presenttion, 2, 2, () n of D 2n is

12 Covering Spces Section necklce of 2n circles, otined from the Cyley grph of Z 2 Z 2 y fctoring out trnsltion. It is not hrd to see the generliztion of the Z 2 Z 2 exmple to Z m Z n with the presenttion, m, n. In this cse X G consists of n infinite union of copies of the Cyley complexes for Z m nd Z n constructed in Exmple 1.37, rrnged in tree-like pttern. The cse of Z 2 Z 3 is pictured elow. 2 e 2 Groups Acting on Spheres Exmple 1.41: RP n. The ntipodl mp of S n, x x, genertes n ction of Z 2 on S n with orit spce RP n, rel projective n spce, s defined in Exmple 0.4. The ction is covering spce ction since ech open hemisphere in S n is disjoint from its ntipodl imge. As we sw in Proposition 1.14, S n is simply-connected if n 2, so from the covering spce S n RP n we deduce tht π 1 (RP n ) Z 2 for n 2. A genertor for π 1 (RP n ) is ny loop otined y projecting pth in S n connecting two ntipodl points. One cn see explicitly tht such loop γ hs order two in π 1 (RP n ) if n 2 since the composition γ γ lifts to loop in S n, nd this cn e homotoped to the trivil loop since π 1 (S n ) = 0, so the projection of this homotopy into RP n gives nullhomotopy of γ γ. One my sk whether there re other finite groups tht ct freely on S n, defining covering spces S n S n /G. We will show in Proposition 2.29 tht Z 2 is the only possiility when n is even, ut for odd n the question is much more difficult. It is esy to construct free ction of ny cyclic group Z m on S 2k 1, the ction generted y the rottion v e 2πi/m v of the unit sphere S 2k 1 in C k = R 2k. This ction is free

13 68 Chpter 1 The Fundmentl Group since n eqution v = e 2πil/m v with 0 <l<mimplies v = 0, ut 0 is not point of S 2k 1. The orit spce S 2k 1 /Z m is one of fmily of spces clled lens spces defined in Exmple There re lso noncyclic finite groups tht ct freely s rottions of S n for odd n>1. These ctions re clssified quite explicitly in [Wolf 1984]. Exmples in the simplest cse n = 3 cn e produced s follows. View R 4 s the quternion lger H. Multipliction of quternions stisfies = where denotes the usul Eucliden length of vector R 4. Thus if nd re unit vectors, so is, nd hence quternion multipliction defines mp S 3 S 3 S 3. This in fct mkes S 3 into group, though ssocitivity is ll we need now since ssocitivity implies tht ny sugroup G of S 3 cts on S 3 y left-multipliction, g(x) = gx. This ction is free since n eqution x = gx in the division lger H implies g = 1orx=0. As concrete exmple, G could e the fmilir quternion group Q 8 ={±1,±i, ±j,±k} from group theory. More generlly, for positive integer m, let Q 4m e the sugroup of S 3 generted y the two quternions = e πi/m nd = j. Thus hs order 2m nd hs order 4. The esily verified reltions m = 2 = 1 nd 1 = 1 imply tht the sugroup Z 2m generted y is norml nd of index 2 in Q 4m. Hence Q 4m is group of order 4m, clled the generlized quternion group. Another common nme for this group is the inry dihedrl group D 4m since its quotient y the sugroup {±1} is the ordinry dihedrl group D 2m of order 2m. Besides the groups Q 4m = D 4m there re just three other noncyclic finite sugroups of S 3 : the inry tetrhedrl, octhedrl, nd icoshedrl groups T 24, O 48, nd I 120, of orders indicted y the suscripts. These project two-to-one onto the groups of rottionl symmetries of regulr tetrhedron, octhedron (or cue), nd icoshedron (or dodechedron). In fct, it is not hrd to see tht the homomorphism