Notes on Algebraic Topology
1.5 Theorem of Van Kampen
The theorem of Van Kampen(28) allows the computation of the fundamental group of a space in terms of the fundamental groups of the open subsets of a suitable cover.
Let X be an arcwise connected topological space, (Λ, ) an ordered set, U : λ Λ an ≤ { λ ∈ } open cover of X such that:
(a) all Uλ are arcwise connected; (b) λ µ if and only if U U ; ≤ λ ⊆ µ (c) the family U : λ Λ is stable under finite intersections; { λ ∈ }
(d) there exists x0 λ Λ Uλ. ∈ ∈ Let us denote by ι :U X and ι : U U the canonical inclusions (where λ λ −→ λ,µ λ −→ µ λ µ), and write for short π (U ) instead of π (U ,x ). We then have morphisms ≤ 1 λ 1 λ 0 ι : π (U ) π (X) and ι : π (U ) π (U )(whereλ µ), which simply λ# 1 λ −→ 1 λ,µ# 1 λ −→ 1 µ ≤ associate to the class of a loop the class of the same loop viewed in the larger space. In particular, one has an inductive system π (U ),ι : λ, µ Λ (see Appendix A.1). { 1 λ λ,µ ∈ }
Let M Λsuch that X = λ M Uλ. ⊂ ∈ Proposition 1.5.1. π (X)is generated by ι (π (U )) : λ M . 1 { λ# 1 λ ∈ } 1 Proof. Let γ : I X be a loop at x , δ>0 the Lebesgue number relative to the cover γ− (Uλ):λ M −→ 0 { ∈ } of I,0=t0 Corollary 1.5.2. If the open subsets U (λ M) are simply connected, such is also X. λ ∈ n Example. S is simply connected for n 2. Namely, let N = en+1 be the North pole, S = N the n ≥ n − South pole, and set U = S N and V = S S :notingthatU V is arcwise connected and that \{ } \{ } ∩ both U and V are simply connected, just apply Corollary 1.5.2. Note that this argument does not apply 1 for S (in that case U V is not arcwise connected). ∩ In general one has the following result (see Appendix A.1 for the notion of “inductive limit” of an inductive system). Theorem 1.5.3. (Van Kampen) In the category Groups it holds π1(X)=limπ1(Uλ). λ Λ −→∈ (28)The result has been proved independently also by Karl Seifert in the 30s of last century; in fact, it is often referred to as “Seifert - Van Kampen theorem”. Corrado Marastoni 22 Notes on Algebraic Topology Proof. Consider a group L and a family of morphisms ψλ : π1(Uλ) L such that ψλ = ψµ ιλ,µ# for λ µ, −→ ◦ ≤ and let us see if there exists a unique morphism ψ : π1(X) L such that ψλ = ψ ιλ# for any λ Λ. So let −→ ◦ ∈ [γ] π1(X): by Proposition 1.5.1 we may write [γ]=ιλ #([γ1]) ιλ #([γk]) with [γj ] π1(Uλ ). If ψ ∈ 1 ····· k ∈ j exists, it must be necessarily unique because ψ([γ]) = ψ(ιλ #[γ1]) ψ(ιλ #[γk]) = ψλ ([γ1]) ψλ ([γk]) 1 ··· k 1 ··· k (the products are in L). We are left with showing that this is actually a well-posed definition for ψ,i.e.thatif [c ]=ι ([σ ]) ι ([σ ]) then also ψ ([σ ]) ψ ([σ ]) = e (where e is the identity element of L). x0 λ1 # 1 λk # k λ1 1 λk k ····· j 1 j ··· Let σ = σ1 σk be in X (hence, if t [ −k , k ]itholdsσ(t)=σj (kt (j 1))), and let h : I I X be a ····· ∈ − − 1× −→ homotopy rel ∂I between σ and c .Letε>0 the Lebesgue number relative to the cover h− (U ):λ Λ x0 λ √2 i 1 i j 1{ j ∈ } of I I,andletr N be such that kr <ε.Hence,settingRi,j = k−r , kr k−r , kr I I (for × r ∈ ×i j ⊂ × i, j =1,...,k ), there exists λi,j Λsuchthath(Ri,j ) Uλi,j .Letvi,j = kr , kr (hence Ri,j is the r ∈ ⊂ square with side k− and opposed vertices vi 1,j 1 and vi,j ), Uµ(i,j) the intersection of the (one, two or − − t+(i 1) j four) Uλ such that vi,j Rl,m,andγi,j apathinU from x to h(vi,j ). Let αi,j (t)=h − , l,m ∈ µ(i,j) 0 kr kr i t+(j 1) (path from h(vi 1,j )toh(vi,j )) and βi,j (t)=h r , r− (path from h(vi,j 1)toh(vi,j )): note that − k k − α r 1 α r 1 =[σ ] (for m =1 ,...,k)andthat α r (t)=β (t)=β r (t) x (for (m 1)k − +1,0 mk − ,0 m i,k 0,j k ,j 0 − ····· r ≡ t I and i, j =1,...,k ). From the equality [αi,j 1 βi,j ]=[βi 1,j αi,j ] one gets (by inserting the paths ∈ − · − · 1 1 γi,j and their inverses to base at x ) the relations (γi 1,j 1 αi,j 1) γ− (γi,j 1 βi,j ) γ− = 0 − − − i,j 1 − i,j 1 1 · · − · · · (γi 1,j 1 βi 1,j ) γ− (γi 1,j αi,j ) γ− in the group π1(Uλ ). Applying ψλ and setting − − − i 1,j − i,j i,j i,j · · − · 1 · · 1 ai,j = ψλ (γi 1,j αi,j ) γ− and bi,j = ψµ(i,j) (γi,j 1 βi,j ) γ− ,onethenhastheequality i,j − · · i,j − · · i,j ai,j 1 bi,j = bi 1,j ai,j in L.Knowingthata1,kr akr ,kr = e and that b0,j = bkr ,j = e (for any j = − r − ··· 1,...,k ), one has e = a1,kr akr ,kr =(b0,kr a1,kr )a2,kr akr ,kr = a1,kr 1(b1,kr a2,kr ) akr ,kr = ··· ··· − ··· = a1,kr 1 akr ,kr 1; by repeating the procedure one obtains a1,0 akr ,0 = e, as required. ··· − ··· − ··· Now the problem is to understand what lim π1(Uλ)seemslike. λ Λ −→∈ In general, the free product λ Λ Gλ of a family of groups Gλ : λ Λ is the group ∈ { ∈ } formed by finite “words” a ∗a constructed with “letters” a G (j =1,...,k; k 1), 1 ··· k j ∈ λj ≥ where any letter is different from the identity element in the respective group and where two adjacent letters must belong to different groups (one often says “reduced letters”); also the “empty word” is considered to be an element. The operation is given by the natural juxtaposition (a a ) (b b )=a a b b where, in the case a and 1 ··· k · 1 ··· h 1 ··· k 1 ··· h k b1 belong to a same group, the expression “akb1” should be replaced by their product in that group (and possibly removed if akb1 is the identity, causing then the same procedure for ak 1b2, and so on); the identity element is clearly the empty word. − Example. The free product of any number of copies of Z is called free group, in the sense that there is one generator for each copy of Z and the elements of the group are words formed by powers of these generators. For example, Z Z is formed by the words r1s1 rksk where all rj ’s and sj ’s are integer (the ∗ ··· rj ’s are meant to belong to the first copy of Z,andthesj ’s to the second); or also, in abstract notation, by ak1 bh1 akr bhr where a and b denote the two generators and the exponents are integers. ··· Note that for any µ Λthere is a natural monomorphism Gµ λ Λ Gλ; in fact one sees ∈ −→ ∈ that the free product λ Λ Gλ is the inductive limit in Groups of∗ the system Gλ : λ Λ ∈ { ∈ } with trivial preorder, i.e.∗ without considering morphisms(29). More generally, if morphisms f : G G are given for some pairs (λ, µ)withf f = f whenever defined, λ,µ λ −→ µ λ,µ ◦ µ,ν λ,ν then to obtain the inductive limit of the system G ,f : λ, µ Λ one must quotient { λ λ,µ ∈ } out the previous free product λ Λ Gλ by its normal subgroup N generated by all the elements of type f (a)f (a)∗1 ∈for a G whenever the morphisms f and f are λ,µ λ,ν − ∈ λ λ,µ λ,ν (29) Namely, given a family of morphisms ψλ : Gλ L, the definition (necessary, hence unique) −→ ψ(a1 ak)=ψλ (a1) ψλ (ak) is a morphism. ··· 1 ··· k Corrado Marastoni 23 Notes on Algebraic Topology defined(30): this procedure is said to be an amalgamation of the free product with respect to the given morphisms fλ,µ. When applying the above notions to the framework of Van Kampen, the groups are π1(Uλ) and the morphisms are the maps ι : π (U ) π (U ), which send the class of a loop λ,µ# 1 λ −→ 1 µ in the small open subset Uλ to the class of same loop viewed in the larger open subset Uµ: by Proposition 1.5.1 and the subsequent discussion it is then enough to consider the free product of the groups π1(Uλ)’s of a selected family of open subsets Uλ with indices λ M which cover X and are not contained in each other, and then to amalgamate only ∈ with respect to the double intersections U U for λ, µ M.(31) λ ∩ µ ∈ Example. (Wedge sums) Consider a family of pointed spaces (Xλ,xλ) for λ M,andletX = λ M Xλ ∈ ∈ be their wedge sum (see p. 9). For each λ M let Vλ be a open subset of xλ in Xλ which has xλ ∈ as deformation retract, and set Uλ = Xλ ( µ=λ Vµ). It is clear that the Uλ’s cover X; moreover, any ∨ intersection of two or more of them is always λ M Vλ, which is arcwise connected and, since it deformation- ∈ retracts to the base point, has trivial fundamental group and hence causes no effective amalgamation. Finally, since each Uλ deformation-retracts to the corresponding Xλ, by Van Kampen theorem it follows that π1(X) λ M π1(Xλ). ∗ ∈ The Theorem of Van Kampen is used mainly in the case of two open subsets X = U V , with U, V and U V arcwise connected: in that case π (X)willbetheinductivelimit∪ ∩ 1 of the system π1(U V ),π1(U),π1(V ); ,ιU V,U,,ιU V,V , and its universal property is expressed by the{ following∩ diagram: ∩ ∩ } (1.4) L ψU !ψ ∃ π1(U) ι π1(X) U # ψU V ψV ∩ ιU V # ιU V,U # ∩ ιV # ∩ ιU V,V # π (U V ) ∩ π (V ) 1 ∩ 1 One usually denotes the free product of two groups G and H by G H and, given another ∗ group K and morphisms f : K G and g : K H, the free product of G and H −→ −→ amalgamated on K by G H.