A Generalized Approach to the Fundamental Group
Author(s): Daniel K. Biss
Source: The American Mathematical Monthly, Vol. 107, No. 8 (Oct., 2000), pp. 711-720
Published by: Mathematical Association of America
Stable URL: http://www.jstor.org/stable/2695468
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A Generalized Approach to the
Fundamental Group
Daniel K. Biss
1. INTRODUCTION.The classical theory of fundamentalgroups and covering
spaces is among the most elegant stories in the mathematicalcanon. The subject,
which bears a striking relationship to the study of Galois groups and field
extensions, provides historical motivation for a significant portion of modern
algebraictopology.The basic idea is to associate a group,the fundamentalgroup,
to each topological space, and then to extract topological informationfrom the
lattice of subgroupsof the fundamentalgroup.
More specifically,elements of the fundamentalgroup of a space X correspond
to loops in X modulo the relationthat if one loop can be continuouslydeformed
into another,then the two are consideredequivalent.A "cover"of a space X is a
map from anotherspace to X such that X can be coveredby open sets Ui whose
inverse images are simply disjoint copies of Ui. It then turns out that each
subgroupof the fundamentalgroupof a space correspondsto a cover.Intuitively,a
subgroupof the fundamentalgroup describessome loops or holes in X, and the
correspondingcover gives a procedurefor unwrappingthe loops. This beautiful
relationshipis useful in manyways, both because coveringspaces can be used to
computethe fundamentalgroupand because knowledgeof the fundamentalgroup
tells us when to search for potentiallyinterestingcovers of a space.
However, the theory works only for certain relatively nice spaces, namely
semi-locallysimplyconnected spaces. Worse yet, it can be shown that the correspondencefails for all spaces that are not semi-locallysimplyconnected!The aim
of this paper is to present a slightly modified version of the definitions that
generalizesthe standardcorrespondencebetween connectedcoversand subgroups
of the fundamentalgroup to a much broader setting. In doing this, we need to
introducesome additionalstructureto the fundamentalgroup that is able to see
subtlerdifferencesbetween some spaces.This extrastructureallowsus to state the
usual theoremsof the subjectin a languagethat generalizesappropriatelyto more
exotic spaces. This paper is meant to be a non-technicalcompanionto the more
precise [2], in whichthe details not includedhere are filled out. A similarprogram
is being carriedout from a slightlydifferentpoint of view by Bogley and Sieradski;
see [3], [4], and [7].
We begin, in Section 2, by briefly reviewingthe classical theory. No previous
knowledgebesides general point-set topology and basic group theory is assumed.
In Section 3, we present the new constructionsand ideas that are necessary to
develop the generalizedtheory. Finally, in Section 4, we sketch (mostly without
proofs) the new resultsthat are extensionsof the theoremsdiscussedin Section 2.
2. THE CLASSICALTHEORY.The fundamentalgroup was first introducedby
Poincarein the groundbreakingpaper [6] in which he essentiallyfounded the field
of topology.Poincareoutlined the theory from its very inception, albeit in a style
that is somewhatless precise than that of the typicalpiece of modernmathematics.
October 2000]GENERALIZEDAPPROACH TO THE FUNDAMENTAL GROUP
711
In this paper,he beganby definingthe objectsof study,which are what would now
be called submanifoldsof Rn, modulo some imprecisionin Poincare'sdefinition.
Later in the article, he defined homology and the fundamentalgroup, with the
object of determiningwhen certain manifolds are not homeomorphicto one
another. The motivatingconcept is a simple one: to find a systematicway of
measuringthe "holes" in a manifold. For example, how can we prove that the
spaces R2 and R2 - {(0, 0)} are not homeomorphic?
