Allen Hatcher
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Algebraic topology, Manuais, Projetos, Pesquisas de Matemática
LIVRO SOBRE TOPOLOGIA ALGEBRICA (EM INGLES)
Tipologia: Manuais, Projetos, Pesquisas
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Allen Hatcher
Copyright c 2002 by Cambridge University Press Single paper or electronic copies for noncommercial personal use may be made without explicit permission from the author or publisher. All other rights reserved.
Chapter 2. Homology....................... 97
2.1. Simplicial and Singular Homology............. 102
∆ Complexes 102. Simplicial Homology 104. Singular Homology 108. Homotopy Invariance 110. Exact Sequences and Excision 113. The Equivalence of Simplicial and Singular Homology 128.
2.2. Computations and Applications.............. 134
Degree 134. Cellular Homology 137. Mayer-Vietoris Sequences 149. Homology with Coefficients 153.
2.3. The Formal Viewpoint.................... 160
Axioms for Homology 160. Categories and Functors 162.
Additional Topics
2 .A. Homology and Fundamental Group 166.
2 .B. Classical Applications 169.
2 .C. Simplicial Approximation 177.
Chapter 3. Cohomology..................... 185
3.1. Cohomology Groups..................... 190
The Universal Coefficient Theorem 190. Cohomology of Spaces 197.
3.2. Cup Product.......................... 206
The Cohomology Ring 211. A K¨unneth Formula 218. Spaces with Polynomial Cohomology 224.
3.3. Poincar´e Duality........................ 230
Orientations and Homology 233. The Duality Theorem 239. Connection with Cup Product 249. Other Forms of Duality 252.
Additional Topics
3 .A. Universal Coefficients for Homology 261.
3 .B. The General K¨unneth Formula 268.
3 .C. H–Spaces and Hopf Algebras 281.
3 .D. The Cohomology of SO(n) 292.
3 .E. Bockstein Homomorphisms 303.
3 .F. Limits and Ext 311.
3 .G. Transfer Homomorphisms 321.
3 .H. Local Coefficients 327.
Chapter 4. Homotopy Theory................. 337
4.1. Homotopy Groups...................... 339
Definitions and Basic Constructions 340. Whitehead’s Theorem 346. Cellular Approximation 348. CW Approximation 352.
4.2. Elementary Methods of Calculation............ 360
Excision for Homotopy Groups 360. The Hurewicz Theorem 366. Fiber Bundles 375. Stable Homotopy Groups 384.
4.3. Connections with Cohomology.............. 393
The Homotopy Construction of Cohomology 393. Fibrations 405. Postnikov Towers 410. Obstruction Theory 415.
Additional Topics
4 .A. Basepoints and Homotopy 421.
4 .B. The Hopf Invariant 427.
4 .C. Minimal Cell Structures 429.
4 .D. Cohomology of Fiber Bundles 431.
4 .E. The Brown Representability Theorem 448.
4 .F. Spectra and Homology Theories 452.
4 .G. Gluing Constructions 456.
4 .H. Eckmann-Hilton Duality 460.
4 .I. Stable Splittings of Spaces 466.
4 .J. The Loopspace of a Suspension 470.
4 .K. The Dold-Thom Theorem 475.
4 .L. Steenrod Squares and Powers 487.
Appendix............................... 519 Topology of Cell Complexes 519. The Compact-Open Topology 529. The Homotopy Extension Property 532. Simplicial CW Structures 533.
Bibliography.............................. 539
Index................................... 545
x Preface
constraints of a first course. Altogether, these additional topics amount to nearly half the book, and they are included here both to make the book more comprehensive and to give the reader who takes the time to delve into them a more substantial sample of the true richness and beauty of the subject.
Not included in this book is the important but somewhat more sophisticated topic of spectral sequences. It was very tempting to include something about this marvelous tool here, but spectral sequences are such a big topic that it seemed best to start with them afresh in a new volume. This is tentatively titled ‘Spectral Sequences in Algebraic Topology’ and is referred to herein as [SSAT]. There is also a third book in progress, on vector bundles, characteristic classes, and K–theory, which will be largely independent of [SSAT] and also of much of the present book. This is referred to as [VBKT], its provisional title being ‘Vector Bundles and K–Theory.’