(32) By what has been said we get: ∗K (30) Namely, if in the previous notation ψλ αλ = ψµ αµ, for any λ, µ ΛthenN ker(ψ)andhenceψ ◦ ◦ ∈ ⊂ factorizes uniquely through the quotient λ Λ Gλ/N . (31) ∈ Namely, a typical element of λ Λ ∗π1(Uλ)is[γ1] [γk] where γj is a loop in Uλ but the class ∈ ··· j [γj ]istakenasloopinX (in fact,∗ we should have written more precisely iλj #([γj ]U )): hence, by the λj compatibility of the various morphisms of type i#,theclass[γj ] can be thought as coming from some Uλ with λ M,andthisshowsthat λ Λ π1(Uλ) λ M π1(Uλ). Similar considerations hold for the ∈ ∈ 1 ∈ amalgamation: the objects iλ,µ#([γ]) ∗iλ,ν #([γ])− coming∗ from intersections of three or more Uλ with · λ M can be thought as having already come from some double intersection. (32∈) In the language of categories, the group G K H is usually called the pushout of the morphisms ∗ f : K G and g : K H.NotethatthenotationG K H does not show explicitly what are f and g, −→ −→ ∗ but of course it is important to take them into account. Corrado Marastoni 24 Notes on Algebraic Topology Corollary 1.5.4. Let X be an arcwise connected topological space, X = U V an open ∪ cover with U, V and U V arcwise connected open subsets. Then ∩ π1(X) π1(U) π (U V ) π1(V ). ∗ 1 ∩ In particular, let us emphasize the following two cases. (i) If U V is simply connected, then π (X) π (U) π (V ); ∩ 1 1 ∗ 1 (ii) If V is simply connected, and N is the normal subgroup generated by the image (by ιU V,U#) of π1(U V ) in π1(U), then π1(X) π1(U)/N . ∩ ∩ 2 Examples. (1) (Plane with k holes) Let x1,...,xk be a family of k distinct points of the plane R ,and 2 { } let X = R x1,...,xk .Thenπ1(X) is the free group with k generators. Namely for k = 1 we already \{ } (33) know that π1(X) Z;givenk>1, let sj be closed half lines with origin xj and with empty intersection , set U = X s1,...,sk 1 and V = X sk .Now,X = U V , all the sets are arcwise connected and \{ − } \{ } ∪ U V is simply connected (even contractible), and so π1(X) π1(U) π1(V ); but U is homotopically ∩ 1 ∗ equivalent to S ,whileπ1(V ) is free with k 1 generators by inductive hypothesis. The same holds for 2 − 2 (34) Y = S y1,...,yk+1 (with y1,...,yk+1 distinct points of S ). (2) (Bouquet of k circles) The \{ } { } fundamental group of a “bouquet” of k circles (i.e. the wedge sum of k circles) is again a free group with k generators: this follows immediately from what has been said in general for wedge sums. Alternatively one could also note that the bouquet is in fact a strong deformation retract of the plane with k holes; another proof is to use induction and Van Kampen, by choosing for any circle Cj apointxj different from the center of the bouquet (j =1,...,k), and then taking U = X x1 and V =(C1 C2) x2 : \{ } ∪ \{ } then U V is contractible, V has the homotopy of a circle and U of a bouquet of (k 1) circles. (3) ∩ 3 1 3 2 2 − (Removing an annulus from R ) Let S(x,y) = (x, y, z) R : z =0,x + y =1,andletuscompute 3 1 { 3∈ } π1(X) where X = R S(x,y).LetRz = (x, y, z) R : x = y =0 (the z-axis), and set U = X Rz \ { ∈ } \ and V = (x, y, z) X : x2 + y2 < 1 .ObviouslyX = U V , all are arcwise connected and V is simply { ∈ } ∪ 1 connected (even contractible). On the other hand, U V is homotopically equivalent to S and hence 3 ∩ 1 π1(U V ) Z,while,settingT = (x, y, z) R : x>0,y =0 (1, 0, 0) , U is homeomorphic to T S ∩ { ∈1 2 }\{ } × (exercise) and hence π1(U) π1(T ) π1(S ) Z Z = Z ; moreover, one may identify the morphism 1 × × 2 π1(U V ) π1(U) π1(T ) π1(S ) with the morphism Z Z ,1 (0, 1) (the generator of π1(U V ) ∩ −→ × −→ → 2 ∩ goes into the generator of the second factor). One therefore has π1(X) Z /Z Z. (4) (Torus) On the 2 1 2 3 surface of X = T =(S ) (the 2-dimensional torus viewed as a doughnut in R ,seeExample1.4)make a small circular hole F ,andletU = X F (open); let V be an open neighborhood of F in X (a “patch” \ above F ). We are in fact in the hypotheses of Van Kampen’s theorem; it is evident that V is contractible and that U V is homotopically equivalent to a circle (hence π1(V ) is trivial and π1(U V ) Z). On ∩ ∩ the other hand U is homotopically equivalent to two tangent circles (this can be easily understood in the 2 interpretation of T as a square modulo identifications, as recalled in the cited Example 1.4: making a hole in the interior of the square, the latter deformation-retracts radially on its boundary; as an useful exercise, we suggest to interpret this retraction on the doughnut), and then to a plane with two holes: hence π1(U) is free on two generators. Now, the normal subgroup of π1(U) Z Z generated by the image (35) ∗ of π1(U V ) is the subgroup of commutators of Z Z (in the interpretation of the square, a generator ∩ ∗ of π1(U V,x0) is a loop based at a vertex which surrounds the hole: such loop is clearly homotopic rel ∩ ∂I to the boundary of the square run twice forth and back, which is exactly the commutator of the two (33) For example, draw the lines rl,m = xl + t(xm xl):t R (for 1 l Corrado Marastoni 25 Notes on Algebraic Topology 2 generators γ1 and γ2 of π1(U, x0)), so one has π1(X) Z Z/[Z Z, Z Z] Z , as we have already seen. (5) 1 ∗ ∗ ∗ (Real projective line) Let P be the real projective line, endowed with the quotient topology with respect to 2 1 1 1 natural map p : R P .Letq = p 1 (Hopf map): since q : S P is continuous, surjective and closed, × −→ |S 2 1 1 −→ then q is still quotient. The map ( ) : S S is also quotient for the same reason, and has exactly · 1 −→ 1 1 the same fibers of q: it follows that S and P are canonically homeomorphic. hence π1(P ) Z. (6) 2 (Real projective plane) Let P be the projective plane, endowed with the quotient topology with respect 2 2 2 1 2 to the map p : B P obtained by identifying in P the pairs of antipodal points on S ∂B (if 2 −→ 2 x =(x, y) B and [x0,x1,x2] are homogeneous coordinates in P one can set p(x)=[1 x ,x,y]): note ∈ −| | that p( 1) 1 (one can also identify p with the Hopf map). The quotient map p is closed but not S P S1 2 2 | 2 1 2 open; nevertheless, if B = B 0 and B˙2 = B S ,thenU = p(B )andV = p(B˙2) are open in × × 2 (36) \{1 } ˙2 \ P . From now on let us choose 2 B = U V C as base point for the computation of fundamental ∈ × ∩ ⊂ groups. The set V (homeomorphic image of B˙2) is clearly contractible, while U V (homeomorphic image ˙2 ∩ e2πit of B ) has fundamental group Z generated by [γ]obtainedby(theimagebyp of) t 2 . As for 2 × 1 x → B , it strong deformation-retracts to S by the affine homotopy h(x,t)=(1 t)x + t ,homotopywhich × − x 1 1 (37) | | 1 descends via p to a strong deformation retraction h˜ of U a p(S ) P . Let r = h˜( , 1) : U p(S ): 1 ∼ 1 · −→ by r# : π1(U, 2 ) π1(p(S ), 1) Z, the canonical generator of the second member comes from the −→ 1 πit 2 generator [ψ]ofπ1(U, 2 )obtainedbyt p(e /2): hence ιU V,U# sends the generator [γ]in[ψ] ,and 2 →3 ∩ n hence π1(P ) Z/2Z (analogously to P and, as we shall show, to any P with n 2). (7) (Removing 3 3 3 ≥ one or two annuli from R ) If A is an annulus in R and X = R A, we already computed above that \ 2 π1(X) Z: another method is to observe that X can be deformation-retracted first to a 2-sphere S plus 1 2 a diameter, then to the a wedge sum S S (by slowly approaching the endpoints of the diameter along 1 ∨2 an equator), hence π1(X) π1(S ) π1(S ) Z. ∗ X : X : X : Figure 5: Deforming R3 minus one annulus; minus two unlinked annuli; minus two linked annuli. 3 Let us use the same approach for two other similar situations. If B is another annulus of R unlinked 3 • 1 1 2 2 with A,thenX = R (A B) can be deformation-retracted to S S S S and hence π1(X)is \ 3 ∨ ∨ ∨ 3 free on two generators. If C is a third annulus of R linked with A,thenX = R (A C)canbe • 2 2 2 \ deformation-retracted to S T and hence π1(X) is isomorphic to π1(T ), i.e. a free abelian group of ∨ rank two. (8) (Klein bottle) Let us compute the fundamental group of the Klein bottle K by using its description in terms of fundamental polygon (i.e. a quotient of a polygon); the argument will be suitable to 2 compute again the fundamental group of the torus T (see Figure 6). In both fundamental polygons take (36) 2 2 2 The map p is closed since B is compact and P is Hausdorff(the finite points of P have the same 1 neighborhoods of the points of B˙2, while a basis of neighborhoods of a point at infinity p(x)withx S 2 ∈ is given by A ( A) where A = B U with U C a small open ball centered in x;henceitisstill ∪ − 2 ∩ ⊂ 2 possible to separate the points of P ), but p is not open (the above A is open in B ,butitsp-saturated 1 1 2 p− (p(A)) = A ( A S ) is not open: hence p(A)isnotopeninthe(quotient)topologyofP ). On the ∪ − ∩ 2 2 other hand the open subsets B and B˙2 are already p-saturated, hence their images by p are open in P . × (37)Recall the factorization property of quotient functions (Proposition 1.1.14): given a quotient function f : X Y and a continuous function g : X Z, there exists a unique continuous function h : Y Z −→ −→ 2 −→ such that f = h g if and only if g is constant on the fibers of f. Here we mean X = B I, Y = U I, ◦ × × × Z = U, f = p idI and g = p h, and the factorization hypothesis are satisfied. The situation would be × ◦ 2 1 different if we would instead consider the strong deformation retraction of B to αS for a 0 <α<1(e.g. 1 x × α = 2 ), for example the affine one hα(x,t)=(1 t)x + αt x :namely,notethatp hα is not constant on − | | ◦ the fibers of p idI ,since(p idI )(x,t)=(p idI )( x,t)butp(hα(x,t)) = p(hα( x,t)) = p(hα( x,t)) 1× × × − − − − for any x S and t I. ∈ ∈ Corrado Marastoni 26 Notes on Algebraic Topology γ1 γ1 x0 ✲ x0 x0 ✲ x0 2 γ γ γ ❄ γ T = 2 ✻ ✻ 2 K = 2 ❄ ✻ 2 ✲ ✲ x x x x 0 γ1 0 0 γ1 0 Figure 6: The torus, the Klein bottle and a suitable open cover for both. U to be a central open square (yellow) whose edges are at some small distance δ>0 from the boundary; and V to be the open square crown (grey) of the points of the polygon whose distance from the boundary is < 2δ.ItisclearthatU is contractible, and that U V (the overlapped yellow-grey zone) is homotopically ∩ equivalent to a circle; on the other hand, V can be deformation-retracted to the boundary, which can be identified to a “figure eight” (a bouquet of two circles) and hence has fundamental group free on two generators, i.e. Z Z. So, by Corollary 1.5.4, the fundamental group is in both cases (Z Z)/N where ∗ ∗ N is the normal subgroup generated by the image of π1(U V ) Z: hence, what makes the difference 2 ∩ between T and K will be the different images of π1(U V )intoπ1(V ). Namely, a generator of π1(U V ) ∩ ∩ is a square-shaped loop, e.g. run counterclockwise: when deformed on the boundary, this loop becomes 1 1 2 1 2 γ1γ2γ1− γ2− in the case of T ,andγ1γ2γ1− γ2 in the case of K. Hence for T the subgroup N is generated 2 2 by the commutator of [γ1]and[γ2], and hence π1(T )istheabelianizationofZ Z,i.e. Z (as we saw ∗ 1 above); while π1(K) is the group with generators a =[γ1]andb =[γ2] with relation aba− b =id,i.e. bab = a. (9) (g-fold torus) A g-fold torus is a orientable closed surface of genus g; its fundamental polygon is a 4g-gon with pairwise identifications of edges allowing g junctions naturally generalizing the one of the (1-)torus (the Figure 7 shows the case n = 2). To compute the fundamental group of the 2-fold torus from Figure 7: The double torus. its fundamental polygon we can proceed exactly as we did above for the Klein bottle: U is contractible, U