Let I denote the closed unit interval[0, 1]. Poincare'selegant idea was to fix a
point x in a manifold X, and considerthe set of all continuouspaths ry: I -> X
such that -y(O)= -y(l) = x. We identify two such paths if they are homotopic
equivalent,that is, if one can be continuouslydeformedinto the other. Formally,
two paths ryo and -yl are said to be homotopic equivalent if there exists a
continuous map F: I x I -* X with F(O, s) = yo(s), F(1, s) = y1(s), and F(t, 0) =
F(t, 1) = x. Now, given two paths ryo and ryl, we can form a new path ryo* -yl by
concatenatingthe two; explicitly,
(y0*^l)(s)=
Yo(2s)
- 1)
l^i(2s
for 0 < s <
for 2< S < 1.
Although concatenationis not an associativeoperation (that is, the paths Pi =
(YO* Y) * Y2 and P2 = YO*((Y * Y2) need not be equal), it is associativeup to
homotopy (that is, Pi and P2 are homotopic). Thus, we can make the set of
equivalenceclasses of paths into a group,called the fundamentalgroup-rT(X,x) of
the manifoldX. With some work,one can in fact see that the fundamentalgroupof
R2 is trivial and the fundamentalgroup of R2 _ {(0, 0)} is isomorphic to the
integers, Z, with the path -y(s) = e 2lTiS acting as a generator. Thus, in this case, the
fundamentalgroup does manage to distinguishbetween the two spaces in question. In general, the fundamentalgroup of a space is a tool that encodes which
loops in the space can be deformedinto one another.Thus, it providesa technique
ideallysuited to the task of differentiatingbetween R2, in which everyloop can be
contractedto a point, and [R2 - {(0,0)}, in which loops that go aroundthe origin
cannot be contracted.
Of course, the fundamentalgroupis nowherenear a subtle enough invariantto
distinguishbetween all pairs of spaces (for example, every sphere of dimension
greater than one has trivialfundamentalgroup),but it remains a cornerstonein
the foundation of algebraic topology. Unfortunately, in spite of (or perhaps
because of) its enticinglyintuitivedefinition,the fundamentalgroupwas relatively
difficultto computefor manyyears. It was the introductionof coveringspaces into
the picturethat made the fundamentalgroup a well-understoodinvariant.
The historyof the theory of coveringspaces is a convolutedone, not simplified
by the fact that Riemann, perhaps the earliest focal player in the development,
publishedalmostnothingon the subject.His ideas were only graduallydiseminated
by other mathematicians,such as Betti [1], primarilyin correspondence.The
genesis of the idea comes from the followingphenomenon.Considerthe function
f: C -> C defined by f(z) = Z2. For every z, we have f(z) = f( -z), so that every
nonzerocomplexnumberhas two pointsin its inverseimage;however,f- 1(0) = {0}.
Suppose that we would like to constructsome new space X that locally "looks
like"C and a function Sq : C -? X that "behaveslike" the function f but is also
one-to-one.This would be useful because an inverseto the function Sq would play
the role of a square root function. We can constructsuch an X by taking two
copies of the complexplane, C1 and C2, cuttingthem both open along the positive
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real axis, and then gluing them together along the cuts in such a way that for a
positivereal numbera, and small E, a + iE in the first plane is close to a - i E of
the second, and vice versa (see Figure 1). Then, we may define the function Sq by
Sq(z)
{Z2
E
C21
for lm(z)
?
0
Now, since X was constructedout of two copies of C, it retains a projection
mapping p: X
->
C. This map is called a branched covering of C-this
means that
away from a finite collection of branchingpoints (in this case, 0 is the only
branchingpoint), p is a local homeomorphism.More precisely,for every complex
number z # 0, there is an open set U c C containing z such that p-1(U) is
homeomorphicto two disjointcopies of U, and the restrictionof p to either copy
is a homoeomorphism.The general definitionof branchedcoveringdiffersonly in
that the "two"in the previoussentence can be replacedby any positiveinteger (or
infinity).
Figure 1. The construction of the space X.
So far, the promisedconnectionwith the fundamentalgroup has not emerged.
To see the relationshipbetween these two concepts,let p: X -? C be a branched
cover with branching points Pi, P2 ... ., Pn Let C = C - {Pi,* .. , Pnl} X = p (0)
and
=p I
Then p3is a branched covering map with no branching points, or a
coveringmap. Now, let us examine more carefully the example of f(z) = z2.