In terms of prerequisites, the present book assumes the reader has some familiar- ity with the content of the standard undergraduate courses in algebra and point-set topology. In particular, the reader should know about quotient spaces, or identifi- cation spaces as they are sometimes called, which are quite important for algebraic topology. Good sources for this concept are the textbooks [Armstrong 1983] and [J¨anich 1984] listed in the Bibliography.
A book such as this one, whose aim is to present classical material from a rather classical viewpoint, is not the place to indulge in wild innovation. There is, however, one small novelty in the exposition that may be worth commenting upon, even though in the book as a whole it plays a relatively minor role. This is the use of what we call ∆ complexes, which are a mild generalization of the classical notion of a simplicial complex. The idea is to decompose a space into simplices allowing different faces of a simplex to coincide and dropping the requirement that simplices are uniquely de- termined by their vertices. For example, if one takes the standard picture of the torus as a square with opposite edges identified and divides the square into two triangles by cutting along a diagonal, then the result is a ∆ complex structure on the torus having 2 triangles, 3 edges, and 1 vertex. By contrast, a simplicial complex structure on the torus must have at least 14 triangles, 21 edges, and 7 vertices. So ∆ complexes provide a significant improvement in efficiency, which is nice from a pedagogical view- point since it cuts down on tedious calculations in examples. A more fundamental reason for considering ∆ complexes is that they seem to be very natural objects from the viewpoint of algebraic topology. They are the natural domain of definition for simplicial homology, and a number of standard constructions produce ∆ complexes rather than simplicial complexes, for instance the singular complex of a space, or the classifying space of a discrete group or category. Historically, ∆ complexes were first introduced by Eilenberg and Zilber in 1950 under the name of semisimplicial com- plexes. This term later came to mean something different, however, and the original notion seems to have been largely ignored since.
Preface xi
This book will remain available online in electronic form after it has been printed in the traditional fashion. The web address is
http://www.math.cornell.edu/˜hatcher
One can also find here the parts of the other two books in the sequence that are currently available. Although the present book has gone through countless revisions, including the correction of many small errors both typographical and mathematical found by careful readers of earlier versions, it is inevitable that some errors remain, so the web page will include a list of corrections to the printed version. With the electronic version of the book it will be possible not only to incorporate corrections but also to make more substantial revisions and additions. Readers are encouraged to send comments and suggestions as well as corrections to the email address posted on the web page.
The aim of this short preliminary chapter is to introduce a few of the most com- mon geometric concepts and constructions in algebraic topology. The exposition is somewhat informal, with no theorems or proofs until the last couple pages, and it should be read in this informal spirit, skipping bits here and there. In fact, this whole chapter could be skipped now, to be referred back to later for basic definitions.
To avoid overusing the word ‘continuous’ we adopt the convention that maps be- tween spaces are always assumed to be continuous unless otherwise stated.
Homotopy and Homotopy Type
One of the main ideas of algebraic topology is to consider two spaces to be equiv- alent if they have ‘the same shape’ in a sense that is much broader than homeo- morphism. To take an everyday example, the letters of the alphabet can be writ- ten either as unions of finitely many straight and curved line segments, or in thickened forms that are compact regions in the plane bounded by one or more simple closed curves. In each case the thin letter is a subspace of the thick letter, and we can continuously shrink the thick letter to the thin one. A nice way to do this is to decompose a thick letter, call it X , into line segments connecting each point on the outer boundary of X to a unique point of the thin subletter X , as indicated in the figure. Then we can shrink X to X by sliding each point of X − X into X along the line segment that contains it. Points that are already in X do not move. We can think of this shrinking process as taking place during a time interval
0 ≤ t ≤ 1 , and then it defines a family of functions ft : X→X parametrized by t ∈ I =
[ 0 , 1 ] , where f (^) t (x) is the point to which a given point x ∈ X has moved at time t.