S 3 SO(3) sending u S 3 H to the isometry v u 1 vu of R 3, viewing R 3 s the pure imginry quternions v = i + j + ck, is surjective with kernel {±1}. Then the groups D 4m, T 24, O 48, I 120 re the preimges in S3 of the groups of rottionl symmetries of regulr polygon or polyhedron in R 3. There re two conditions tht finite group G cting freely on S n must stisfy: () Every elin sugroup of G is cyclic. This is equivlent to sying tht G contins no sugroup Z p Z p with p prime. () G contins t most one element of order 2. A proof of () is sketched in n exercise for 4.2. For proof of () the originl source [Milnor 1957] is recommended reding. The groups stisfying () hve een completely clssified; see [Brown 1982], section VI.9, for detils. An exmple of group stisfying () ut not () is the dihedrl group D 2m for odd m>1. There is lso much more difficult converse: A finite group stisfying () nd () cts freely on S n for some n. References for this re [Mdsen, Thoms, & Wll 1976] nd [Dvis & Milgrm 1985]. There is lso lmost complete informtion out which

14 Covering Spces Section n s re possile for given group. One More Exmple Let us illustrte how one might uild simply-connected covering spce of given spce y gluing together simply-connected covering spces of vrious simpler pieces of the spce. Exmple For integers m, n 2, let X m,n e the quotient spce of cylinder S 1 I under the identifictions (z, 0) (e 2πi/m z, 0) nd (z, 1) (e 2πi/n z, 1). Let A X nd B X e the quotients of S 1 [0, 1 / 2 ] nd S 1 [ 1 / 2, 1], soand B re the mpping cylinders of z z m nd z z n, with A B = S 1. The simplest cse is m = n = 2, when A nd B re Möius nds nd X 2,2 is the Klein ottle. The complexes X m,n ppered erlier in this chpter in connection with torus knots, in Exmple The figure for Exmple 1.29 t the end of the preceding section shows wht A looks like in the typicl cse m = 3. We hve π 1 (A) Z, nd the universl cover à is homeomorphic to product C m R where C m is the grph tht is cone on m points, s shown in the figure to the right. The sitution for B is similr, nd B is homeomorphic to C n R. Now we ttempt to uild the universl cover X m,n from copies of à nd B. Strt with copy of Ã. Its oundry, the outer edges of its fins, consists of m copies of R. Along ech of these m oundry lines we ttch copy of B. Ech of these copies of B hs one of its oundry lines ttched to the initil copy of Ã, leving n 1 oundry lines free, nd we ttch new copy of à to ech of these free oundry lines. Thus we now hve m(n 1) + 1 copies of Ã. Ech of the newly ttched copies of à hs m 1 free oundry lines, nd to ech of these lines we ttch new copy of B. The process is now repeted d infinitim in the evident wy. Let X m,n e the resulting spce. The product structures à = C m R nd B = C n R give X m,n the structure of product T m,n R where T m,n is n infinite grph constructed y n inductive scheme just like the construction of X m,n. Thus T m,n is the union of sequence of finite sugrphs, ech otined from the preceding y ttching new copies of C m or C n. Ech of these finite sugrphs deformtion retrcts onto the preceding one. The infinite conctention of these deformtion retrctions, with the k th grph deformtion retrcting to the previous one during the time intervl [1/2 k, 1/2 k 1 ], gives deformtion retrction of T m,n onto the initil stge C m. Since C m is contrctile, this mens T m,n is contrctile, hence lso X m,n, which is the product T m,n R. In prticulr, X m,n is simply-connected.

