by p , and let eU1 and eU2 be the eUα’s containing ef1(y) and ef2(y) , respectively. By continuity of ef1 and ef2 there is a neighborhoodN of y mapped into eU1 by ef1 and into eU2 by ef2. If ef1(y) ≠ ef2(y) then eU1 ≠ eU2, hence eU1 and eU2 are disjoint and fe1≠ ef2 throughout the neighborhood N . On the other hand, if ef1(y)= ef2(y) then Ue1 = eU2 so ef1= ef2 on N since p ef1 = p ef2 and p is injective on eU1= eU2. Thus the set of points where ef1 and ef2 agree is both open and closed in Y . tu
We shall now show how to construct a simply-connected covering space of X if X is path-connected, locally path-connected, and semilocally simply-connected. To motivate the construction, supposep : ( eX,xe0)
→
(X, x0) is a simply-connected cover-ing space. Each pointxe∈ eX can then be joined to xe0 by a unique homotopy class of paths, by Proposition 1.6, so we can view points of eX as homotopy classes of paths starting atxe0. The advantage of this is that, by the homotopy lifting property, homo-topy classes of paths in eX starting atxe0 are the same as homotopy classes of paths in X starting at x0. This gives a way of describing eX purely in terms of X .Given a path-connected, locally path-connected, semilocally simply-connected spaceX with a basepoint x0∈ X , we are therefore led to define
Xe=
[γ] |||| γ is a path in X starting at x0
where, as usual, [γ] denotes the homotopy class of γ with respect to homotopies that fix the endpointsγ(0) and γ(1) . The function p : eX
→
X sending [γ] to γ(1) is then well-defined. SinceX is path-connected, the endpoint γ(1) can be any point of X , so p is surjective.Before we define a topology on eX we make a few preliminary observations. Let U be the collection of path-connected open sets U ⊂ X such that π1(U)
→
π1(X) istrivial. Note that if the mapπ1(U)
→
π1(X) is trivial for one choice of basepoint in U , it is trivial for all choices of basepoint since U is path-connected. A path-connected open subset V ⊂ U ∈U is also in U since the composition π1(V )→
π1(U)→
π1(X)will also be trivial. It follows that U is a basis for the topology on X if X is locally path-connected and semilocally simply-connected.
Given a set U ∈U and a path γ in X from x0 to a point in U , let U[γ]=
[γ η] |||| η is a path in U with η(0) = γ(1)
As the notation indicates, U[γ] depends only on the homotopy class [γ] . Observe that p : U[γ]
→
U is surjective since U is path-connected and injective since differ-ent choices of η joining γ(1) to a fixed x ∈ U are all homotopic in X , the map π1(U)→
π1(X) being trivial. Another property is(∗)
U[γ] = U[γ0] if [γ0]∈ U[γ]. For if γ0 = γ η then elements of U[γ0] have the form [γ η µ] and hence lie in U[γ], while elements of U[γ] have the form [γ µ]= [γ η η µ] = [γ0 η µ] and hence lie in U[γ0].
This can be used to show that the sets U[γ] form a basis for a topology on eX . For if we are given two such sets U[γ], V[γ0] and an element [γ00]∈ U[γ]∩ V[γ0], we have U[γ]= U[γ00]andV[γ0]= V[γ00]by(∗). So if W ∈Uis contained inU∩V and contains γ00(1) then W[γ00]⊂ U[γ00]∩ V[γ00] and [γ00]∈ W[γ00].
The bijectionp : U[γ]
→
U is a homeomorphism since it gives a bijection between the subsetsV[γ0]⊂ U[γ] and the setsV ∈Ucontained inU . Namely, in one direction we have p(V[γ0]) = V and in the other direction we have p−1(V )∩ U[γ] = V[γ0] forany[γ0]∈ U[γ] with endpoint inV , since V[γ0]⊂ U[γ0]= U[γ] and V[γ0] maps ontoV by the bijection p .
The preceding paragraph implies that p : eX
→
X is continuous. We can also de-duce that this is a covering space since for fixedU ∈U, the setsU[γ] for varying [γ]partitionp−1(U) because if [γ00]∈ U[γ]∩ U[γ0] then U[γ]= U[γ00]= U[γ0]by (∗).