V is homotopically equivalent to a circle, while V can be deformation-retracted to the boundary, which ∩ in this case can be identified to a bouquet of four circles and hence has fundamental group free on four generators. Since a counterclockwise loop generating π1(U V ), when deformed on the boundary, becomes 1 1 1 1 ∩ bab− a− cdc− d− , we get that the fundamental group of the 2-fold torus is the free group generated by 1 1 1 1 a, b, c, d modulo the normal subgroup generated by bab− a− cdc− d− . More generally, the fundamental group of the g-fold torus is the free group generated by a1,b1, ,ag,bg modulo the normal subgroup 1 1 1 1 ··· generated by a1b1a− b− agbgag− bg− . (10) (Spaces with fundamental group Z/nZ) Given any n N, 1 1 ··· ∈ using the above technique of fundamental polygons it is then easy to construct a space whose fundamental group is Z/nZ: just consider a regular n-gon and identify all its edges with a chosen direction (e.g. conterclockwise). Namely, here we have π1(V ) Z (say with generator a) and the image of a generator n of π1(U V ) Z into π1(V )isa , hence the quotient π1(V )/N is isomorphic to Z/nZ. (11) (Graphs) ∩ In a connected graph, a tree is a contractible subgraph; a tree is called maximal if it contains all vertices of X.IfT is a maximal tree in a connected graph X,let dλ : λ Λ be the family of edges of X T : { ∈ } − then π1(X) is a free group with generators [γλ] corresponding to each edge dλ. This can be proved by Corrado Marastoni 27 Notes on Algebraic Topology Figure 8: Graphs and maximal trees. considering, for any λ Λ, an open neighborhood Uλ of T + dλ which deformation-retracts to T + dλ:then ∈ each Uλ deformation-retracts into a circle, and the intersections of two or more Uλ’s is contractible since it deformation-retracts to T . For example, the fundamental group of the graph X on the left of Figure 8 is free on four generators, each one corresponding to a loop containing only one of the edges not in any chosen maximal tree (whose edges are represented in black). Similarly, the graph Y on the right of Figure 8 — which can also be interpreted as the suspension of the three red vertices Pj with j =1, 2, 3—has fundamental group free on two generators (a maximal tree is depicted in black). As for this last example note that, setting Uj = Y Pj (for j =1, 2, 3), then the open cover U1,U2 is suitable for applying \{ } { } Van Kampen and confirms that π1(Y ) Z Z,whiletheopencover U1,U2,U3 is not suitable since ∗ { } U1 U2 U3 = Y P1,P2,P3 is not arcwise connected (hence one cannot conclude that π1(Y ) Z Z Z, ∩ ∩ \{ } ∗ ∗ a statement that would be false). Corrado Marastoni 28 Notes on Algebraic Topology 1.6 Covering spaces The prototype of a covering space is the exponential map : R S1 −→ given by (t)=e2πit : the key property is that any small open interval of S1 is “well-covered” by this map, i.e. its inverse image is a family of pairwise disjoint homeomorphic copies of itself. We have already used this map to prove (see Proposition 1.4.2) that the fundamental group of S1 is free with one generator: in fact, we shall see that there is a deep relation between the classification of the covering spaces of some topological space X and the structure of the fundamental group of X. 1.6.1 Fiber bundles and covering spaces Definition 1.6.1. Let X be a topological space. A space on X is a pair (Y,π)whereY is a topological space and π : Y X is a surjective continuous function. A morphism from −→ (Y ,π )to(Y ,π ) is a continuous function f : Y Y such that π = π f. 1 1 2 2 1 −→ 2 1 2 ◦ Given x X and a space (Y,π) on X, we denote by Y = π 1(x)thefiber on x. Note that a ∈ x − morphism of spaces on X respects the fibers, in the sense that f(Y ) Y ; in particular, 1,x ⊂ 2,x if f is a isomorphism, for any x X it is induced a homeomorphism f : Y ∼ Y . ∈ x 1,x −→ 2,x The simplest case of space on X is the one of type (X F, p )whereF is a topological × X space and pX the projection on X. More generally: Definition 1.6.2. A space (Y,π) on X is called trivial if there exists a topological space F and an isomorphism f :(Y,π) ∼ (X F, p ): in this case, such an isomorphism of −→ × X spaces on X is called a trivialization of (Y,π). Anyway, the most important notion is the one of “locally trivial space”, or “fiber bundle”. Definition 1.6.3. Given a space (Y,π) on X and an open subset U X,therestriction ⊂ 1 of (Y,π)toU (sometimes denoted by Y ) is the space on U given by (π− (U),π 1 ). |U |π− (U) The space (Y,π) on X is called locally trivial (or also fiber bundle, or bundle) on X if there exists an open cover = Uλ : λ Λ of X such that Y is trivial for any λ Λ; i.e., U { ∈ } |Uλ ∈ for any x X there exists an open neighborhood U X of x such that Y is trivial. A ∈ ⊂ |U local trivialization of (Y,π) on Uλ is a trivialization of Y . |Uλ If the space (Y,π) on X is trivial, then obviously the map π is open and all fibers of (Y,π) on X are homeomorphic.(38) This is still true for any bundle on an arcwise connected space: (38)If f :(Y,π) ∼ (X F, p ) is a trivialization, then all fibers of Y are homeomorphic to F (the fiber of −→ × X X F ). As for the openness, since f is a homeomorphism we are left with proving that p : X F X is × X × −→ open. Let V be an open subset of X F ,and(x, f) V : then there exist open subsets U X and W F × ∈ ⊂ ⊂ such that (x, f) U W V ,andhenceU = p (U W ) p (V ). Therefore p (V ) is a neighborhood ∈ × ⊂ X × ⊂ X X of pX (x, f)=x because it contains U (an open neighborhood of x), and this proves that pX (V )isopen. Corrado Marastoni 29 Notes on Algebraic Topology Proposition 1.6.4. If (Y,π) is a bundle on X, then π is open. Moreover, if X is arcwise connected, then all fibers of (Y,π) on X are homeomorphic. Proof. Let = Uλ : λ Λ be an open cover of X such that (Y,π) is trivial on every U .LetV Y U { ∈ } 1 ∈U ⊂ be open, y V and U such that x = π(y) U:thenW = π− (U) V is an open neighborhood of ∈ ∈U ∈ 1 ∩ y,andsincethemapπ 1 is open and W π− (U), one has that π(W ) U π(V ) π(V )isan |π− (U) ⊂ ⊂ ∩ ⊂ open neighborhood of x.Henceπ(V )isopeninX.NowletX be arcwise connected; given x0 ,x1 X, k ∈ let γ : I X be a path between them and let λ0,...,λk Λbesuchthatγ(I) Uλ , x Uλ , −→ ∈ ⊂ j=0 j 0 ∈ 0 x1 Uλk and Uλj Uλj+1 = ∅. We are left with proving that Y U and Y U are isomorphic if further ∈ ∩ | λj | λj+1 1 ∼ restricted to Uλj Uλj+1 : it is enough to observe that, given local trivializations ψj : π− (Uλj ) Uλj Fj , ∩ 1 −→ × the isomorphism ψj+1 ψ− of trivial spaces on Uλ Uλ induces a homeomorphism Fj ∼ Fj+1. ◦ j j ∩ j+1 −→ If (Y,π) is a bundle on X,thenY is usually referred to as the “total space” and X as the “base” of the bundle; moreover, if X is arcwise connected, thanks to Proposition 1.6.4 one directly talks about “bundle with fiber F ” or “F -bundle”, where F is a topological space homeomorphic to the fibers of π. Remark 1.6.5. Since here we are interested only in topological matters, in the previous brief exposition of the notion of bundle we have not paid so much attention to the structure of the fiber, which has been required to be nothing more than a topological space. In other words: we saw that, if (Y,π) is a bundle on the arcwise connected space X with fiber F 1 and U1,U2 are two open subsets of X with trivializations ψj : π− (Uj) ∼ Uj F ,then 1 −→ × for any x U U there is an induced homeomorphism (ψ ψ− )(x, ):F ∼ F , and we ∈ 1 ∩ 2 2 ◦ 1 · −→ have not requested this homeomorphism to respect also possible further structures of F (hence, for example, that it should be a linear map if F is a vector space, or a orthogonal transformation if F is a sphere). In a more motivated exposition the structure of F has to be respected by these homeomorphisms, which are commonly called transition functions. In fact, the proper notion of bundle requires also a group structure operating effectively on the fiber.(39) More precisely, a bundle of base X with total space Y , fiber F and group structure G (where X, Y and F are topological spaces and G is a topological group) is the datum of: (1) a space (Y,π) on X with fibers homeomorphic to F ; (2) an effective action of G as a group of homeomorphisms on F ; (3) an open cover = U : λ Λ of X with a family of local trivializations ψ : U { λ ∈ } λ π 1(U ) ∼ U F (the map ψ is usually called a local chart of Y over U ); − λ −→ λ × λ λ (4) for any λ, µ Λsuch that Uλ Uµ = ∅, a continuous transition function αλ,µ : ∈ ∩ U U G such that ψ ψ 1(x, t)=(x, α (x) t) for any x U U and t F . λ ∩ µ −→ λ µ− λ,µ · ∈ λ ∩ µ ∈ In particular, the bundle will be called: (a) vector bundle if F is a real or complex euclidean space (for example, real vector bundle of rank n if the fiber is Rn) and G is the general linear group (or a subgroup of it) of the same euclidean space; (b) sphere bundle if F is a sphere in an euclidean space and G is the orthogonal group (or a subgroup of it) of the same euclidean space; (c) principal bundle if F is the same group G operating on itself (39)Recall that a (left) action of a group G on a topological space F is a morphism from G to the group of autohomeomorphisms of F : in other words, the identity element of G acts as the identity of F ,and g1(g2(f)) = (g1g2)(f) for any g1,g2 G and f F .Theactioniscalledeffective if the only element of G ∈ ∈ which operates trivially on F is the identity. Corrado Marastoni 30 Notes on Algebraic Topology by right translation. In the case of particular structures on Y , X, F and G (e.g. if they are real or complex manifolds, or algebraic varieties) it is usual to require more regularity than continuity to the trivializations and to the transition functions, and so one can also talk about continuous, differentiable, holomorphic, algebraic ... bundles. Examples. (1) If H G are closed subgroups of GL(n; C)andπ : G G/H is the canonical projection, ⊂ −→ then (G,π)isabundlewithfiberH and structure group (H)/H, where (H) is the normalizer of H N N in G (see for example Bredon [2, II.14, pp. 110-111]). (2) Given a real manifold M of class 1 and C dimension n,thetangentbundleTM = (x, v):x M, v TxM and the cotangent bundle T ∗M = { ∈ ∈ } (x, α):x M,α T ∗M are real vector bundles on M of rank n;ifN M is a submanifold { ∈ ∈ x } ⊂ of dimension k, there are real vector bundles on N of rank n k by considering the normal bundle − TN M = (x, v):x N, v TxM/TxN and the conormal bundle TN∗ M = (x, α):x N,α (TxN)⊥ { ∈ ∈ } 1 { 1 ∈ 1 ∈ ⊂1 Tx∗M .NotethatthetangentbundleT S is trivial, being homeomorphic to S R by S R ∼ T S , } × × −→ ((x, y),t) ((x, y), ( ty, tx)). → − From now on we shall assume that the topological space X is arcwise connected. Definition 1.6.6. A covering space of X is a bundle (Y,π) on X with discrete fiber. In this case one also says that π : Y X is a covering map. The cardinality of the fibers −→ (well-defined thanks to Proposition 1.6.4) is called multiplicity of the covering (if such multiplicity is finite, say n, one also talks about a “n-sheet covering”). A morphism of covering spaces on X is a morphism as spaces on X. In other words, the fact that π : Y X is a covering map −→ means that for any x X there exists a neighborhood U X of 1 ∈ ⊂ x such that π− (U) is the disjoint union of homeomorphic copies 1 of U,i.e.