15 70 Chpter 1 The Fundmentl Group The mp tht projects ech copy of à in X m,n to A nd ech copy of B to B is covering spce. To define this mp precisely, choose point x 0 S 1, nd then the imge of the line segment {x 0 } I in X m,n meets A in line segment whose preimge in à consists of n infinite numer of line segments, ppering in the erlier figure s the horizontl segments spirling round the centrl verticl xis. The picture in B is similr, nd when we glue together ll the copies of à nd B to form X m,n, we do so in such wy tht these horizontl segments lwys line up exctly. This decomposes X m,n into infinitely mny rectngles, ech formed from rectngle in n à nd rectngle in B. The covering projection X m,n X m,n is the quotient mp tht identifies ll these rectngles. The rectngles define cell structure on X m,n lifting cell structure on X m,n with two vertices, three edges, nd one 2 cell. Suppose we orient nd lel the three edges of X m,n nd lift these orienttions nd lels to the edges of X m,n. The symmetries of X m,n preserving the orienttions nd lels of edges form group G m,n. The ction of this group on X m,n is covering spce ction, nd the quotient X m,n /G m,n is just X m,n since for ny two rectngles in X m,n there is n element of G m,n tking one rectngle to the other. By Theorem 1.34 the group G m,n is therefore isomorphic to π 1 (X m,n ). From vn Kmpen s theorem pplied to the decomposition of X m,n into the two mpping cylinders we hve the presenttion, m n for this group G m,n = π 1 (X m,n ). The element for exmple cts on X m,n s screw motion out n xis tht is verticl line {v } R with v vertex of T m,n, nd cts similrly for n djcent vertex v. Since the ction of G m,n on X m,n preserves the cell structure, it lso preserves the product structure T m,n R. This mens tht there re ctions of G m,n on T m,n nd R such tht the ction on the product X m,n = T m,n R is the digonl ction g(x,y) = ( g(x), g(y) ) for g G m,n. If we mke the rectngles of unit height in the R coordinte, then the element m = n cts on R s unit trnsltion, while cts y 1 / m trnsltion nd y 1 / n trnsltion. The trnsltion ctions of nd on R generte group of trnsltions of R tht is infinite cyclic, generted y trnsltion y the reciprocl of the lest common multiple of m nd n. The ction of G m,n on T m,n hs kernel consisting of the powers of the element m = n. This infinite cyclic sugroup is precisely the center of G m,n, s we sw in Exmple There is n induced ction of the quotient group Z m Z n on T m,n, ut this is not free ction since the elements nd nd ll their conjugtes fix vertices of T m,n. On the other hnd, if we restrict the ction of G m,n on T m,n to the kernel K of the mp G m,n Z given y the ction of G m,n on the R fctor of X m,n, then we do otin free ction of K on T m,n. Since this ction tkes vertices to vertices nd edges to edges, it is covering spce ction, so K is free group, the

16 Covering Spces Section fundmentl group of the grph T m,n /K. An exercise t the end of the section is to determine T m,n /K explicitly nd compute the numer of genertors of K. The Clssifiction of Covering Spces Our ojective now is to develop the necessry tools to clssify ll the different covering spces of fixed pth-connected spce X. The min thrust of the clssifiction will e the Glois correspondence etween connected covering spces of X nd sugroups of π 1 (X), ut when this is finished we will lso descrie different method of clssifiction tht includes disconnected covering spces s well. The Glois correspondence will e clssifiction up to