2 Chapter 0 Some Underlying Geometric Notions
Naturally we would like ft (x) to depend continuously on both t and x , and this will be true if we have each x ∈ X − X move along its line segment at constant speed so as to reach its image point in X at time t = 1 , while points x ∈ X are stationary, as remarked earlier. Examples of this sort lead to the following general definition. A deformation
retraction of a space X onto a subspace A is a family of maps ft : X → X , t ∈ I , such
that f 0 = 1 1 (the identity map), f 1 (X) = A , and ft || A = 1 1 for all t. The family ft
should be continuous in the sense that the associated map X × I → X , (x, t) , ft (x) ,
is continuous. It is easy to produce many more examples similar to the letter examples, with the deformation retraction f (^) t obtained by sliding along line segments. The figure on the left below shows such a deformation retraction of a M¨obius band onto its core circle.
The three figures on the right show deformation retractions in which a disk with two smaller open subdisks removed shrinks to three different subspaces. In all these examples the structure that gives rise to the deformation retraction can
be described by means of the following definition. For a map f : X → Y , the mapping
cylinder M (^) f is the quotient space of the disjoint union (X × I) q Y obtained by iden- tifying each (x, 1 ) ∈ X × I with f (x) ∈ Y. In the let-
X × I
X
Y Y
f (^ X ) Mf
ter examples, the space X is the outer boundary of the thick letter, Y is the thin
letter, and f : X → Y sends
the outer endpoint of each line segment to its inner endpoint. A similar description applies to the other examples. Then it is a general fact that a mapping cylinder Mf deformation retracts to the subspace Y by sliding each point (x, t) along the segment { x }× I ⊂ Mf to the endpoint f (x) ∈ Y. Not all deformation retractions arise in this way from mapping cylinders, how- ever. For example, the thick X deformation retracts to the thin X , which in turn deformation retracts to the point of intersection of its two crossbars. The net result is a deformation retraction of X onto a point, during which certain pairs of points follow paths that merge before reaching their final destination. Later in this section we will describe a considerably more complicated example, the so-called ‘house with two rooms,’ where a deformation retraction to a point can be constructed abstractly, but seeing the deformation with the naked eye is a real challenge.
4 Chapter 0 Some Underlying Geometric Notions
A space having the homotopy type of a point is called contractible. This amounts to requiring that the identity map of the space be nullhomotopic, that is, homotopic to a constant map. In general, this is slightly weaker than saying the space deforma- tion retracts to a point; see the exercises at the end of the chapter for an example distinguishing these two notions. Let us describe now an example of a 2 dimensional subspace of R^3 , known as the house with two rooms , which is contractible but not in any obvious way. To build this
space, start with a box divided into two chambers by a horizontal rectangle, where by a ‘rectangle’ we mean not just the four edges of a rectangle but also its interior. Access to the two chambers from outside the box is provided by two vertical tunnels. The upper tunnel is made by punching out a square from the top of the box and another square directly below it from the middle horizontal rectangle, then inserting four vertical rectangles, the walls of the tunnel. This tunnel allows entry to the lower chamber from outside the box. The lower tunnel is formed in similar fashion, providing entry to the upper chamber. Finally, two vertical rectangles are inserted to form ‘support walls’ for the two tunnels. The resulting space X thus consists of three horizontal pieces homeomorphic to annuli plus all the vertical rectangles that form the walls of the two chambers. To see that X is contractible, consider a closed ε neighborhood N(X) of X. This clearly deformation retracts onto X if ε is sufficiently small. In fact, N(X) is the mapping cylinder of a map from the boundary surface of N(X) to X. Less obvious is the fact that N(X) is homeomorphic to D^3 , the unit ball in R^3. To see this, imagine forming N(X) from a ball of clay by pushing a finger into the ball to create the upper tunnel, then gradually hollowing out the lower chamber, and similarly pushing a finger in to create the lower tunnel and hollowing out the upper chamber.
Mathematically, this process gives a family of embeddings h t : D^3 →R^3 starting with
the usual inclusion D^3 ↩ R^3 and ending with a homeomorphism onto N(X).