A natural basepoint for eX is [x0] , the homotopy class of the constant path at x0. Any [γ]∈ eX can be joined to [x0] by a path in eX by restricting γ to progressively shorter segments [0, t]⊂ [0, 1], so eX is path-connected. To see that π1( eX)= 0, let f : I
→
X be a loop based at [xe 0] . The composition pf is then a loop γ in X based atx0. Letγt be the path inX obtained by restricting the loop γ to [0, t] . Then [γt] for t varying from 0 to 1 forms a path in eX lifting the loop γ . This lift [γt] starts at[γ0]= [x0]= f (0), so by the unique lifting property of the cover space eX→
X wemust have [γt]= f (t) for all t . In particular, [γ1]= f (1) = [x0] . Since γ1= γ , this says the loop γ = pf is nullhomotopic. Thus p∗([f ]) = 0 in π1(X) . Since p∗ is injective, [f ]= 0 and hence π1( eX)= 0.
This completes the construction of a simply-connected covering space eX
→
X .In concrete cases one usually constructs a simply-connected covering space by more direct methods. For example, supposeX is the union of subspaces A and B for which simply-connected covering spaces eA
→
A and eB→
B are already known. Then one can attempt to build a simply-connected covering space eX→
X by assembling copies of eA and eB . For example, for X= S1∨ S1, if we take A and B to be the two circles, then eA and eB are each R, and we can build the simply-connected cover eX described earlier in this section by glueing together infinitely many copies of eA and B , the horizontal and vertical lines in ee X . Here is another illustration of this method:E
xample 1.35. For integers m, n ≥ 2, let Xm,n be the quotient space of a cylinder S1×I under the identifications (z, 0) ∼ (e2π i/mz, 0) and (z, 1) ∼ (e2π i/nz, 1) . Let A ⊂ X and B ⊂ X be the quotients of S1×[0,1/2] and S1×[1/2, 1], so A and B are the mapping cylinders of z,
zm and z,
zn, withA∩ B = S1. The simplest case is m = n = 2, when A and B are M¨obius bands and X2,2 is the Klein bottle. We encountered the complexesXm,n previously in analyzing torus knot complements in Example 1.24.The figure for Example 1.29 at the end of the preceding section shows whatA looks like in the typical case m= 3. We have π1(A)≈ Z, and the universal cover eA is homeomorphic to a product Cm×R where Cm is the graph that is a cone on m points, as shown in the figure to the right. The situation for B is similar, and eB is homeomorphic to Cn×R. Now we attempt to build the universal cover eXm,n from copies of eA and eB . Start with a copy of eA . Its boundary, the outer edges of its fins, consists of m copies of R. Along each of these m boundary
lines we attach a copy of eB . Each of these copies of eB has one of its boundary lines attached to the initial copy of eA , leaving n− 1 boundary lines free, and we attach a new copy of eA to each of these free boundary lines. Thus we now have m(n− 1) + 1 copies of eA . Each of the newly attached copies of eA has m− 1 free boundary lines, and to each of these lines we attach a new copy of eB . The process is now repeated ad infinitim in the evident way. Let eXm,n be the resulting space.
The product structures eA = Cm×R and eB = Cn×R give eXm,nthe structure of a productTm,n×R where Tm,n
is an infinite graph constructed by an inductive scheme just like the construction of eXm,n. ThusTm,nis the union of a sequence of finite subgraphs, each obtained from the preceding by attaching new copies of Cm or Cn. Each of these finite subgraphs deformation retracts onto the preceding one. The infinite concatenation of these
defor-mation retractions, with the kth graph deformation retracting to the previous one during the time interval [1/2k, 1/2k−1] , gives a deformation retraction of Tm,n onto the initial stage Cm. SinceCm is contractible, this means Tm,n is contractible, hence also eXm,n, which is the product Tm,n×R. In particular, eXm,n is simply-connected.