Ignoring the point z = 0 (which we need to do anywayin order to make the
branchedcover we constructedinto a cover),we have a two-to-onefunction, and
so we have troubleconstructinga squareroot functionthat is to be its inverse.Of
course, one can try to solve this problemas it is solved in the real case, namelyby
simplydeclaringthe squareroot of a positive real numberto be positive,but this
brings up a new difficultythat did not present itself in the real case. Indeed,
consider a circle of radius one going counterclockwise about the origin: y(t) =
e2Tit*
Since we declared that the square root of a positive real numberis positive, we
find that -y(O)= 1. Now, of course we must alwayshave y(tE) {e-ie, -eTit},
but since we have already declared y (0) = +e0, in order to ensure that our
square root function is continuous, as t increases, it must remain the case that
y(t) = +ewit.However,takingthe limit as t approaches1, we find a contradiction: 1 = - 1! Thus, the existenceof this branchedcoverX is closelyrelatedto the
existenceof a nontrivialelement of the fundamentalgroupof C - {0},represented
by y.
There is also informationto be uncovered by looking more carefully at the
relationshipbetween the fundamentalgroupsof X and C - {0}.Notice first of all
October 2000]GENERALIZEDAPPROACHTO THE FUNDAMENTAL GROUP
713
that if we have two pointed spaces (Y1,yl) and (Y2, Y2) and a continuousfunction
f : Y1 -> Y2 such that f(Y1) = Y2' we get an induced map f*: ir1(Y1,yl) -71(Y2, Y2). This is done simplyby taking a loop y: I -* Y1 and composingwith f
to get a loop f o y : I -> Y2. We next try to understandthe fundamentalgroup of
X. Recall that X is made up of two copies of C - {0}, so we start at the point 1 in
one of the copies and begin following the path y that goes around the origin
counterclockwise.Because of the way the two planes in X are glued together,after
going aroundthe originonce, we find ourselveson the other plane, and need to go
aroundagainto get back to the point where we started.Denote by jo this path that
goes aroundboth planesbefore comingbackto its initialpoint. One can check that
the fundamental group of X is Z, generated by the path y. Thus, the projection
Z that is multiplicationby 2, since po
f :X -> C - {0} inducesthe map p*: Z
goes aroundthe originin C twice. Thus, the image of the map fip*is the subgroup
2Z c Z. As we will see, this fits into a very general picture:the coveringspaces
lying over a space can be understoodin terms of the subgroupsof the fundmental
group of the space.
Now, a space X is said to be simplyconnectedif 71(X, x) is the trivialgroupfor
all x c X. Furthermore,for any open set U c X, we have the obvious inclusion
map i: U " X, which, for any x
c
U, induces a map i * : ir1(U, x)
-s
1(X, x). We
say that X is semi-locallysimplyconnectedif every x E X has a neighborhoodU
such that the induced map i* is the trivial map. Intuitively speaking, X is
semi-locallysimplyconnectedif it does not have a collection of loops that "bunch
up" (for a more precise descriptionof such phenomena, see Section 4). We are
now ready to list the theorems that make up the relationship between the
fundamentalgroup and coveringspaces. Throughoutthe rest of this section, all
spaces are assumedto be semi-locallysimplyconnected.
Proposition 1. Let p: X -* Y be a covering map. Then for all x
map p* : ir1(X, x) -- 'nr1(Y,p(x))is injective.
E
X, the induced
Thus, to understand what a covering map does to fundamental groups, it
suffices to know the image of the induced map on 7r1.It turns out that the image
of the induced map tells us everythingthere is to know about the coveringmap.
That is, two covers P1 : X, -> Y and P2 : X2 -* Y are said to be isomorphic if there
is a homeomorphism h: X1 -* X2 such that P2 o h = Pi, and using the fundamental group, it can be determinedpreciselywhen two covers are isomorphic.
Theorem 1. For everysubgroupH c ir1(Y, y), there is a coveringmap p : X
Yand
a point x E X such that p. (r1(X, x)) = H. Furthermore,two coversp1: X1
Y and
P2 : X2 -* Y are isomorphic if and only if (P1):* ( r(X,, x)) and (P2)* (7r1(X1I X2))
are conjugate subgroupsof 7r1(Y,y).