isomorphism, where this term hs its most nturl definition: An isomorphism etween covering spces p 1 : X 1 X nd p 2 : X 2 X is homeomorphism f : X 1 X 2 such tht p 1 = p 2 f. This condition mens exctly tht f preserves the covering spce structures, tking p 1 1 (x) to p 1 2 (x) for ech x X. The inverse f 1 is then lso n isomorphism, nd the composition of two isomorphisms is n isomorphism, so we hve n equivlence reltion. The correspondence etween isomorphism clsses of connected covering spces of X nd sugroups of π 1 (X) will e given y the function Γ sending covering spce ( p : ( X, x 0 ) (X, x 0 ) to the sugroup Γ ( X, x 0 ) = p π1 ( X, x 0 ) ) of π 1 (X, x 0 ). Let us mke few preliminry oservtions out the sugroups Γ ( X, x 0 ). Proposition The mp p : π 1 ( X, x 0 ) π 1 (X, x 0 ) induced y covering spce p : ( X, x 0 ) (X, x 0 ) is injective. The imge sugroup Γ ( X, x 0 ) in π 1 (X, x 0 ) consists of the homotopy clsses of loops in X sed t x 0 whose lifts to X strting t x 0 re loops. Proof: An element of the kernel of p homotopy γ t : I X of γ 0 = p γ 0 to the trivil loop γ 1. This homotopy lifts to homotopy of loops γ t strting with γ 0 nd ending with constnt loop since the only lift of constnt loop is constnt loop. Hence [ γ 0 ] = 0inπ 1 ( X, x 0 ) nd p is injective. For the second sttement of the proposition, loops t x 0 lifting to loops t x 0 certinly represent elements of the imge of p : π 1 ( X, x 0 ) π 1 (X, x 0 ). Conversely, loop representing n element of the imge of p is homotopic to loop with lift to loop t x 0, so y lifting the homotopy we see tht the originl loop must itself lift to loop t x 0. is represented y loop γ 0 : I X with Proposition The numer of sheets of covering spce p : ( X, x 0 ) (X, x 0 ) with X nd X pth-connected equls the index of Γ ( X, x 0 ) in π 1 (X, x 0 ). Proof: For loop γ in X sed t x 0, let γ e its lift to X strting t x 0. A product η γ with [η] H = Γ ( X, x 0 ) hs the lift η γ ending t the sme point s γ since η

17 72 Chpter 1 The Fundmentl Group is loop. Thus we my define function Φ from cosets H[γ] to p 1 (x 0 ) y sending H[γ] to γ(1). The pth-connectedness of X implies tht Φ is surjective since x 0 cn e joined to ny point in p 1 (x 0 ) y pth γ projecting to loop γ t x 0. To see tht Φ is injective, oserve tht Φ(H[γ 1 ]) = Φ(H[γ 2 ]) implies tht γ 1 γ 2 lifts to loop in X sed t x 0,so[γ 1 ][γ 2 ] 1 H nd hence H[γ 1 ] = H[γ 2 ]. Now we descrie how the sugroup Γ ( X, x 0 ) depends on the choice of x 0. Lemm Given covering spce p : ( X, x 0 ) (X, x 0 ), let γ e loop in X t x 0 representing clss g π 1 (X, x 0 ) nd lifting to pth γ strting t x 0 nd ending t point x 1 p 1 (x 0 ). Then Γ ( X, x 0 ) = gγ( X, x 1 )g 1. Thus ny sugroup of π 1 (X, x 0 ) conjugte to Γ ( X, x 0 ) corresponds to the sme covering spce X with different choice of sepoint x 0 in p 1 (x 0 ). Conversely, if X is pth-connected we cn choose γ to e the projection of pth joining ny two choices of sepoint in p 1 (x 0 ) to deduce tht the conjugcy clss of Γ ( X, x 0 ) is independent of the choice of x 0 within p 1 (x 0 ). Proof: Represent n element of Γ ( X, x 1 ) y loop η lifting to loop η t x 1. Then γ η γ is loop t x 0 lifting γηγ,sogγ( X, x 1 )g 1 Γ ( X, x 0 ). The opposite inclusion is equivlent to g 1 Γ ( X, x 0 )g Γ ( X, x 1 ) which holds y the sme resoning, replcing γ with γ nd interchnging x 0 nd x 1. To proceed further with the clssifiction of covering spces we need two sic propositions on the existence nd uniqueness of lifts of generl mps. For the existence