π− (U)= Vλ,withπ : Vλ ∼ U: such an open λ Λ |Vλ −→ subset U is said to be∈evenly covered. It is then clear that, if (Y,π) is a covering space of X and U X is open, then Y (see ⊂ |U Definition 1.6.3) is a covering space of U. Proposition 1.6.7. Any covering space is a local homeomorphism.(40) Conversely, a local homeomorphism (Y,π) where Y is Hausdorffand whose fibers are finite sets with the same cardinality is a covering. Proof. The first statement follows immediately from the definitions and Proposition 1.6.4. For the second, 1 let k be the cardinal of the fibers; fixed x0 X,letπ− (x0 )= y1,...,yk .Thankstothehypotheses,we ∈ { } ∼ may choose by recurrence some neighborhoods Vyj Y of yj such that π V : Vyj Uyj = π(Vyj )and ⊂ | yj −→ j 1 k 1 1 k Vy Y − Vy .SosetU = Uy X and Vj = π− (U) Vy :itclearlyholdsπ− (U)= Vj , j ⊂ \ i=1 i j=1 j ⊂ ∩ j j=1 and π : Vj U is a homeomorphism. |Vj −→ n j Examples. (1) Let p(z)= j=0 aj z be any polynomial with complex coefficients and an =0,andlet 1 1 Γ=p(p− (0)) (the set of critical values of p). Then, setting X = C ΓandY = p− (X), the space (Y,p)is \ a n-sheet covering space of X. For example, if p(z)=zn the open subset which are evenly covered are those U X = C× such that the inclusion map j : U X is nullhomotopic, i.e. those which do not contain ⊂ → (40)Aspace(Y,π)onX is called local homeomorphism if π is open and if for any y Y there exists an ∈ open neighborhood V Y of y such that π : V π(V ) is a homeomorphism. ⊂ |V −→ Corrado Marastoni 31 Notes on Algebraic Topology loops of nonzero index in 0.(41) Among them we find all simply connected open subsets of X, for example iθ also the “tickened spiral” z = re X : θ R>0,θ φ(θ) Figure 9: Connected covering spaces of the circle. (4) Let H be a discrete topological group operating on the left on a topological space Y in a properly discontinuous way (i.e., for any y Y there exists a open neighborhood V Y of y such that g1V g2V = ∅ ∈ ⊂ ∩ if g1 = g2). Let X be the space of orbits of H in Y ,endowedwiththequotienttopology:thenthecanonical projection π : Y X is a covering space. Namely, given y Y let V be an open neighborhood of y with −→ ∈ the properties of discontinuity just defined, and let U = π(V ) (an open neighborhood of π(y), because 1 (42) π is open): one has π− (U)= g H gV ,andπ gV : gV U is a homeomorphism. For example, ∈ | −→ let Y = G be a topological group and be H a discrete subgroup operating by multiplication on the left: such action is properly discontinuous, and the projection π : Y = G X = G/H is a covering.(43) In −→ the case (2), we had Y = C and H =2πiZ operating by translation; in the case (3), we had Y = R 1 2πit and H = Z. (5) Setting Y =]0, 2[ and X = S ,themapπ = : Y X, π(t)=e is a local |]0,2[ −→ (41)Suppose that U X is an open subset containing a loop γ (say based at a point z) of index nonzero ⊂ 1 n in 0. We shall show that, for w p− (z)(i.e.w = z), there exists a unique “lifting” of γ based at w, ∈ 1 i.e. a path δ completely contained in the inverse image V = p− (U)inY = C such that δ(0) = w and p δ = γ (here the computation can be performed also explicitly: if δ(t)=r(t)eiθ(t) and γ(t)=ρ(t)eiϕ(t) ◦ with γ(0) = γ(1) = z, from p δ = γ one gets r(t)= n ρ(t)andθ(t)= ϕ(t)+2kπ for some 0 k n 1, ◦ n ≤ ≤ − and the good k can be found by requiring that δ(0) = w), whose extremity is another w in the inverse image of z certainly different from w (namely, if w = w then δ should be nullhomotopic because C is simply connected, and hence also γ = p δ would be nullhomotopic): but then V could not be a disjoint ◦ union of copies homeomorphic to U by p,henceU is not evenly covered. Conversely, if U is not evenly 1 covered there exist a point z in U,twodistinctpointsw and w in the inverse image p− (z)ofz and a path α from w to w completely contained in V .Now,p α is surely a loop in U based at z;ontheother ◦ hand, if ψ is the shortest path from w to w along the circle containing both of them, it is clear that α and ψ are paths homotopic rel ∂I (because C is simply connected) hence also p α and p ψ are loops ◦ ◦ homotopic rel ∂I:butp ψ is the loop based at z which describes the circle one or more times, hence its ◦ index in 0 is nonzero, and hence also the index in 0 of p α is nonzero. (42) ◦ It is clearly continuous, open and surjective; it is also injective, because from π(gy1)=π(gy2)onegets gy2 = hgy1 for some h H,hencegU hgU = ∅,henceg = hg,i.e.h = e and gy1 = gy2. ∈ ∩ (43)This fact will explain in a more general framework the properties of the maps of canonical projection (see 1.4) of lifting uniquely paths and homotopies. In that case we were considering the right classes (i.e. § G/H = gH : g G ), hence H was acting on G on the right instead than on the left. { ∈ } Corrado Marastoni 32 Notes on Algebraic Topology homeomorphism with discrete fibers, but it is not a covering (the cardinality of the fibers is not the same for any point). The same conclusion holds with Y = R>0 (no neighborhood of 1 is evenly covered). (6) 2 2 2 2 Setting Y = (z, w) C : z = w and V = (z,w) C : z = w ,z=0 (open subset of Y ), the first { ∈ } { ∈ } projection p1 : Y C ((z,w) z) is not a covering space, while p1 : V C× is: namely in the first case −→ → −→ the map is even not a local homeomorphism, while in the second there is a local homeomorphism with fibers finite and of the same cardinality.(44) 1.6.2 Liftings and the Monodromy lemma A crucial feature of covering maps π : Y X is that they are able to lift maps from the −→ base X to the full space Y in a unique way. Let X be an arcwise connected topological space. Definition 1.6.8. Let (Y,π) be a space on X, f : Z X a continuous function. A −→ lifting of f by π is a continuous function f˜ : Z Y such that f = π f˜. In particular, if −→ ◦ Z = A X and f = ι is the canonical inclusion, a lifting of ι is called a (continuous) ⊂ A A section of π over A. Y f˜ π Z X f Example. If X is a differential manifold and U X is open, a section of the tangent bundle on U is a ⊂ vector field in U. Proposition 1.6.9. If π : Y X is a local homeomorphism, two liftings of f : Z −→ −→ X which coincide in one point, coincide in a whole neighborhood of the point itself. If moreover Z is connected and π is a covering space, or if Z is connected and Y is Hausdorff, then they are equal. ˜ ˜ ∼ Proof. If f1(z0 )=f2(z0 )=y0 ,andifV Y is a neighborhood of y0 on which π : V U = π(V )isa ˜ ˜ ⊂ −→1 ˜ 1 homeomorphism, then f1 and f2 must necessarily coincide on the neighborhood W = f − (U) f1− (V ) 1 ∩ ∩ f˜− (V )ofz .ThissaysthatZ = z Z : f˜1(z)=f˜2(z) is an open subset of Z.Ifπ is a covering space, 2 0 { ∈ } or if Y is of Hausdorff, Z is also a closed subset of Z (in the first case, if z Z Z let U X be an 1 ˜ ∈ \ ˜ ⊂ evenly covered neighborhood of f(z): then π− (U)= λ Λ Vλ with f1(z) Vλ1 and f2(z) Vλ2 (where 1 1 ∈ ∈ ∈ λ1 = λ2), so that z f˜− (Vλ ) f˜− (Vλ ) Z Z,i.e.Z Z is open; in the second see Lemma 1.2.2), ∈ 1 1 ∩ 2 2 ⊂ \ \ and this implies that f˜1 = f˜2 because Z is connected. Definition 1.6.10. The space (Y,π) on X has the property of lifting paths (uniquely) if for any path γ : I X and any initial point y π 1(γ(0)) there exists a (unique) path −→ 0 ∈ − (44) π1 is not a local homeomorphism in (0, 0); while it is in (z0,w0) V , by taking as neighborhood a ∈ small open ball not containing (0, 0). Corrado Marastoni 33 Notes on Algebraic Topology γ˜y : I Y such that γ = π γ˜y andγ ˜y (0) = y ;ifZ is a topological space, we say 0 −→ ◦ 0 0 0 that (Y,π) has the property of lifting homotopies (uniquely) with respect to Z if for any homotopy h : Z I X and any lifting α : Z Y of the base h (i.e., π α = h ) × −→ 0 −→ 0 ◦ 0 0 there exists a (unique) homotopy h˜ : Z I Y such that h = π h˜ and (h˜ ) = α . α0 × −→ ◦ α0 α0 0 0 Proposition 1.6.11. Let (Y,π) be a space on X which lifts paths uniquely, and let γ,φ : I X be two paths with x = γ(0) and x1 = γ(1) = φ(0). Then, given y Yx and set −→ 0 0 ∈ 0 y1 =˜γy (1) Yx1 ,itholds(γ φ)y =˜γy φy1 . 