In particular,there is a unique cover p : X -* Y that correspondsto the trivial
subgroup-equivalently, this is the unique cover with X simply connected. Furthermore,if p': X' -> Y is another cover, then the cover of X' correspondingto
the trivial subgroup of 7r1(X')will be a covering map q : X -> X' such that
p = p' oq. For this reason, the unique simply connected cover X is called the
universalcover of Y. For example, the map p : C
?->
- {(0, 0)1 given by p(z) = eZ
is the universalcover of C - (0, 0)1.To see the "universality"propertyin action,
consider the covering map p': : - ((0, 0)1 -> ? - (0, 0)1 defined by p'(z) = z2.
Then our theory predicts the existence of a covering map q : C -? C - (O,0)}
satisfyingp = p' o q. The map q(z) = ez/2 turns out to do the trick.
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Now, a homeomorphism h: X -> X such that p o h = p is called a deck transfor-
mationof X. The set of all deck transformationscan easily be seen to be a group
under composition.Magically,it turns out that this group is completely understood.
Theorem 2. Let p: X -> Y be a universal cover and y
transformationsof X is isomorphic to i1(Y, y).
E
Y. Then the group of deck
We now illustrate all of these phenomena with an example. Although the
motivation behind this theory comes from complex analysis, the most easily
visualizedcase comes from a more basic space. So, let Y = S1, the circle given by
the equation {z E C: lzI = 11. It can be checked that iT1(Y,1) = Z (generatedby
the loop y(s) = e2wis). Now, the subgroupsof Z are all of the form nZ for some
nonnegativeinteger n. First, suppose that n > 0. Theorem 1 predictsthat there is
a unique cover p: X
-*
Y with p*(X1(X, x)) = nZ. To find such a cover, we first
need to understandthe elements of the subgroupnZ c r-1(Y,1). This subgroupis
generatedby the path yn(s) = e2n,is. Thus our coveringmap must somehow"wrap
around"Y exactly n times. Considerthe map p: Y -- Y given by p(z) = zn. Since
p oy, = yn, we find that this map has preciselythe desired property.
We still have not dealt with the case n = 0, that is, the case in which X is
simplyconnected.To find this universalcover of Y, we need to somehowunwrap
the circle Y to get a simply connected space. Indeed, define p R -: Y by
p(x) = e2 ix. This does precisely what we hoped for-it is a local homeomorphism
in which the preimageof each point is countablyinfinite. We now try to compute
the group of deck transformations.Suppose h: R 1Ris a homeomorphismsuch
that p o h = p. Then for all x, we know that p(h(x)) = p(x), or e2Tih(x) = e2wix, or
= 1, SO that x - h(x) must be an integer. But then the function
e2wi(x-h(x))
x - h(x) is a continuousinteger-valuedfunction,so it must be constant.Thus, we
have h(x) = x + h(O). Furthermore, since 0 - h(O) is an integer, h(O) must be an
integer, so h(x) = x + k for an arbitraryinteger k. Therefore, the group of all
such h is preciselythe group Z, which is also the group ,T1(Y,1), as predictedby
Theorem2.
3. SOME EXTENDED DEFINITIONS. In the rest of this article, we set up a
frameworkthat gives us naturalgeneralizationsof the resultspresentedin Section
2. The basic observationis that the fundamentalgroupis not only a group,but also
a topologicalspace. A topologicalgroupis a group G that is a topologicalspace in
which the multiplication map G x G
-*
G and the inversion map G
-*
G (sending
x to x-1) are continuous.It turnsout that the fundamentalgroupcan be naturally
given the structureof a topologicalgroup.Then, the conditionof semi-localsimple
connectednessof a topologicalspace will correspondto a topologicalconditionon
its fundamentalgroup, and this motivatesall our new definitions.
We begin by introducingthe compact-opentopology. Let X and Y be two
topologicalspaces. The set Hom(X, Y) of continuousmaps from X to Y can be
endowed with a topology,called the compact-opentopology,in the followingway.