question n nswer is provided y the following lifting criterion: Proposition Suppose p : ( X, x 0 ) (X, x 0 ) is covering spce. Then mp f : (Y, y 0 ) (X, x 0 ) whose domin Y is pth-connected nd loclly pth-connected hs lift f ( : (Y, y 0 ) ( X, x 0 ) iff f π1 (Y, y 0 ) ) ( p π1 ( X, x 0 ) ). Proof: The only if sttement is ovious since f = p f. For the converse, let y Y nd let γ e pth in Y from y 0 to y. The pth fγ in X strting t x 0 hs unique lift fγ strting t x 0. Define f(y) = fγ(1). To show this is well-defined, independent of the choice of γ, let γ e nother pth from y 0 to y. Then (f γ ) (fγ) is f loop t x 0 representing n element of ( f π1 (Y, y 0 ) ) ( p π1 ( X, x 0 ) ) γ f. y y By Proposition 1.43 the loop 0 γ (f γ ) (fγ) lifts to loop t x 0 fγ f ( y) f γ p f γ f ( y ) x0 f γ x 0. By the uniqueness of lifted pths, the first hlf of this lift is fγ nd the second

18 Covering Spces Section hlf is fγ trversed ckwrds, with the common midpoint fγ(1) = fγ (1). This shows tht f is well-defined. To see tht f is continuous, let U X e n open neighorhood of f(y) hving lift Ũ X contining f(y) such tht p : Ũ U is homeomorphism. Choose pth-connected open neighorhood V of y with f(v) U. For pths from y 0 to points y V we cn tke fixed pth γ from y 0 to y followed y pths η in V from y to the points y. Then the pths (f γ) (f η) in X hve lifts ( fγ) ( fη) where fη = p 1 fη nd p 1 : U Ũ is the inverse of p : Ũ U. Thus f(v) Ũ nd f V = p 1 f, hence f is continuous t y. An exmple showing the necessity of the locl pth-connectedness ssumption on Y is descried in Exercise 7 t the end of this section. Next we hve the unique lifting property: Proposition Given covering spce p : X X nd mp f : Y X with two lifts f 1, f 2 : Y X tht gree t one point of Y, then if Y is connected, these two lifts must gree on ll of Y. Proof: For point y Y, let U e n open neighorhood of f(y) in X for which p 1 (U) is disjoint union of open sets Ũ α ech mpped homeomorphiclly to U y p, nd let Ũ 1 nd Ũ 2 e the Ũ α s contining f 1 (y) nd f 2 (y), respectively. By continuity of f 1 nd f 2 there is neighorhood N of y mpped into Ũ 1 y f 1 nd into Ũ 2 y f 2. If f 1 (y) f 2 (y) then Ũ 1 Ũ 2, hence Ũ 1 Ũ 2 = nd f 1 f 2 throughout the neighorhood N. On the other hnd, if f 1 (y) = f 2 (y) then Ũ 1 = Ũ 2 so f 1 = f 2 on N since p f 1 = p f 2 nd p is injective on Ũ 1 = Ũ 2. Thus the set of points where f 1 nd f 2 gree is oth open nd closed in Y, so it must e ll of Y if Y is connected. Here is the uniqueness hlf of the Glois correspondence: Theorem If X is pth-connected nd loclly pth-connected, then two pthconnected covering spces p 1 : X 1 X nd p 2 : X 2 X re isomorphic vi n isomorphism f : X 1 X 2 tking sepoint x 1 p 1 1 (x 0 ) to sepoint x 2 p 1 2 (x 0 ) iff Γ ( X 1, x 1 ) = Γ ( X 2, x 2 ). The covering spces X 1 nd X 2 re isomorphic without regrd to sepoints iff Γ ( X 1, x 1 ) nd Γ ( X 2, x 2 ) re conjugte sugroups of π 1 (X, x 0 ). Proof: If there is n isomorphism f : ( X 1, x 1 ) ( X 2, x 2 ), then from the two reltions p 1 = p 2 f nd p 2 = p 1 f 1 it follows tht Γ ( X 1, x 1 ) = Γ ( X 2, x 2 ). Conversely, suppose tht Γ ( X 1, x 1 ) = Γ ( X 2, x 2 ). By the lifting criterion, we my lift p 1 to mp p 1 : ( X 1, x 1 ) ( X 2, x 2 ) with p 2 p 1 = p 1. Similrly, we otin p 2 : ( X 2, x 2 ) ( X 1, x 1 ) with p 1 p 2 = p 2. Then y the unique lifting property, p 1 p 2 = 11 nd p 2 p 1 = 11 since these composed lifts fix the sepoints. Thus p 1 nd p 2 re inverse isomorphisms. The lst sttement follows immeditely using Lemm 1.45.