0 ∈ · 0 0 · Proof. Obvious. Lemma 1.6.12. The covering spaces have the property of lifting paths uniquely. Proof. Uniqueness is given by Proposition 1.6.9; as for the existence, let π : Y X be the covering space, 1 −→ γ : I X apathinX with γ(0) = x ,andlety π− (x ). Let us define the liftingγ ˜y piecewise. −→ 0 0 ∈ 0 0 There exist 0 = t0 Lemma 1.6.13. Any local homeomorphism which has the property of lifting paths uniquely has also the property of lifting homotopies uniquely. Proof. Let π : Y X be a local homeomorphism with the property of lifting paths uniquely, and −→ let h : Z I X be a homotopy and α0 : Z Y with π α0 = h0 .Foranyz Z,thepath z × z−→ z −→ ◦ z ∈ γ : I X, γ (t)=h(z,t)liftsuniquelyto˜γ : I Y such thatγ ˜ (0) = α0(z): this leads necessarily −→ −→ to define h˜ : Z I Y as h˜(z, t)=˜γz(t). We are left with showing the continuity of h˜.Now,since × −→ π is a local homeomorphism, any continuous function with values in X admits locally liftings around z any point: given (z ,s) Z I,letVs Y be an open neighborhood of h˜(z ,s)=˜γ 0 (s)suchthat 0 ∈ × ⊂ 0 π : Vs ∼ Us = π(Vs), let Ws Js be a neighborhood of (z ,s)suchthath(Ws Js) Us,and |Vs −→ × 0 × ⊂ ˜ 1 ˜ z0 ˜ define hs =(π V )− h W J : Ws Js Vs. Observe that hs(z0 ,s)=˜γ (s)=h(z0 ,s): this | s ◦ | s× s × −→ implies that h˜s(z ,t)=h˜(z ,t) for any t Js (the paths h˜s(z , )andh˜(z , )onJs are both liftings of 0 0 ∈ 0 · 0 · h(z0 , ) and coincide for t = s). Since z0 I is compact, there exist 0 Proposition 1.6.14. The covering spaces have the property of lifting homotopies uniquely. Proof. Follows from Lemmas 1.6.12 and 1.6.13. As a consequence, homotopic paths are lifted to paths ending at the same point, and even homotopic (see Figure 10): Corrado Marastoni 34 Notes on Algebraic Topology Corollary 1.6.15. (Monodromy lemma) Let π : Y X be a covering space, α and β be −→ two paths in X with [α]=[β]. Then, if α˜ and β˜ are liftings of α and β with α˜(0) = β˜(0), it holds also α˜(1) = β˜(1) and, even, [˜α]=[β˜]. Proof. Let h : I I X be a homotopy rel ∂I between α and β: by Proposition 1.6.14 (with Z = I in the × −→ second factor), there exists a unique homotopy h˜ : I I Y such that h = π h˜ and h˜(0,τ) y =˜α(0) = × −→ ◦ ≡ 0 β˜(0) for any τ I. By the property of lifting paths uniquely, one getsα ˜(t)=h˜(t, 0) and β˜(t)=h˜(t, 1). ∈ Now, the map h˜(1, ):I Y is continuous; since h = π h˜, that map must take values in the (discrete) · −→ ◦ fiber of π on α(1) = β(1): hence it is constant, and in particularα ˜(1) = h˜(1, 0) = h˜(1, 1) = β˜(1). Figure 10: The Monodromy lemma. From the Monodromy lemma it follows that the only connected covering of a simply connected space is, up to homeomorphisms, the space itself: Corollary 1.6.16. Let X be a simply connected topological space, π : Y X a covering −→ with Y arcwise connected. Then π is a homeomorphism. Proof. We already know that π is a local homeomorphism: it is enough to prove that π is injective. So let y1,y2 Yx ,andletγ : I Y be a path from y1 to y2.Thepathπ γ is a loop based at x ,hence ∈ 0 −→ ◦ 0 [π γ]=[cx ] by hypothesis. By the Monodromy lemma (with α = π γ, β = cx ,˜α = γ and β˜ = cy )we ◦ 0 ◦ 0 1 get y2 = γ(1) = cy1 (1) = y1. 1.6.3 Classification of covering spaces Let X be an arcwise connected topological space and x X. Let us see how the subgroups 0 ∈ of π1(X, x0 ) are in corrispondence with the covering spaces of X. Proposition 1.6.17. Let π : Y X be a covering space, and y Yx . Then the −→ 0 ∈ 0 morphism π : π (Y,y ) π (X, x ) is injective. # 1 0 −→ 1 0 Proof. Let π#([γ]) = [π γ]=[cx ]: observing that (π γ) = γ and (cx ) = cy , by the Monodromy 0 y 0 y 0 ◦ ◦ 0 0 lemma one has [γ]=[c ]. y0 Corrado Marastoni 35 Notes on Algebraic Topology Definition 1.6.18. We denote by G(Y,y0 )(characteristic subgroup of the covering space) the isomorphic image of π1(Y,y0 )inπ1(X, x0 )byπ#: G(Y,y )=π#(π1(Y,y )) = [γ] π1(X, x ):˜γy is a loop based at y . 0 0 { ∈ 0 0 0 } Proposition 1.6.19. Let π : Y X be a covering space, and y Yx . Then the −→ 0 ∈ 0 subgroups of π (X, x ) conjugated to G(Y,y ) are exactly the subgroups G(Y,y ) with y 1 0 0 1 1 ∈ Y in the same arc-component of y in Y . x0 0 Proof. Exercise. Remark 1.6.20. (Monodromy action) The group π1(X, x0 ) acts on the right on the fiber Yx : in other words, there is a monodromy morphism µ : π1(X, x ) SY ,whereSY 0 0 −→ x0 x0 denotes the group of permutations of the fiber Y . This action is described as follows: x0 given y Yx and γ a loop in X based at x ,let˜γy be the lifting of γ with starting point ∈ 0 0 y, and define µ([γ])(y)=y [γ]:=˜γ (1). Hence the stabilizer of some y Y is precisely · y ∈ G(Y,y). Moreover, if the covering space Y is arcwise connected then by Proposition 1.6.19 the subgroup acting trivially on Y is the heart(45) of G(Y,y )inπ (X, x ), for any x0 0 1 0 y Yx . 0 ∈ 0 iθ Examples. (1) The fiber of the covering space exp : C C× over z0 = re is the set of complex −→ 2πit logarithms wk =logr + i(θ +2kπ) for k Z, and the generator re of π1(C×,z0)sendswk to wk+1. (2) ∈ 1 Let us consider the following 3-sheet covering spaces of S : p1 p2 p3 −→ −→ −→ 1 1 3 1 1 1 2 where p1 : S S , p1(z)=z ; p2 : S S S , p2(z)=z or p2(z)=z according to the fact that z −→ 1−→ 1 1 1 1 belongs to the first or to the second copy of S ;andp3 : S S S S , p3(z)=z.Then,denotingthe −→ fiber of pj always by y1,y2,y3 (where y1 is the external one, y2 the middle one and y3 the internal one), { }1 the action of a generator of π1(S ) on the fiber is (in the standard notation of S3) the cyclic permutation (1 2 3) for p1, the transposition (2 3) for p2, and the identity for p3. Lemma 1.6.21. Let π : Y X be a covering space, α and β two paths in X from x −→ 0 to x1, and let y Yx . Then α˜y and β˜y have the same ending point if and only if 0 ∈ 0 0 0 [α β 1] G(Y,y ). · − ∈ 0 Proof. Exercise (apply Proposition 1.6.11). We saw that every covering space π : Y X has the property of lifting paths uniquely: −→ given a path (continuous function) f : I X and a starting point in the covering space −→ (i.e. a point y Y in the fiber of x = f(0)), there exists a unique path (continuous 0 ∈ 0 function) f˜ : I Y such that π f˜ = f and f˜(0) = y . If we aim to replace (I,0) −→ ◦ 0 (45)If G is a group and H is a subgroup of G,theheart of H is the largest normal subgroup of G contained 1 in H:henceitis g G gHg− . Dually, the smallest normal subgroup of G containing H is the normal ∈ 1 subgroup generated by the subset gHg− . The latter shoul not be confused with the normalizer g G 1 ∈ H = g G : gHg− = H , which is the largest subgroup of G containing H as a normal subgroup. N { ∈ } Hence, H is normal in G if and only if the heart of H and the normal subgroup generated by H coincide with H,and H = G. N Corrado Marastoni 36 Notes on Algebraic Topology by any pointed topological space (Z, z0 ) the solution of the same problem depends on the properties of the function f to be lifted and also of the space Z, for which we must introduce a topological notion. Definition 1.6.22. A topological space is said to be locally (arcwise) connected if any point has a basis of open and (arcwise) connected neighborhoods. Example. The “comb space” (Figure 2(a)) is arcwise connected but not locally arcwise connected. Proposition 1.6.23. (Lifting criterion) Let X, Y and Z be topological spaces with Z arcwise connected and locally arcwise connected, π :(Y,y ) (X, x ) a covering space 0 −→ 0 and f :(Z, z ) (X, x ) a continuous function. Then f admits a unique lifting f˜ : 0 −→ 0 (Z, z ) (Y,y ) if and only if f (π (Z, z )) G(Y,y ). 0 −→ 0 # 1 0 ⊂ 0 Proof. Necessity is an immediate consequence of the functoriality of π1 (exercise); let us see now the sufficience. The uniqueness of f˜ comes from Proposition 1.6.9. As for the existence, given any z Z let ∈ us choose a path α : I Z from z to z:itsimagef α : I X is a path from x to f(z), which lifts to −→ 0 ◦ −→ 0 ˜ auniquepath(f α)y .Thedefinitionf(z)=(f α)y (1) is well-posed: if β is another path in Z from ◦ 0 ◦ 0 z0 to z,thenf α and f β are two paths in X from x0 to f(z), and their liftings from y0 have the same ◦ ◦ 1 1 1 endpoint if and only if (by Lemma 1.6.21) [(f α) (f β)− ]=[f (α β− )] = f#([α β− ]) G(Y,y ), ◦ · ◦ ◦ · · ∈ 0 a fact ensured by the hypotheses. We are left with showing the continuity of f˜.Letz Z and V Y ∈ ⊂ be an open neighborhood of f˜(z): we may assume that the open U = π(V ) X is evenly covered. Let 1 ⊂ W f − (U) be open, arcwise connected and containing z (such a W exists since Z is locally arcwise ⊂ connected): let us prove that f˜(W ) V .Letζ W ,andletβζ be a path in W from z to ζ.Wehave ⊂ ∈ ˜ ˜ f(ζ)=((f (α βζ ))y (1) = (f α)y (f βζ )f˜(z)(1): now, (f βζ )f˜(z) is a path from f(z) V which ◦ · 0 ◦ 0 · ◦ ◦ ∈ lifts f βζ (path in U), and hence also the endpoint (fβζ ) (1) is in V (recall that U = π(V )isevenly ◦ ◦ f˜(z) covered, and V is one of the disjoint sheets above U). Figure 11: AnonlocallyarcwiseconnectedspaceforwhichtheLiftingcriteriondoesnotwork. Remark 1.6.24. The hypothesis of locally arcwise connectedness cannot be dropped in the proof of the Lifting criterion (Proposition 1.6.23). For example, let Z be the “quasi- circle” of Figure 11 (starting with a vertical straight segment which will be later approached by a part of type “sin 1 ”), and let f : Z S1 be a quotient map which collapses all the x −→ points of Z contained in the dashed box into the point 1 of S1. If we consider the usual exponential covering : R S1 given by (t)=e2πit, since the fundamental group of −→ Z is trivial(46) the hypothesis on the characteristic subgroup is satisfied; however, if we lift f by to f˜ : Z R with base point 0 in R, then the upper point of the straight −→ (46)Namely, given a path α : I Z, the support α(I) Z is compact and hence it cannot collapse on the −→ ⊂ vertical straight segment: then α is nullhomotopic, in other words Z is simply connected. Corrado Marastoni 37 Notes on Algebraic Topology 1 ˜ segment goes to 0 while the part of type “sin x ” is sent to 1, but this shows that f is not continuous. We now show that, in the hypothesis of local arcwise connectedness, a covering space of X is covered by any other covering space of X having a smaller characteristic subgroup. Proposition 1.6.25. Let π :(Y,y ) (X, x ) and p :(Z, z ) (X, x ) be two covering 0 −→ 0 0 −→ 0 spaces of X with Z arcwise connected and locally arcwise connected. Then there exists a morphism of covering spaces ϕ :((Z, z ),p) ((Y,y ),π) if and only if G(Z, z ) 0 −→ 0 0 ⊂ G(Y,y0 ), and in such a case ϕ itself is a covering space. Proof. The first statement follows from the Lifting criterion (Proposition 1.6.23); we are left with proving that ϕ itself is a covering space. Given y Y ,letβ be a path in Y from y to y,andconsiderthepath ∈ 0 π β in X from x0 to π(y), which lifts uniquely to α = (π β)z in Z:sinceπ (ϕ α)=p α = π β, ◦ ◦ 0 ◦ ◦ ◦ ◦ using the Monodromy lemma one has ϕ(α(1)) = β(1) = y.Henceϕ is surjective. Let Vy Y be 0 ⊂ an open neighborhood of y such that U = π(V ) is an arcwise connected open neighborhood of x 0 y0 0 1 ∼ evenly covered both for p and for π:onethenhasπ− (U)= y π 1(x ) Vy (with π V : Vy U for ∈ − 0 | y −→ 1 1 1 any y π− (x )) and p− (U)= 1 Wz (with p : Wz ∼ U for any z p− (x )). One 0 z p (x ) Wz 0 ∈ − 0 ∈ 1 1| 1−→ ∈ 1 then shows that ϕ− (V )= W (note that ϕ− (y ) p− (x )): namely, if z ϕ− (y )then y0 z ϕ 1(y ) z 0 0 0 ∈ − 0 ⊂ ∈ (47) 1 ϕ : Wz ∼ Vy , and hence ζ Wz for a certain z ϕ− (y )ifandonlyifϕ(ζ) Vy ,i.e.ifand Wz 0 0 0 | −→ 1 ∈ ∈ ∈ only if ζ ϕ− (Vy ). ∈ 0 Corollary 1.6.26. (Uniqueness theorem) Two arcwise connected and locally arcwise con- nected covering spaces of a (connected and locally arcwise connected) topological space are isomorphic if and only if they have the same characteristic subgroup.