Let K c X be a compactsubset of X, and U c Y an open subset of Y. Denote by
KK, U) the set of all continuous maps f: X -* Y such that f(K) c U. Then the
compact-opentopologyon Hom(X, Y) is the topologywith sub-basis{(K, U) IK
c X compact,U c Y openi, that is, the minimaltopologycontainingall <K, U) as
open sets. In particular,we can furnishthe space Hom(S1,X) with the compactopen topology.Now, for any x E X, the space Hom((S1,1),(X, x)) is a subspaceof
Hom(S1,X), and can accordinglybe giventhe subspacetopology.Lastly,the group
October 2000]GENERALIZEDAPPROACH TO THE FUNDAMENTAL GROUP
715
mT(X,x) is simplya quotient of the space Hom((S1,1), (X, x)) (by the homotopy
relation),and hence also comes with a naturaltopology.It is not too hard to check
that in this topology,the multiplicationand inversionmaps are continuous,so that
m1(X,x) is actuallya topologicalgroup. When we want to stress the topological
structureof the fundamentalgroup,we denote it by r-t0P(X,x).
Viewing the fundamentalgroup as a topologicalgroup gives us a slightlyfiner
invariant that is more naturally suited to the study of spaces that are not
semi-locallysimply connected. However, to have any hope of generalizing the
results of Section 2, we need also to modify somehow the definition of the other
majorplayerin that story,namelythe coveringspace. Recall that a coveringmap is
a map p: X -- Y such that every point y E Y has a neighborhoodU such that
p- (U) is just a collectionof disjointcopies of U. Anotherway of sayingthis is that
p1'(U) is homeomorphicto the space U x F for some discrete space F. Of
course, we can make an analogous definition for an arbitrary(not necessarily
discrete) space F. A map p: X -- Y is said to be a fiber bundle with fiber F if
everypoint y E Y has a neighborhoodU such that there exists a homeomorphism
h:p-1(U) -> UxF such that iToh =p Ip-1(U), where iT:-:UxF -* U is the
projection map. Now, a covering map is simply a fiber bundle with discrete
fiber-to complete our new picture, it suffices to replace the notion of "discrete
fiber"with an appropriatelyrelaxedconstraint.
In order to decide how preciselyto weaken the definitionof coveringspaces, let
us take stock of our situation.First of all, we would like to produce a theory that
"works"for some spaces that are not semi-locallysimplyconnected.In orderto do
this, we must allow fiber bundleswith non-discretefibers.Secondly,we must insist
that the results of Section 2 hold in some guise in our new setting. Thus, it is
essentialto isolate preciselywhat propertiesof coveringmaps are needed to make
the machine turn. It turns out that most of these proofs (see [5], or, for an
especially systematic and general exposition, [8]), hinge on one fundamental
property:the fact that coveringmaps satisfy the uniquepath-liftingproperty.This
property says that if p: X -* Y is a covering map, y: I -* Y is a path in Y, and
x E X is a point such that y(O) = p(x), then there is a unique path : I -* X such
that j(O) = x and p o y = y. Furthermore,it can be shown (again, see [8]) that as
long as F is a totally disconnectedspace, then any fiber bundle p: X -- Y with
fiber F satisfiesthe unique path-liftingproperty.Thus,we choose as our objectsof
study rigid covering bundles p: X
-*
Y, or fiber bundles with totally disconnected
fiber. As we will soon see, shiftingour attentionfrom the classificationof coversof
a given space Y as in Section 2 to a study of rigid coveringbundles enables us to
obtain some resultsfor spaces that are not semi-locallysimplyconnected.