The map that projects each copy of eA in eXm,n to A and each copy of eB to B is a covering space. To define this map precisely, choose a point x0 ∈ S1, and then the image of the line segment{x0}×I in Xm,nmeetsA in a line segment whose preimage in eA consists of an infinite number of line segments, appearing in the earlier figure as the horizontal segments spi-raling around the central vertical axis. The picture in eB is similar, and when we glue together all the copies of eA and eB
to form eXm,n, we do so in such a way that these horizontal segments always line up exactly. This decomposes eXm,n into infinitely many rectangles, each formed from a rectangle in an eA and a rectangle in a eB . The covering projection eXm,n
→
Xm,n is thequotient map that identifies all these rectangles.
Now we return to the general theory. The hypotheses for constructing a simply-connected covering space ofX in fact suffice for constructing covering spaces realiz-ing arbitrary subgroups of π1(X) :
P
roposition 1.36. Suppose X is path-connected, locally path-connected, and semilo-cally simply-connected. Then for every subgroup H⊂ π1(X, x0) there is a covering space p : XH→
X such that p∗ π1(XH,xe0)= H for a suitably chosen basepoint e
x0∈ XH.
P
roof: For points [γ] , [γ0] in the simply-connected covering space eX constructed above, define [γ] ∼ [γ0] to mean γ(1) = γ0(1) and [γγ0] ∈ H . It is easy to seethat this is an equivalence relation sinceH is a subgroup; namely, it is reflexive since H contains the identity element, symmetric since H is closed under inverses, and transitive since H is closed under multiplication. Let XH be the quotient space of eX obtained by identifying [γ] with [γ0] if [γ]∼ [γ0] . Note that if γ(1)= γ0(1) , then [γ]∼ [γ0] iff [γη]∼ [γ0η] . This means that if any two points in basic neighborhoods U[γ] andU[γ0]are identified inXH then the whole neighborhoods are identified. Hence the natural projection XH
→
X induced by [γ],
γ(1) is a covering space.If we choose for the basepointxe0∈ XH the equivalence class of the constant path c at x0, then the image of p∗:π1(XH,xe0)
→
π1(X, x0) is exactly H . This is because for a loop γ in X based at x0, its lift to eX starting at [c] ends at [γ] , so the image of this lifted path in XH is a loop iff [γ]∼ [c], or equivalently, [γ] ∈ H . tu Having taken care of the existence of covering spaces of X corresponding to all subgroups of π1(X) , we turn now to the question of uniqueness. More specifically, we are interested in uniqueness up to isomorphism, where anisomorphism between covering spaces p1: eX1→
X and p2: eX2→
X is a homeomorphism f : eX1→
Xe2 suchthat p1 = p2f . This condition means exactly that f preserves the covering space structures, takingp1−1(x) to p−12 (x) for each x∈ X . The inverse f−1 is then also an isomorphism, and the composition of two isomorphisms is an isomorphism, so we have an equivalence relation.
P
roposition 1.37. If X is connected and locally connected, then two path-connected covering spaces p1: eX1→
X and p2: eX2→
X are isomorphic via an isomor-phism f : eX1→
Xe2 taking a basepoint xe1∈ p1−1(x0) to a basepointxe2∈ p2−1(x0) iff p1∗ π1( eX1,xe1)= p2∗ π1( eX2,xe2) .
P
roof: If there is an isomorphism f : ( eX1,xe1)→
( eX2,xe2) , then from the two relations p1= p2f and p2 = p1f−1 it follows that p1∗ π1( eX1,xe1)= p2∗ π1( eX2,xe2) . Con-versely, suppose that p1∗ π1( eX1,xe1)
= p2∗ π1( eX2,xe2)
. By the lifting criterion, we may lift p1 to a map pe1:( eX1,xe1)
→
( eX2,xe2) with p2pe1= p1. Symmetrically, we obtain pe2:( eX2,xe2)→
( eX1,xe1) with p1pe2 = p2. Then by the unique lifting property,e
p1pe2=11 and pe2pe1=11 since these composed lifts fix the basepoints. Thus pe1 and e
p2 are inverse isomorphisms. tu
We have proved the first half of the following classification theorem:
T
heorem 1.38. Let X be path-connected, locally path-connected, and semilocally simply-connected. Then there is a bijection between the set of basepoint-preserving isomorphism classes of path-connected covering spaces p : ( eX,xe0)→
(X, x0) and theset of subgroups of π1(X, x0) , obtained by associating the subgroup p∗ π1( eX,xe0) to the covering space ( eX,xe0) . If basepoints are ignored, this correspondence gives a bijection between isomorphism classes of path-connected covering spaces p : eX
→
Xand conjugacy classes of subgroups of π1(X, x0) .