(48) The statement of a Existence theorem for a covering space with prescribed characteristic subgroup requires a slightly stronger topological hypothesis. Definition 1.6.27. A topological space X is said locally simply connected if any x X ∈ has a basis of open simply connected neighborhoods; more generally, X is said semi-locally simply connected if any x X admits a open neighborhood U X such that any loop ∈ ⊂ in U based at x is nullhomotopic in X (i.e., with homotopies not necessarily with values only in U). Examples. (1) Obviously, manifolds are locally (hence also semi-locally) simply connected. (2) (Shrinking wedge of circles) Let C = n Cn, where Cn is the circle of center ( 1/n, 0) and radius 1/n:thenC is ∈N − locally arcwise connected but neither locally nor semi-locally simply connected. The fundamental group 2 of C turns out to be very complicated. In fact, the topology of C is the one induced from R ,sothat a neighborhood of (0, 0) must contain all but a finite number of the Cn:hencethistopologyismuch 1 weaker than the wedge sum topology of . In particular, since the Cn collapse to (0, 0), this allows N S also infinite junctions of loops on different Cn to be continuous as loops in C based at 0, and hence to contribute to the complication of π1(C). In particular, for each sequence (rn) of integers one can construct (47) 1 From U = p(Wz)=π(ϕ(Wz)) one has that the connected subset ϕ(Wz)isinp− (U)= y π 1(x ) Vy ∈ − 0 and contains y ,henceϕ(W ) V and it holds even ϕ(W )=V because otherwise p = π ϕ 0 z y0 z y0 Wz Vy Wz ⊂ | | 0 ◦ | would not be an isomorphism. (48)It must be noted that we are always working in the framework of pointed topological spaces,sowe are keeping track of a base point both in the base space (i.e. a x0), and in the covering (i.e. a point in the fiber of x0). In the case we do not keep track of base points, the result should be stated as follows: Two arcwise connected and locally arcwise connected covering spaces of a (connected and locally arcwise connected) topological space are isomorphic if and only if they have conjugated characteristic subgroups. Corrado Marastoni 38 Notes on Algebraic Topology Figure 12: The sets C (shrinking wedge of circles), T (a cone of C)andX (union of two copies of T ). aloopγ(rn) in C winding rk times at each Ck, and these loops are mutually nonhomotopic: this fact provides a a surjective morphisms π1(C) and so, the direct product being uncountable, N Z N Z −→ 1 also π1(C) is uncountable and hence deeply different from π1( ) (which has countably many N S N Z 3 generators, and hence is countable). (3) Let T be a cone in Rwith base∗C (where all the Cn are meant to be e.g. in the plane (x, y)withcenter( 1/n, 0, 0)): then T is locally arcwise connected and clearly − contractible, hence simply connected; in particular T is semi-locally simply connected, but not locally simply connected. (4) Let X be the union of two copies of T at the base point, e.g. X = T ( T ) where 3 ∪ − T = (x, y, z) R :( x, y, z) T is the opposite to T (hence T and T have only the point (0, 0, 0) − { ∈ − − − ∈ } − in common): then X is connected and locally arcwise connected, but neither simply connected nor semi- locally (hence, nor locally) simply connected. The argument is as follows: using the notation introduced above for C,theloopsγ(rn) with all but a finite number of the rn equal to zero are nullhomotopic in X 2 (namely, if N N is the largest number such that rN = 0, then all extremities ( n , 0, 0) of the loops ∈ 1 ± which constitute γ(rn) keep being at “security distance” N > 0 from (0, 0, 0), hence they can be sent to the vertices of T for 0 t 1 ,andthendownto(0, 0, 0) 1 t 1 without breaking the continuity of ± ≤ ≤ 2 2 ≤ ≤ the homotopy), while the γ(rn) with infinitely many rn different from zero are not. Proposition 1.6.28. (Existence theorem) Given an arcwise connected, locally arcwise connected and semi-locally simply connected topological space (X, x ) and a subgroup H 0 ⊂ π (X; x ), there exists a unique (up to a canonical isomorphism) covering space π : Y X 1 0 −→ such that G(Y,y0 )=H. Proof. (Sketch) The idea is to consider on the set Ω of paths from x to x X the equivalence relation x0 ,x 0 1 ∈ given by α β if [α β− ] H,thentodefineY = Ω / , y as the class of c in Ω and x X x0 ,x 0 x0 x0 ,x0 ∼ · ∈ ∈ ∼ π : Y X given by [γ] γ(1), then finally to endow Y with a suitable topology using the hypotheses on −→ −→ X. For more details we refer for example to J¨anich [9, from p. 144]. 1 Examples. (1) The subgroups of π1(S ) Z are Z itself, nZ (for n N)and 0 : they correspond to the 1 1 n ∈ { } coverings (S , id), (S ,z ) (for n N)and(R,) (recall that (t)=exp(2πit)), which therefore represent ∈ 1 —up to isomorphism— all arcwise connected and locally acwise connected covering spaces of S . (2) As 1 1 for the bouquet X = S S , the family of subgroups of π1(X) Z Z is much richer than the one of ∨ ∗ Z, and hence the classification of arcwise connected and locally arcwise connected covering spaces of X is much more interesting (see e.g. [8, 1.3], or the example at p. 41). § Remark 1.6.29. The hypothesis of semi-local simple connectedness is necessary for the proof of Proposition 1.6.28. For example, the above double cone of shrinking wedge of circles X = T ( T ) has been proved to have nontrivial fundamental group, but it is ∪ − Corrado Marastoni 39 Notes on Algebraic Topology possible to prove that any arcwise connected covering space of X is necessarily trivial(49): hence the proper subgroups of π1(X; x0 ) do not correspond to any covering space of X. 1.6.4 Covering automorphisms Let X and Y be topological spaces, π : Y X a covering space, and consider the set of −→ endomorphisms of (Y,π), i.e. End(Y X)= ϕ : Y Y : π ϕ = π .Thesubset | { −→ ◦ } Aut(Y X)= ϕ : Y Y : ϕ homeomorphism,π ϕ = π | { −→ ◦ } (the “covering automorphisms”, or deck transformations) has a natural structure of group, given by the composition. 1 2πit Examples. (1) The deck transformations of : R S (where (t)=e ) are the translations τk : R R 1 −→ n 1 1−→ given by τk(t)=t + k for k Z,henceAut(R S ) Z. (2) The deck transformations of z : S S are ∈ | 1 n 1 −→ the rotations of multiples of 2π/n,henceAut((S ,z ) S ) Z/nZ. | An immediate consequence of Corollary 1.6.26 is the following Proposition 1.6.30. Let π : Y X be a covering space with X and Y arcwise connected −→ and locally arcwise connected topological spaces, x X, y ,y1 Yx . Then there exists 0 ∈ 0 ∈ 0 ϕ Aut(Y X) with ϕ(y )=y if and only if G(Y,y )=G(Y,y ). ∈ | 0 1 0 1 What does this condition mean? By Proposition 1.6.19 we know that G(Y,y1) is conju- gated to G(Y,y )inπ (X, x ): if γ is a path in Y from y to y ,settingα = π γ it holds 0 1 0 0 1 ◦ G(Y,y )=[α 1] G(Y,y ) [α]. Hence, the condition G(Y,y )=G(Y,y ) is equivalent to 1 − · 0 · 0 1 [α] G(Y,y ) (the normalizer of G(Y,y )inπ (X, x )). ∈N 0 0 1 0 Theorem 1.6.31. Let π :(Y,y ) (X, x ) be a covering space with X and Y arcwise 0 −→ 0 connected and locally arcwise connected topological spaces. Then for any [α] G(Y,y ) ∈N 0 there exists one and only one covering automorphism ϕ such that ϕ (y )=˜αy (1). [α] [α] 0 0 The application G(Y,y ) Aut(Y X) obtained in this way is a surjective morphism N 0 −→ | of groups with kernel G(Y,y ), and provides an isomorphism of groups 0 G(Y,y0 ) N ∼ Aut(Y X). G(Y,y ) −→ | 0 Proof. Let y1 =˜αy (1): then the hypothesis [α] G(Y,y ) is equivalent to the fact that G(Y,y )= 0 ∈N 0 0 G(Y,y ), and the result follows from the Lifting criterion (Proposition 1.6.23). In particular, one constructs 1 explicitly ϕ Aut(Y X) as follows: given y Y and a path βy from y to y,onesetsϕ (y)= [α] ∈ | ∈ 0 [α] (πβy) (1) (note that ϕ[α](y )=cy (1) = y1, as required). ◦ y1 0 1 (49)The idea is that a arcwise connected covering space π : Y X induces on T and T (which are simply −→ − connected) trivial covering spaces ( T ) F ;ifF would not be a point (i.e., if such induced covering spaces ± × would not be homeomorphisms), two points of the same fiber would stay in different arcwise connected components, and that would contradict the fact that π is arcwise connected (just think that X = T ( T ), ∪ − and that x =(0, 0, 0) is the only point in common between T and T ...); hence F = pt ,andsince 0 − { } X = T ( T )thisimpliesthatπ is a homeomorphism. ∪ − Corrado Marastoni 40 Notes on Algebraic Topology Definition 1.6.32. A connected and locally arcwise connected covering space π :(Y,y ) 0 −→ (X, x0 ) is called normal, or Galois,ifG(Y,y0 ) is a normal subgroup of π1(X, x0 ). The following proposition shows the properties of normal covering spaces. In particular they act transitively on the fibers, and hence these covering spaces can be viewed as those having a complete symmetry among different sheets. Proposition 1.6.33. A connected and locally arcwise connected normal covering space π :(Y,y ) (X, x ) has the following properties. 0 −→ 0 (i) π (X, x )/G(Y,y ) ∼ Aut(Y X). 