4. THE EXTENDEDTHEORY.In this section, we finally demonstratehow to
extend the results of Section 2 to some spaces that are not semi-locallysimply
connected. In order to give the results some conceptual meaning, it is useful to
have a comparativelysimple example in the back of one's head at all times. One
such exampleis the Hawaiianearringspace HE (see Figure 2). The space HE is a
subset of the plane consistingof infinitelymany circles of decreasingradii, all of
which are tangent to one another. That is, HE is the union over all positive
integers n of the circle of radius 1 and center (, 0). This space is not semi-locally
simplyconnected,for any open neighborhoodof the origincontainsinfinitelymany
of the circles, each of which contributesnon-trivialelements of the fundamental
group. It is precisely this type of phenomenonthat prevents a space from being
semi-locallysimplyconnected.
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Figure 2. The Hawaiian Earring HE.
The idea that motivatesthe results of this section is very simple. First, notice
that a universalcoverof a space Y is a rigidcoveringbundlewhose fiber is discrete
and correspondsbijectivelyto r1(Y). Second, such a gadget exists if and only if Y
is semi-locallysimply connected. In other words, in the semi-locallysimply connected case, we know that there exists a universalcover, and its fiber has two
properties:it is discrete,and there exists a bijectionbetween it and 'n-1(Y).Hence,
if only we could view the elements of the fiber of the universalcover as corresponding naturally not just to those of T1(Y),but also to those of T?oP(Y), then a
natural syllogismwould suggest itself: Y has a universal cover if and only if
1 toP(Y) is discrete.This indeed turns out to be the case.
Theorem 3. Let Y be a topological space. Then Y is semi-locally simply connected if
and only if iTOP(Y) is discrete. Thus, if Y has a universal cover, then its fiber is
homeomorphic to r1OP(Y).
This observationgives us the vantagepoint fromwhichwe can shift our point of
view to the more general one we now pursue. Indeed, instead of viewing the
universalcover as the objectof study,we can seek out a rigidcoveringbundlewith
fiber Tt0P(Y)-clearly,this objectwould be the universalcover in the semi-locally
simplyconnected case. Unfortunately,this type of rigid coveringbundle need not
alwaysexist, but the conditionunder which it does exist is far weaker than that of
semi-localsimple connectedness.
Proposition 2. Let Y be a topological space. Then there exists a rigid coveringbundle
p : X -> Ywith fiber <TtP(Y) if and only if the path-components of s?oP(Y) arepoints.
Although the proof of existence is relativelyinvolved and dry, note that the
condition that iTtP(Y) be totally path-disconnected is clearly necessary, simply by
the definitionof rigidcoveringbundles.Thus, this result is the best that one could
hope for. In fact, we can go slightlyfurtherand obtaina completegeneralizationof
Theorem 1. First, we need to point out that an extension of Proposition1 also
holds in the new setting.
Proposition 3. Let p: X
induced map p*:
-*
Y be a rigid covering bundle. Then for all x E X, the
iT1(X, x) -- iT1(Y,p(x)) is injective.
Now we can state the classificationof rigid coveringbundles. Of course, we
need a notion of isomorphismof rigid coveringbundles;this definitioncan also be
October2000]GENERALIZEDAPPROACHTO THE-FUNDAMENTAL GROUP
717
made in analogy with concepts already introduced. That is, an isomorphism
-' Y and P2: X2 -* Y is a homeomor-
between two rigid covering bundles Pi: X1
phism h: X1 -* X2 such that Pi = P2 o h.
Theorem 4. Let H c Ttp(Y, y) be a subgroup such that the left coset space
s?oP(Y, y)/H is totally path-disconnected. Then there exists a rigid covering bundle
p: X -> Y and a point x E X such that p*(Xr1(X, x)) = H. Furthermore, two rigid
covering bundles Pi: X1 -* Y and P2: X2 -* Y are isomorphic if and only if
of 'i1(Y,y).
(p1):O
*T1(X1, x1)) and (P2)* (T1(X1, X2)) are conjugatesubgroups
Incidentally,lest it seem aestheticallydissatisfyingfor a criterionto depend on
topologicalpropertiesof the left coset space of a subgroupof i1tP, notice that the
left and right coset spaces are homeomorphicvia the inversionmap.