P
roof: It remains only to prove the last statement. We show that for a covering space p : ( eX,xe0)→
(X, x0) , changing the basepoint xe0 within p−1(x0) corresponds exactly to changing p∗ π1( eX,xe0)to a conjugate subgroup of π1(X, x0) . Suppose that xe1 is another basepoint inp−1(x0) , and let eγ be a path from ex0 toxe1. Then eγ projects to a loop γ in X representing some element g∈ π1(X, x0) . Set Hi= p∗ π1( eX,xei) for i= 0, 1. We have an inclusion g−1H0g ⊂ H1 since for ef a loop at xe0, eγ efeγ is a loop at xe1. Similarly we have gH1g−1 ⊂ H0. Conjugating the latter relation by g−1 gives H1⊂ g−1H0g , so g−1H0g= H1. Thus, changing the basepoint from xe0 to xe1 changes H0 to the conjugate subgroup H1= g−1H0g .
Conversely, to change H0 to a conjugate subgroup H1= g−1H0g , choose a loop γ representing g , lift this to a path eγ starting at ex0, and letxe1= eγ(1). The preceding argument then shows that we have the desired relationH1= g−1H0g . tu
A consequence of the lifting criterion is that a simply-connected covering space of a path-connected, locally path-connected space X is a covering space of every other path-connected covering space of X . A simply-connected covering space of X is therefore called auniversal cover. It is unique up to isomorphism, so one is justified in calling itthe universal cover.
More generally, there is a partial ordering on the various path-connected covering spaces of X , according to which ones cover which others. This corresponds to the partial ordering by inclusion of the corresponding subgroups ofπ1(X) , or conjugacy classes of subgroups if basepoints are ignored.
Representing Covering Spaces by Permutations
We wish to describe now another way of classifying the different covering spaces of a connected, locally path-connected, semilocally simply-connected space X , with-out restricting just to connected covering spaces. To give the idea,
con-sider the 3 sheeted covering spaces of S1. There are three of these, Xe1, eX2, and eX3, with the subscript indicating the number of compo-nents. For each of these covering spacesp : eXi
→
S1the three different lifts of a loop inS1generating π1(S1, x0) determine a permutation of p−1(x0) sending the starting point of the lift to the ending point of the lift. For eX1 this is a cyclic permutation, for eX2 it is a transposition of two points fixing the third point, and for eX3 it is the identity permu-tation. These permutations obviously determine the covering spaces uniquely, up to isomorphism. The same would be true for n sheeted covering spaces of S1 for arbitrary n , even for n infinite.The covering spaces of S1∨ S1 can be encoded using the same idea. Referring back to the large table of examples near the beginning of this section, we see in the covering space (1) that the loopa lifts to the identity permutation of the two vertices and b lifts to the permutation that transposes the two vertices. In (2), both a and b
lift to transpositions of the two vertices. In (3) and (4),a and b lift to transpositions of different pairs of the three vertices, while in (5) and (6) they lift to cyclic permutations of the vertices. In (11) the vertices can be labeled byZ, with a lifting to the identity permutation and b lifting to the shift n
,
n+ 1. Indeed, one can see from these examples that a covering space ofS1∨ S1 is nothing more than an efficient graphical representation of a pair of permutations of a given set.This idea of lifting loops to permutations generalizes to arbitrary covering spaces.