19 74 Chpter 1 The Fundmentl Group It remins to discuss the question of whether there exist covering spces of given pth-connected, loclly pth-connected spce X relizing ll sugroups of π 1 (X, x 0 ). There is specil cse tht is esily delt with, tht X is the orit spce X/G for covering spce ction of group G on simply-connected spce X. From Theorem 1.34 we know tht G is isomorphic to π 1 (X, x 0 ) y the mp sending g G to the imge in X/G of pth in X from the sepoint x 0 to g( x 0 ). A sugroup of π 1 (X, x 0 ) corresponds to sugroup H of G, nd the covering spce X X/G fctors s the composition of two mps X q X/H p X/G, ech of which is oviously covering spce. We hve π 1 ( X/H,q( x 0 )) H, nd in fct Γ ( X/H,q( x 0 )) is the sugroup of π 1 (X, x 0 ) corresponding to H under the isomorphism G π 1 (X, x 0 ) since loop in X t x 0 lifts to loop in X/H t q( x 0 ) iff its lift to X strting t x 0 ends t point h( x 0 ) for some h H. Thus we otin ll possile pth-connected covering spces of X/G, up to isomorphism, s orit spces X/H for sugroups H G. Our strtegy for generl X will e to show tht this specil cse is relly the generl cse. The esier prt will e to show tht if X hs simply-connected covering spce X then there is covering spce ction of π 1 (X, x 0 ) on X with orit spce just X itself. For n ritrry covering spce p : X X one cn consider the isomorphisms from this covering spce to itself. These re clled deck trnsformtions or covering trnsformtions. They form group G( X) under composition. For exmple, for the covering spce R S 1 projecting verticl helix onto circle, the deck trnsformtions re the verticl trnsltions tking the helix onto itself, so G( X) Z in this cse. Lemm For covering spce p : X X with X nd X pth-connected nd loclly pth-connected, the ction of G( X) on X is covering spce ction. Proof: By locl pth-connectedness, ech point in X hs pth-connected open neighorhood U such tht p 1 (U) is disjoint union of copies of U projecting homeomorphiclly to U y p, so these re the pth-components of p 1 (U). Any deck trnsformtion just permutes these pth-components. The result now follows from the fct tht the ction of G( X) is free, since deck trnsformtion of pth-connected covering spce is uniquely determined y where it sends point, y the unique lifting property. Now let us specilize the lemm to the cse tht X is simply-connected. By the lifting criterion there exists deck trnsformtion tking the sepoint x 0 to ny other point in p 1 (x 0 ). This mens tht the orit spce X/G( X) is just X, or more precisely tht p induces homeomorphism from X/G( X) onto X. in prticulr G( X) is isomorphic to π 1 (X, x 0 ). Thus we hve shown:

20 Covering Spces Section Proposition If pth-connected, loclly pth-connected spce X hs simplyconnected covering spce, then every sugroup of π 1 (X, x 0 ) is relized s Γ ( X, x 0 ) for some covering spce ( X, x 0 ) (X, x 0 ). A consequence of the preceding constructions is tht simply-connected covering spce of pth-connected, loclly pth-connected spce X is covering spce of every other pth-connected covering spce of X. This justifies clling simply-connected covering spce of X universl cover. It is unique up to isomorphism, so one cn in fct sy the universl cover. More generlly, there is prtil ordering on the vrious pth-connected covering spces of X, ccording to which ones cover which others. This corresponds to the prtil ordering y inclusion of the corresponding sugroups of π 1 (X), or conjugcy clsses of sugroups if sepoints re ignored. There remins the question of when spce X hs simply-connected covering spce. A necessry condition is the following: Ech point x X hs neighorhood U such tht the inclusion-induced mp π 1 (U, x) π 1 (X, x) is trivil. One sys X is semiloclly simply-connected if this holds. To see the necessity of this condition, suppose p : X X is covering spce with X simply-connected. Every point x X hs neighorhood U hving lift Ũ X projecting homeomorphiclly to U y p. Ech loop in U lifts to loop in Ũ, nd the lifted loop is nullhomotopic in X since π 1 ( X) = 0. So, composing this nullhomotopy with p, the originl loop in U is nullhomotopic in X. A loclly simply-connected