Finally,we providea generalizationof Theorem2 to the land of rigid covering
bundles.As with the other concepts introducedin Section 2, the definitionof the
groupof deck transformationsmust be modifiedslightlyin orderto state the result
in its strongestform. So, suppose p: X -* Y is a rigid coveringbundle with fiber
s-?oP(Y)
(or, equivalently,with X simply connected). Then much as before, the
group of deck transformationsof the rigid coveringbundle is defined to be the
group of homeomorphisms h: X -> X satisfying p o h = p. However, once again,
we notice that this groupis actuallya subset of the set of all continuousmaps from
X to itself, and as such, inherits a compact-opentopology.Thus, we can view the
groupof deck transformationsas a topologicalgroup.
Theorem 5. Let p: X -* Y be a rigid covering bundle with X simply connected and
locallypath-connected. Then iTOP(Y) is isomorphicas a topologicalgroup to the group
of deck transformationsof the bundle.
Once again,let us look at some examplesto see how this theoryworks,and how
the criteria that need to be satisfied to guarantee the existence of certain rigid
coveringbundlescan fail. First,recall the exampleof the HawaiianearringHE. In
order to see whether our machinerygives us interestingresults for this space, we
first need to understand the fundamentalgroup of HE. For each n > 1, let
,yn: I -> HE be the loop that circumnavigates the nth circle, that is, -yn(t)=
sin (2'nTt)).
Then each yngives an element of iT1(HE,(0, 0)). It
(1 - cos (2-t)),
0
is not too hard to see that there are no relations among the yn that is, that
T1(HE,(0,0)) containsa free groupwith generators{y1, 2 ... }.
However, it also contains more elements: for example, consider the path that
for 1 - 1/2n-1 < t < 1 - 1/2n traces the loop yn at 2n times the ordinaryspeed.
This provides a new element of iT1(HE,(0,
0)) that is not an element of the free
0)) can be describedas the set of
group {yl, Y29 ... }. It turns out that iT1(HE,(0,
infinite (formal) products of the yn and their inverses, subject to the condition
that in any particularproduct, each yn appears only finitely many times. Thus,
(x2x4x6 ...)( ... x4x3x2x1) is an element of iT1(HE,(0,0)), but x1x2xlx3xlx4xl .1
0)) as having a set of "formal"generators
is not. This descriptionof iT1(HE,(0,
requiressome care:for example,althoughthe words x1x2x3 ... and ... x-1x-lxj1
are inverses, it is not immediatelyobvious what cancellation laws lead to the
= e. A modified approachto combinatorial
formula (x1x2x3 ...)( *... x1xj1xj1)
grouptheorythat irons out these difficultiesin a way that is applicableto topology
is developedin [7].
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In any case, regardless of the technical difficulties, it is pretty clear that
(0,0)) is a (non-discrete)topological space with no nonconstantpaths.
(Non-discretebecause any open neighborhoodof the origin contains infinitely
many loops, and hence each KK,U) contains infinitely many xi.) Thus, by
Proposition 2, there exists a rigid covering bundle p: X -> HE with fiber
i4tP(HE,(0, 0)). It turns out that not all spaces enjoy this property.For example,
considerthe exampleof the HarmonicarchipelagoI1 (see Figure3). This example
is due to Bogley and Sieradski;see also [3]. To constructIA, begin with the disc
of unit radius centered at the point (1, 0) in the plane; notice that HE is a
subspace of this disc. Now (thinking of the x,y-plane a subset of 3-space), we
stretch the surface of the disc to form very narrowspires of unit height at each
i4tP(HE,
point of the form
(n
+
n
9
1,0),
that is, along the x-axis between every pair of
circlesbelongingto HE.
Figure 3. The Harmonic Archipelago H.
Now, althoughintuition might suggest that w,(IL4, (0, 0, 0)) ought to be trivial
(in fact that I4 ought to be contractible),this turns out to not be the case. For
example,consider the loop yl. If it can be contractedto the constant loop, then
certainlyas it is being deformed,it must pass throughthe top of each spire, that is,
through each point of the form (n + n 0, 1). If F: I x I -* A is a homotopy
between yi and the constantpath, then I x I must contain a sequence of points
a1, a2,... such that F(an) = ( + n 1 0,1). But by compactnessof I x I, the
has a convergent subseqence, whereas no subsequence of
sequence a,, a2,...