For a covering space p : eX
→
X , a path γ in X has a unique lift eγ starting at a given point of p−1(γ(0)) , so we obtain a well-defined map Lγ:p−1(γ(0))→
p−1(γ(1)) bysending the starting point eγ(0) of each lift eγ to its ending point eγ(1). It is evident that Lγ is a bijection since Lγ is its inverse. For a composition of paths γη we have Lγη = LηLγ, rather than LγLη, since composition of paths is written from left to right while composition of functions is written from right to left. To compensate for this, let us modify the definition by replacing Lγ by its inverse. Thus the new Lγ is a bijection p−1(γ(1))
→
p−1(γ(0)) , and Lγη = LγLη. Since Lγ depends only on the homotopy class of γ , this means that if we restrict attention to loops at a basepoint x0∈ X , then the association γ,
Lγ gives a homomorphism fromπ1(X, x0) to the group of permutations ofp−1(x0) . This is called the action of π1(X, x0) on the fiber p−1(x0) .Let us see how the covering space p : eX
→
X can be reconstructed from the asso-ciated action of π1(X, x0) on the fiber F = p−1(x0) , assuming that X is connected, path-connected, and semilocally simply-connected, so it has a universal cover eX0→
X .We can take the points of eX0 to be homotopy classes of paths in X starting at x0, as in the general construction of a universal cover. Define a map h : eX0×F
→
X send-eing a pair ([γ],xe0) to eγ(1) where eγ is the lift of γ to eX starting at xe0. Then h is continuous, and in fact a local homeomorphism, since a neighborhood of([γ],xe0) in Xe0×F consists of the pairs ([γη], ex0) with η a path in a suitable neighborhood of γ(1) . It is obvious that h is surjective since X is path-connected. If h were injec-tive as well, it would be a homeomorphism, which is unlikely since eX is probably not homeomorphic to eX0×F . Even if h is not injective, it will induce a homeomorphism from some quotient space of eX0×F onto eX . To see what this quotient space is, suppose h([γ],xe0)= h([γ0],xe00) . Then γ and γ0 are both
paths from x0 to the same endpoint, and from the figure we see that xe00 = Lγ0γ(xe0) . Letting λ be the loop γ0γ , this
γ
γ x0
f
f γf
x0 xf0
0 0
γ0 means that h([γ],xe0) = h([λγ], Lλ(xe0)) . Conversely, for
any loop λ we have h([γ],xe0) = h([λγ], Lλ(xe0)) . Thus h induces a well-defined map to eX from the quotient space of Xe0×F obtained by identifying ([γ], ex0) with ([λγ], Lλ(xe0))
for each [λ]∈ π1(X, x0) . Let this quotient space be denoted eXρ where ρ is the ho-momorphism fromπ1(X, x0) to the permutation group of F specified by the action.
Notice that the definition of eXρ makes sense whenever we are given an action ρ of π1(X, x0) on a set F . There is a natural projection eXρ
→
X sending ([γ],xe0) to γ(1) , and this is a covering space since if U ⊂ X is an open set over which the universal cover eX0 is a product U×π1(X, x0) , then the identifications defining eXρ simply collapseU×π1(X, x0)×F to U ×F .Returning to our given covering space eX
→
X with associated action ρ , the map Xeρ→
X induced by h is a bijection and therefore a homeomorphism since h was ae local homeomorphism. Since this homeomorphism eXρ→
X takes each fiber of ee Xρ to the corresponding fiber of eX , it is an isomorphism of covering spaces.If two covering spaces p1: eX1
→
X and p2: eX2→
X are isomorphic, one may ask how the corresponding actions of π1(X, x0) on the fibers F1 and F2 over x0 are related. An isomorphism h : eX1→
Xe2 restricts to a bijection F1→
F2, and evidently Lγ(h(ex0))= h(Lγ(xe0)) . Using the less cumbersome notation γxe0 for Lγ(xe0) , this relation can be written more concisely asγh(xe0)= h(γ ex0) . A bijection F1→
F2 withthis property is what one would naturally call anisomorphism of sets with π1(X, x0) action. Thus isomorphic covering spaces have isomorphic actions on fibers. The converse is also true, and easy to prove. One just observes that for isomorphic actions ρ1 and ρ2, an isomorphism h : F1
→
F2 induces a map eXρ1→
Xeρ2 and h−1 induces a similar map in the opposite direction, such that the compositions of these two maps, in either order, are the identity.This shows that n sheeted covering spaces of X are classified by equivalence classes of homomorphisms π1(X, x0)