spce is certinly semiloclly simply-connected. For exmple, CW complexes hve the much stronger property of eing loclly contrctile, s we show in the Appendix. An exmple of spce tht is not semiloclly simplyconnected is the shrinking wedge of circles, the suspce X R 2 consisting of the circles of rdius 1 / n centered t the point ( 1 / n, 0) for n = 1, 2,, introduced in Exmple On the other hnd, the cone CX = (X I)/(X {0}) is semiloclly simplyconnected since it is contrctile, ut it is not loclly simply-connected. Proposition A spce tht is pth-connected nd loclly pth-connected hs simply-connected covering covering spce iff it is semiloclly simply-connected. Proof: It remins to prove the if impliction. To motivte the construction, suppose p : ( X, x 0 ) (X, x 0 ) is simply-connected covering spce. Ech point x X cn then e joined to x 0 y unique homotopy clss of pths, y Proposition 1.6, so we cn view points of X s homotopy clsses of pths strting t x 0. The dvntge of this is tht, y the homotopy lifting property, homotopy clsses of pths in X strting t x 0 re the sme s homotopy clsses of pths in X strting t x 0. This gives wy of descriing X purely in terms of X. Given pth-connected, loclly pth-connected, semiloclly simply-connected spce X with sepoint x 0 X, we re therefore led to define X = { } [γ] γ is pth in X strting t x 0

21 76 Chpter 1 The Fundmentl Group where, s usul, [γ] denotes the homotopy clss of γ with respect to homotopies tht fix the endpoints γ(0) nd γ(1). The function p : X X sending [γ] to γ(1) is then well-defined. Since X is pth-connected, the endpoint γ(1) cn e ny point of X,sop is surjective. Before we define topology on X we mke few preliminry oservtions. Let U e the collection of pth-connected open sets U X such tht π 1 (U) π 1 (X) is trivil. Note tht if the mp π 1 (U) π 1 (X) is trivil for one choice of sepoint in U, it is trivil for ll choices of sepoint since U is pth-connected. A pth-connected open suset V U U is lso in U since the composition π 1 (V ) π 1 (U) π 1 (X) will lso e trivil. It follows tht U is sis for the topology on X if X is loclly pth-connected nd semiloclly simply-connected. Given set U U nd pth γ in X from x 0 to point in U, let U [γ] = { [γ η] η is pth in U with η(0) = γ(1) } As the nottion indictes, U [γ] depends only on the homotopy clss [γ]. Oserve tht p : U [γ] U is surjective since U is pth-connected nd injective since different choices of η joining γ(1) to fixed x U re ll homotopic in X, the mp π 1 (U) π 1 (X) eing trivil. Another property is ( ) U [γ] = U [γ ] if [γ ] U [γ]. For if γ = γ η then elements of U [γ ] hve the form [γ η µ] nd hence lie in U [γ], while elements of U [γ] hve the form [γ µ] = [γ η η µ] = [γ η µ] nd hence lie in U [γ ]. This cn e used to show tht the sets U [γ] form sis for topology on X. For if we re given two such sets U [γ], V [γ ] nd n element [γ ] U [γ] V [γ ], we hve U [γ] = U [γ ] nd V [γ ] = V [γ ] y ( ). SoifW Uis contined in U V nd contins γ (1) then W [γ ] U [γ ] V [γ ] nd [γ ] W [γ ]. The ijection p : U [γ] U is homeomorphism since it gives ijection etween the susets V [γ ] U [γ] nd the sets V U contined in U. Nmely, in one direction we hve p(v [γ ] ) = V nd in the other direction we hve p 1 (V ) U [γ] = V [γ ] for ny [γ ] U [γ] with endpoint in V, since V [γ ] U [γ ] = U [γ] nd V [γ ] mps onto V y the ijection p. The preceding prgrph implies tht p : X X is continuous. We cn lso deduce tht this is covering spce since for fixed U U, the sets U [γ] for vrying [γ] prtition p 1 (U) ecuse if [γ ] U [γ] U [γ ] then U [γ] = U [γ ] = U [γ ] y ( ). It remins only to show tht X is simply-connected. For point [γ] X let γ t e the pth in X otined y restricting γ to the intervl [0,t]. Then the function t [γ t ] is pth in X lifting γ tht strts t [x 0 ], the homotopy clss of the constnt pth t x 0, nd ends t [γ]. Since [γ] ws n ritrry point in X, this shows tht X is pth-connected. To show tht π 1 ( X,[x 0 ]) = 0 it suffices to show tht the imge of this group under p is trivil since p is injective. Elements in the imge of p re represented y loops γ t x 0 tht lift to loops in X t [x 0 ]. We hve oserved tht