(
+
n
1,
0,
1) can converge to a point in A. This provides us with a contradic-
tion: thus, T1(1L4,
(0, 0, 0)) is not trivial,since Yl representsa non-trivialelement.
(0, 0, 0)) is isomorphicto 17T(HE,(0, 0)) modulo
Indeed, it turnsout that ,1(1L4,
the relation xn xm for all n, m ? 1. Equivalently,a word a is identified with
anyword that can be obtainedby choosingfinitelymanyletters in a and changing
them to any other letters. In particular,notice that any open set U in 14
containingthe origin contains the images of all but finitely many of the yn, say
and any word a contains only finitely many letters of the form
-yn 1,.,
)nk
xn1,... ,xnk. Therefore, a is equivalentto some word none of whose letters are of
the form xn1, . .,. xnk, and so for any K, we see thata E (K, U>. Since a was an
arbitrary word, it follows that for any compact K, (K, U> = v,1(1L4,(0,0,0)).
Therefore,ITtoP(IL4,(0, 0, 0)) has the trivialtopology-that is, its only open subsets
October2000]GENERALIZEDAPPROACHTO THE FUNDAMENTAL GROUP
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are itself and the empty set. Thus, for this disastrousexample,Theorem 4 tells us
that no rigidcoveringbundlesexist! So, our new theory,while more powerfulthan
the old, still has considerablelimitationsin sufficientlyexotic settings.
The appeal of the new theory lies in the fact that much of its reasoningfollows
in quite the same vein as the classicalsetting, once we observethat the fundamental group can be viewed as a topologicalspace. Thus, without doing a significant
amountof extrawork,we are able to removethose old hypothesesthat turn out to
be unnecessary,and at the same time gain a clearer understandingof what
obstructionslie in the way of the constructionof coveringspaces.
ACKNOWLEDGMENTS. I thank Bill Bogley and Al Sieradski for sharing with me their work on this
subject, and especially for showing me the striking example of the Harmonic Archipelago. Also, many
thanks to Jim Davis and Kent Orr for their insight and advice.
REFERENCES
1. E. Betti, Sopra gli spazi di unnumero qualunquedi dimensioni, Ann. Mat. Pura Appl. 4 (1871)
140-158.
2. D. K. Biss, The topological fundamental group and generalized covering spaces, in preparation.
3. W. A. Bogley and A. J. Sieradski, Weightedcombinatorial group theory and wild metric complexes,
Groups-Korea 98, (A. C. Kim, ed.), de Gruyter, Pusan, 2000, pp. 53-80.
4. W. A. Bogley and A. J. Sieradski, Universalpath spaces, http: // osu. orst. edu/- bogleyw.
5. J. R. Munkres, Topology: a first course, Prentice-Hall, Englewood Cliffs, NJ, 1975.
6. H. Poincare, Sur l'analysis situs, C. R. Acad. Sci. Paris. 115 (1892) 663-636.
edu/-bogleyw.
7. A. J. Sieradski, Omega-groups,http: // osu.orst.
8. E. H. Spanier, Algebraic Topology, McGraw-Hill, New York, 1966.
DANIEL BISS was born in 1977 in Akron, OH, and grew up in Bloomington, IN. He received an A.B.
summa cum laude in mathematics from Harvard University in 1998, and is now a second-year topology
graduate student at MIT. Among his awards are the Thomas Wendel Hoopes prize for an outstanding
undergraduate thesis at Harvard, and the 1998 AMS-MAA-SIAM Morgan prize for undergraduate
research. He has for several years taken a great interest in the teaching and exposition of mathematics,
during which period he has several times been a research advisor at the summer Research Experience
for Undergraduates in Duluth, MN, a course assistant at Harvard, and a frustrated member of
Harvard's undergraduate mathematics curriculum committee. Daniel also avidly pursues interests in
photography, music, writing, and travel.
Room 2-251, MassachusettsInstitute of Technology, Cambridge,MA 02139
daniel@math.mit.edu
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