1 0 0 −→ | (ii) Aut(Y X) operates on the left on Y in a properly discontinuous way (hence freely)(50), | and the orbits are the fibers of π: in particular, the multiplicity of the covering space is equal to the index of G(Y,y0 ) in π1(X, x0 ). (iii) Denoted by Y/Aut(Y X) the space of orbits of Aut(Y X) in Y (i.e., the space of | | fibers of π) with the quotient topology, the natural bijection Y/Aut(Y X) X is a | −→ homeomorphism. (iv) Either all liftings of loops in X based at x0 are loops in Y , or no one of them is. (Such condition is also sufficient in order that a covering space be Galois.) Proof. (i) follows from Theorem 1.6.31, as well as the fact that the action on the left of Aut(Y X)hasthe | 1 fibers of π as orbits. Let U X be a evenly covered open neighborhood of x : hence one has π− (U)= ⊂ 0 V ,andlety V .Ifϕ (V ) ϕ (V ) = ,lety ,y V be such that ϕ (y )=ϕ (y ) U; λ Λ λ 0 λ0 1 λ0 2 λ0 ∅ 1 2 λ0 1 1 2 2 ∈ ∈ ∩ ∈ ∈ in particular y1 and y2 belong to the same fiber of π,andalsotothesamesheetVλ :hencey1 = y2 y, 0 ≡ which implies ϕ1 = ϕ2 because they coincide in the point y (uniqueness of lifting). This shows (ii). The bijection in (iii) is continuous by definition of quotient topology, and is open because such is also π.Finally, to be Galois is equivalent to the fact that G(Y,y1)=G(Y,y2) for any y1 and y2 in the same fiber of π, and this is equivalent to (iv). Remark 1.6.34. From Remark 1.6.20 it follows immediately that, if the covering space π :(Y,y ) (X, x ) is Galois, the subgroup of π1(X, x ) acting trivially on the fiber Yx 0 −→ 0 0 0 is G(Y,y0 ). Example. If π1(X, x0 ) is commutative, then obviously all arcwise connected and locally arcwise connected covering spaces of X are normal. On the other hand, let us show an example of connected and locally arcwise connected covering space which is not normal. Consider the function ✛✛ ✛ ✛✛ ✬ ✩✬ ✩✬ ✩✬ ✩ Y = y1 ✲ y0 ✲✲ y2 ✲ π ✫ ✪✫ ✪✫ ✪ ✫ ✪✛βα✛ ✛ ❄ X = ✬ ✩✬ ✩ x0 ✫ ✪✫ ✪ (50)Recall that the action of a group G on a set Z si said properly discontinuous if any z Z has a ∈ neighborhood U Z such that g1U g2U = ∅ if g1 = g2,andfree (a weaker notion) if any point has ⊂ ∩ trivial stabilizer, i.e. Gz = 1 for any z Z. { } ∈ Corrado Marastoni 41 Notes on Algebraic Topology (We mean that π(yj )=x (j =0, 1, 2) and that the arcs of Y denoted by (resp. by ) are sent into 0 −→ α (resp. into β) in the specified direction. Note that π is a surjective local homeomorphism with fiber of cardinality 3: by Proposition 1.6.7, π is a 3-sheet covering space. Consider the morphism of pointed spaces π :(Y,y ) (X, x ), and the injective morphism π# : π1(Y,y ) π1(X, x ). The space X is the bouquet 0 −→ 0 0 −→ 0 of two circles and hence π1(X, x0 ) is free on two generators [α]and[β]; on the other hand, π1(Y,y0 )is generated by the classes of loops ✛✛ ✛ ✛ ✛✛ ✛✛ γ = γ1 = ✓ ✓✏ ✏γ2 = ✓ ✏γ3 = ✓ ✏ 4 ✓ ✓✏ ✏ ✲ ✲ y ✲✲ y ✲✲ ✲ y0 y0 0 0 ✒ ✒✑ ✑ ✒ ✑ ✒ ✑ ✒ ✒✑ ✑ 2 2 and one has π#([γ1]) = [α β α], π#([γ2]) = [α ], π#([γ3]) = [β ], π#([γ4]) = [β α β]: hence the · · · · characteristic subgroup G(Y,y ) is generated by [α β α], [α2], [β2]and[β α β]. Would there exist a 0 · · · · covering automorphism sending y0 for example into y1, the liftings from y0 and y1 of the same loop based at x should be either both loops or no one of them: actuallyα ˜ andα ˜ are not loops (the first one goes from 0 y0 y1 y to y , the second from y to y ), but β˜ is not a loop (goes from y to y )whileβ˜ is. Therefore one 0 1 1 0 y0 0 2 y1 has Aut(Y X)= idY ,whichimplies G(Y,y0 ) = G(Y,y0 ) (hence the covering space is not normal). | { }1 N 1 Hence G(Y,y1)=[α− ] G(Y,y ) [α] = G(Y,y )andG(Y,y2)=[β− ] G(Y,y ) [β] = G(Y,y ). The · 0 · 0 · 0 · 0 heart j=0,1,2 G(Y,yj ) is the subgroup of π1(X, x0 ) formed by the loops whose liftings from the yj ’s are 2 2 1 all loops: it is generated by [α ]and[β ]. The action of π1(X, x )onπ− (x )= y ,y1,y2 is given by 0 0 { 0 } [α]=(01)and[β] = (0 2): hence the morphism π1(X, x ) S3 is surjective. 0 −→ A particular case of Galois covering space is, if it exists, to one with G(Y,y )= 1 . 0 { } Definition 1.6.35. A connected and locally arcwise connected covering space π : Y X −→ is called universal cover of X if Y is simply connected. Theorem 1.6.36. If X is a connected, locally arcwise connected and semi-locally sim- ply connected topological space, there exists a unique —up to canonical isomorphisms— universal cover π˜ :(X,˜ x˜ ) (X, x ), with the following properties. 0 −→ 0 (i) π (X, x ) Aut(X˜ X). 1 0 | (ii) Aut(X˜ X) operates on the left on X˜ in a properly discontinuous way (hence freely), | and the orbits are the fibers of π˜. (iii) if π :(Y,y ) (X, x ) is another connected and locally arcwise connected covering 0 −→ 0 space, there exists one and only one covering space π˜ :(X,˜ x˜ ) (Y,y ) such that Y 0 −→ 0 π˜ = π π˜ . (Hence, the universal cover of X is also the universal cover of any other ◦ Y connected and locally arcwise connected covering space of X.) (iv) The universal cover determines all other arcwise connected and locally arcwise con- nected covering spaces of X, in the following sense: if Γ Aut(X˜ X) is a subgroup, ⊂ | denoted by X/˜ Γ the space of orbits of Γ in X˜ endowed with the quotient topology, the natural map π˜ :(X/˜ Γ, [˜x ]) (X, x ) is a connected and locally arcwise connected Γ 0 −→ 0 covering of X, and moreover all arcwise connected and locally arcwise connected covering spaces of X are obtained in this way, up to a canonical isomorphism. Proof. Existence and uniqueness follow from Proposition 1.6.28 and Corollary 1.6.26. (i) and (ii) follow from Proposition 1.6.33, (iii) from Proposition 1.6.25, (iv) follows from (iii) and Proposition 1.6.33 (which says that Y X/˜ Aut(X˜ Y ), and Γ= Aut(X˜ Y ) Aut(X˜ X)). | | ⊂ | Corrado Marastoni 42 Notes on Algebraic Topology In particular, it is useful to remark that (i) provides a method —often the preferable one— for computing the fundamental group of X: π (X, x ) Aut(X˜ X). 1 0 | Some examples of this method will be shown in the next Section. Corrado Marastoni 43 Notes on Algebraic Topology 1.7 Exercises and complements (1) The universal covering of S1 is the exponential map : R S1 C, (t)=e2πit; −→ ⊂ more generally, the universal cover of the torus Tn =(S1)n is the map : Rn Tn, 2πit1 2πitn n n n n −→ (t1,...,tn)=(e ,...,e ), and it holds Aut(R T )= τ : R R ,τ(t1,...,tn)= n | { −→ n n (t1 + k1,...,tn + kn) for some kj Z Z , which yields again π1(T ) Z . ∈ } (2) The real space projective Pn (with n 1) is defined as the space of orbits of the n+1 ≥ n+1 multiplicative group R× in (R )× (i.e., the family of homogeneous lines of R ), en- dowed with the quotient topology. Consider the Hopf map q : Sn Pn (the restriction n n+1 n −→ n to S of the projection (R )× P ), which is a 2-sheet covering of P .Ifn 2 −→ ≥ this map is the universal cover; since the covering automorphisms of q are idSn ,we n n n ± get π1(P ) Aut(S P )=Z/2Z. On the other hand, for n = 1 we saw that the map | ( )2 : S1 S1 has the same fibers of q, hence there exists a homeomorphism γ : S1 ∼ P1 · −→ 2 1 −→ such that q = γ ( ) . Therefore π1(P ) Z. ◦ · An interesting application is a particular case of the Borsuk-Ulam theorem (if n 2, there ≥ does not exist any continuous functions of Sn into itself which is odd and nullhomotopic): Corollary 1.7.1. (Borsuk-Ulam, particular case) If n 2, there does not exist odd con- ≥ tinuous functions of Sn with values in S1. n 1 j j Proof. By absurd let f : S S be a odd continuous function, and denote by qj : S P the Hopf −→ n 1 −→ map. By Proposition 1.1.14, there exists g : P P continuous such that g qn = q1 f (note that −→ n ◦ n ◦ q1 f is constant on the fibers of qn). Now, fixed x0 S , the morphism g# : π1(P ,qn(x0 )) = Z/2Z ◦ 1 ∈ −→ π1(P ,g(qn(x0 ))) = Z must be zero, hence by the Lifting criterion (Proposition 1.6.23) there exists a unique n 1 liftingg ˜ : P S such that g = q1 g˜ andg ˜(qn(x0 )) = f(x0 ). But one has also f =˜g qn (namely f and −→ ◦ ◦ g˜ qn are two liftings of g qn which coincide in x ): this is a contradiction because f( x)= f(x)while ◦ ◦ 0 − − g˜(qn( x)) =g ˜(qn(x)) = f(x). − A consequence of Corollary 1.7.1 is, for example, that at any particular time there are two antipodal places on the Earth with the same temperature and the same pressure. Namely let, at a certain fixed moment, t : S2 R be the temperature and p : S2 R be the −→ −→ pressure: if the statement would be false, the function ϕ : S2 R2 given by ϕ(x)= −→ (t(x) t( x),p(x) p( x)) would never take the value (0, 0) and hence the function − − − − f : S2 S1 given by f(x)=ϕ(x)/ ϕ(x) would be continuous and odd. −→ | | (3) We now deal with the fundamental group of topological groups. Proposition 1.7.2. Let G a topological group with identity element 1. Then: (i) the fundamental group π1(G, 1) is commutative; (ii) if G is a connected and locally arcwise connected topological group, and π :(E,e) −→ (G, 1) is a connected and locally arcwise connected covering, then there exists one and only one multiplication on E for which (a) E is a topological group with identity element e, (b) π is a morphism. Proof. (The student should verify by exercise the unproven statements.) (i) Let µ : G G G be the × −→ multiplication. This map induces a pointwise product between loops based at 1, by defining (α β)(t)= ∗ µ(α(t),β(t)) for any t I; and this product induces a product in π1(G, 1). We then have two operations ∈ ∗ Corrado Marastoni 44 Notes on Algebraic Topology ( and )inπ1(G, 1). Now, by a elementary algebraic fact, if a set S is endowed with two binary operations ∗ · and with a same identity element and such that (a b) (c d)=(a c) (b d) for any a, b, c, d S, ∗ · ∗ · ∗ · ∗ · ∈ then and coincide and are associative and commutative: such condition is verified in our case. (ii) The ∗ · map µ (π π):(E E,(e, e)) (G, 1) (given n covering spaces πj : Yj Xj ,andsetπ = π1 πn, ◦ × × −→ −→ ×···× Y = Y1 Yn and X = X1 Xn,alsoπ : Y X is a covering) lifts uniquely to a map ×···× ×···× −→ µ˜ :(E E,(e, e)) (E,e): this follows from Proposition 1.6.23, since (µ (π π))#([α], [β]) = [µ(π × −→ ◦ × ◦ α,π β)] = [π α] [π β]=[π α] [π β]=π#([α β]) G(E,e). The uniqueness of lifting shows all ◦ ◦ ∗ ◦ ◦ · ◦ · ∈ the properties which are required toµ ˜ to be the desired operation in E: for example, to show that e is the identity element ofµ ˜ we defineµ ˜e :(E,e) (E,e)by˜µe(y)=˜µ(y, e): since π µ˜e = π,weget˜µe =idE −→ ◦ (because idE andµ ˜e are two morphisms of the covering space π which coincide in e). (4) Let us study the covering spaces of manifolds and, in particular, what happens to the fundamental group when we remove, from a given manifold, a closed submanifold which is “small enough”. Let M be a (arcwise) connected C0 manifold of dimension m, N M a C0 submanifold ⊂ of dimension n m, and let ι : M N M be the open embedding. ≤ \ −→ Proposition 1.7.3. If π : P M is a local homeomorphism, then on P is naturally 0 −→ 1 induced a structure of C manifold of dimension m,andonπ− (N) a structure of sub- manifold of dimension n. 1 Proof. The local charts on P and π− (N) are just the local pullbacks of local charts on M and N. Proposition 1.7.4. The following statements hold. (i) If m n 2, then M N is connected. − ≥ \ (ii) If m n 3, any covering space of M N extends to one of M (i.e., given a − ≥ \ covering space q : P M N there exist a covering spaceq ˜ : P˜ M and a −→ \ −→ injective morphism of manifolds ι : P P such thatq ˜ ι = ι q). P −→ ◦ P ◦ m n Proof. (i) It is clear that R R is connected if and only if m n 2. In general, it is enough to show \ − ≥ that, given x, y M N, there exists a path in M N from x to y.Letα : I M be a path from x to ∈ \ \ −→ n y,and (Uλ,ϕλ):λ Λ be an atlas of M such that Uλ N = ∅ or ϕλ(Uλ N)=R .Bycompactness, { ∈ } ∩ ∩ there exist 0 = t0 Theorem 1.7.5. For x M N consider the morphism of fundamental groups 0 ∈ \ ι : π (M N,x ) π (M,x ). # 1 \ 0 −→ 1 0 Then: (i) if m n 2, then ι is surjective; − ≥ # (ii) if m n 3, then ι is a isomorphism. − ≥ # Corrado Marastoni 45 Notes on Algebraic Topology Proof. (i) It is enough to show that, given x, y M N and a path α : I M from x to y, there ∈ \ −→ exists a path γ : I M N from x to y with [α]=[γ]. Following the proof of the first statement of −→ \ Proposition 1.7.4, we may also require that (3) γj be homotopic (not necessarily rel ∂I)to α .The |[tj ,tj+1] path γ obtained by joining the paths γj has the required properties. (ii) Let H = ker(ι#), and let us prove that H = 1 . Consider the connected covering space q : P M N having H as characteristic { } −→ \ subgroup.(51) From Proposition 1.7.4 we know that there exist a coveringq ˜ : P M and an injective −→ morphism of manifolds ιP : P P such thatq ˜ ιP = ι q.BydefinitionofH,onehas(˜q# ιP #)(π1(P )) = −→ ◦ ◦ ◦ (ι# q#)(π1(P )) = ι#(H)= 1 ,hence(˜q# being injective and ιP # surjective) it holds π1(P )= 1 :i.e., ◦ { } { } P is the universal cover of M.Ontheotherhand,since P ιP (P )isasubmanifoldofcodimension 1 \ 1 m n 3ofP (namely, from ιP (P )=˜q− (M N)wegetP ιP (P )=˜q− (N), and it is enough to recall − ≥ \ \ Proposition 1.7.3), the universal cover π : S P extends toπ ˜: S P ,which(P being simply connected) −→ −→ is a homeomorphism: hence also π is a homeomorphism, because the fiber has cardinality 1. Therefore P is simply connected, i.e. H = 1 . { } (5) We now deal with the fundamental group of some classical real groups. Given n N, ∈ let M(n, K) be the vector space of square matrices of order n with coefficients in K = R, C. t If A M(n, C), we denote by A∗ = A the adjoint matrix, and: ∈ GL(n, C)= A M(n, C):det(A) =0 (complex general linear group) { ∈ } 1 U(n, C)= A GL(n, C):A− = A∗ (unitary group) { ∈ } SU(n, C)= A U(n, C):det(A)=1 (special unitary group) { ∈ } GL(n, R)= M(n, R) GL(n, C) (real general linear group) ∩ GL±(n, R)= A GL(n, R):det(A) 0 { ∈ ≷ } O(n, R)= M(n, R) U(n, C) (real orthogonal group) ∩ SO(n, R)= M(n, R) SU(n, C) (real special orthogonal group) ∩ and moreover H(n, C)= A M(n, C):A = A∗ (hermitian matrices) + { ∈ } H (n, C)= A H(n, C):txAx>0 x Cn 0 (positive definite h. m.) { ∈ ∀ ∈ \{ }} S(n, R)= M(n, R) H(n, C) (symmetric matrices) + ∩ S (n, R)= M(n, R) H+(n, C) (positivedefinites.m.). ∩ We briefly recall the following facts (for further details we refer e.g. to Godbillon [5, II.2]): (1) M(n, C)(resp.M(n, R)) is a vector space on C (resp. on R) of dimension n2, and H(n, C)(resp.S(n, R)) is a real subspace of M(n, C)(resp.M(n, R)) of dimension 2 n (resp. n(n+1)/2). The application A A = tr(AA∗) is a norm on M(n, C), → which induces the norm A = tr(A At ) on M(n, R). (2) GL(n, C) is an open subset of M(n, C) and a multiplicative topological group. n (3) The exponential exp : M(n, C) GL(n, C)(whereexp(A)= ∞ A /n!) satisfies −→ n=0 exp(A + B)=exp(A)exp(B)ifAB = BA, and is also a diffeomorphism between an open neighborhood of 0 M(n, C) and an open neighborhood of the identity ∈ (51)Such a q exists, because the manifolds —and M N is so— are locally simply connected, and in \ particular locally arcwise connected and semi-locally simply connected. Corrado Marastoni 46 Notes on Algebraic Topology (52) 2 1n GL(n, C): this makes GL(n, C) into a real Lie group of dimension 2n . ∈ Moreover, exp induces a homeomorphism of H(n, C) on H+(n, C) and of S(n, R) on S+(n, R). (4) U(n, C), SU(n, C), GL(n, R), GL+(n, R), O(n, R) and SO(n, R) are closed Lie sub- groups of GL(n, C) of dimensions n2, n2 1, n2, n2, n(n 1)/2 and n(n 1)/2. − − − Local charts around the identity 1n of these groups, seen as real submanifolds of GL(n, C), are obtained by inverting the restriction of exp respectively to the real subspaces u(n, C)= A M(n, C):A + A∗ =0, su(n, C)= A M(n, C): { ∈ } { +∈ A + A∗ =0, tr(A)=0, gl(n, R)=M(n, R) (for GL(n, R) and GL (n, R)) and } so(n, R)= A M(n, R):A + At =0 (for O(n, R) and SO(n, R))(53). More- { ∈ } over, U(n, C), SU(n, C), O(n, R) and SO(n, R) are compact (if A U(n, C)then ∈ A = √n). (5) There are isomorphisms of Lie groups U(n, C) S1 SU(n, C) and O(n, R) 1 × {± }× SO(n, R), given by A (det(A), A/ det(A)). Moreover, SU(n, C) and SO(n, R) are → arcwise connected, and hence U(n, C) is arcwise connected while O(n, R) has two connected components. (6) GL(n, C) is homeomorphic to the product H+(n, C) U(n, C)(polar decomposition) × and in particular, by (3) and (5), GL(n, C) is arcwise connected. A homeomorphism is induced between GL(n, R) and S+(n, C) O(n, R) and in particular between × GL+(n, R) and S+(n, C) SO(n, R), hence GL(n, R) has two connected components × GL±(n, R). + + From (3) one has π1(H (n, C), 1n) 1 and π1(S (n, R), 1n) 1 : hence from (5) and { } { } (6) one has π1(GL(n, C)) π1(U(n, C)) Z π1(SU(n, C)), × π1(GL(n, R), 1n) π1(O(n, R), 1n) π1(SO(n, R)). We are left with computing the fundamental groups of SO(n; R) and SU(n, C). cos θ sin θ 1 It holds SO(1; )= 1 , and SO(2; )= : θ R . For n = 3, R R sin θ cos θ ∈ S • { } − note that S3 (intended as the group of quaternions of unitary norm, see Example 3 3 3 1 1.4) operates on R by S R (q, u) quq− : such trasformation is linear and × → preserves the norms, i.e. it is in SO(3; R), and one obtains in this way a morphism S3 SO(3; R), which is surjective and has 1 as kernel. This shows that SO(3; R) −→ {± } (52)A real Lie group of dimension m is a topological group with a structure of real 1 manifold of dimension C m which makes the multiplication and the inversion into differentiable maps. By the way, GL(n, C)would be of course also a complex Lie group, but here we are interested only in its structure of real differentiable manifold. (53)In the terminology of Lie theory, these vector subspaces are the Lie algebras associated to the Lie subgroups: in the ambient vector space M(n, C), they are the tangent space to the Lie subgroups at the identity 1n (hence, one uses also the notation gl(n, C)=M(n, C)). In general, a Lie algebra on a field K is a vector space V on K endowed with a internal multiplication [ , ] which is bilinear, antisymmetric · · and satisfying the Jacobi identity [x, [y, z]] + [y, [z,x]] + [z,[x, y]] = 0 for any x, y, z V : in our case, for ∈ A, B M(n, C), it is [A, B]=AB BA (the commutator of A and B). Note that each one of the real ∈ − subspaces of M(n, C) considered here is stable with respect to such operation. Corrado Marastoni 47 Notes on Algebraic Topology 3 is homeomorphic to P , and hence π1(SO(3; R)) Z/2Z. In the general case, note that SO(n; R) is a submanifold of the open subset GL+(n; R) of M(n; R), of dimension n(n 1)/2. We embed N = SO(n 1; R)intoM = SO(n; R) as those − − orthogonal transformations that fix the North pole e : note that dim M dim N = n − n 1, hence for n 4wehaveπ1(SO(n; R) SO(n 1; R)) π1(SO(n; R)). Now − ≥ \ − we aim to show that SO(n; R) SO(n 1; R) is an open subset of SO(n; R)which \ − deformation-retracts to a manifold homeomorphic to SO(n 1; R), and this will − n 1 imply that π1(SO(n; R)) Z/2Z for n 4. Let ρ : SO(n; R) S − be the n 1 ≥ −→ map ρ(A)=Aen:settingV = S − en ,wehavepreciselySO(n; R) SO(n 1 \{ } 1\ − 1; R)=ρ− (V ). Now define a function f : V SO(n 1; R) ρ− (V ). Let n 1 × − −→ s : S − en SO(n) be the continuous application under which, given x = en \{− }−→ − (the South pole), s(x) induces the identity on x, en ⊥ and the rotation sending en n 1 n 1 into x in the plane x, en . Let a : S − S − be the antipodal map: since n 1 −→ a(V )=S − en , for x V the trasformation s(a(x)) is well defined. So let us \{− } ∈ set f(x, α)=a s(a(x)) α (note that f is well defined because f(x, α)(e )=x = e ). ◦ ◦ n n Well, such f is a homeomorphism: namely it is continuous, and its inverse is given by g(β)=(β(e ),s(a(β(e ))) 1 a β). Since V is contractible, we have proven the n n − ◦ ◦ claim. Summarizing up, one has 1 (n = 1) { } π1(GL(n, R), 1n) π1(O(n, R), 1n) π1(SO(n; R)) Z (n = 2) Z/2Z (n 3) ≥ As for SU(n, C), let us begin by observing that SU(1, C)= 1 is simply connected. • { } Since dim (SU(n, C)) = n2 1, if n 2 one has dim (SU(n, C)) dim (SU(n R − ≥ R − R − 1, C)) = 2n 1 3, and hence π1(SU(n, C) SU(n 1, C)) ∼ π1(SU(n, C)). On the − ≥ \ − −→ other hand, arguing as before one shows that for n 2 there exists a homeomorphism ≥ 2n 1 n 2n f : SU(n, C) SU(n 1, C) V SU(n 1, C), where V = S − e2n C R , \ − −→ × − \{ }⊂ contractible. This implies that SU(n, C) is simply connected and therefore π1(SU(n, C)) = 1 ,π1(GL(n, C)) π1(U(n, C)) Z. { } Corrado Marastoni 48