Thus we have X ' N(X) = D^3 ' point , so X is contractible since homotopy equivalence is an equivalence relation. In fact, X deformation retracts to a point. For
if f t is a deformation retraction of the ball N(X) to a point x 0 ∈ X and if r : N(X) → X
is a retraction, for example the end result of a deformation retraction of N(X) to X , then the restriction of the composition r ft to X is a deformation retraction of X to x 0. However, it is quite a challenging exercise to see exactly what this deformation retraction looks like.
Cell Complexes Chapter 0 5
Cell Complexes
A familiar way of constructing the torus S^1 × S^1 is by identifying opposite sides of a square. More generally, an orientable surface Mg of genus g can be constructed from a polygon with 4 g sides by identifying pairs of edges, as shown in the figure in the first three cases g = 1 , 2 , 3. The 4 g edges of the polygon
b^ a a
a
b b
b
b
b
b c
a
a
a
d
a
c
c
c
c
b
d
d
d d e
e
f
f
a
e
f
c d b
a
become a union of 2 g circles in the surface, all intersect- ing in a single point. The in- terior of the polygon can be thought of as an open disk, or a 2 cell, attached to the union of the 2 g circles. One can also regard the union of the circles as being obtained from their common point of intersection, by attaching 2 g open arcs, or 1 cells. Thus the surface can be built up in stages: Start with a point, attach 1 cells to this point, then attach a 2 cell.
A natural generalization of this is to construct a space by the following procedure: (1) Start with a discrete set X^0 , whose points are regarded as 0 cells. (2) Inductively, form the n skeleton X n^ from X n −^1 by attaching n cells e nα via maps
ϕ α : S n −^1 → X n −^1. This means that X n^ is the quotient space of the disjoint union
X n −^1
α D^ nα of^ X^ n −^1 with a collection of^ n^ disks^ D^ nα under the identifications x ∼ ϕ (^) α (x) for x ∈ ∂D nα. Thus as a set, X n^ = X n −^1
α e^ nα where each^ e^ nα is an open n disk. (3) One can either stop this inductive process at a finite stage, setting X = X n^ for some n < ∞ , or one can continue indefinitely, setting X =
n X^ n^. In the latter case X is given the weak topology: A set A ⊂ X is open (or closed) iff A ∩ X n^ is open (or closed) in X n^ for each n.
A space X constructed in this way is called a cell complex or CW complex. The explanation of the letters ‘CW’ is given in the Appendix, where a number of basic topological properties of cell complexes are proved. The reader who wonders about various point-set topological questions lurking in the background of the following discussion should consult the Appendix for details.
Cell Complexes Chapter 0 7
equivalence relation v ∼ λv for λ ≠ 0. Equivalently, this is the quotient of the unit sphere S^2 n +^1 ⊂ C n +^1 with v ∼ λv for | λ | = 1. It is also possible to obtain CP n^ as a quotient space of the disk D^2 n^ under the identifications v ∼ λv for v ∈ ∂D^2 n^ , in the following way. The vectors in S^2 n +^1 ⊂ C n +^1 with last coordinate real and nonnegative are precisely the vectors of the form (w,
1 − | w |^2 ) ∈ C n^ × C with | w | ≤ 1. Such
vectors form the graph of the function w ,
1 − | w |^2. This is a disk D +^2 n bounded by the sphere S^2 n −^1 ⊂ S^2 n +^1 consisting of vectors (w, 0 ) ∈ C n^ × C with | w | = 1. Each vector in S^2 n +^1 is equivalent under the identifications v ∼ λv to a vector in D^2 + n , and the latter vector is unique if its last coordinate is nonzero. If the last coordinate is zero, we have just the identifications v ∼ λv for v ∈ S^2 n −^1. From this description of CP n^ as the quotient of D^2 + n under the identifications v ∼ λv for v ∈ S^2 n −^1 it follows that CP n^ is obtained from CP n −^1 by attaching a
cell e^2 n^ via the quotient map S^2 n −^1 →CP n −^1. So by induction on n we obtain a cell
structure CP n^ = e^0 ∪ e^2 ∪ ··· ∪ e^2 n^ with cells only in even dimensions. Similarly, CP∞ has a cell structure with one cell in each even dimension.
After these examples we return now to general theory. Each cell e nα in a cell
complex X has a characteristic map Φ α : D αn → X which extends the attaching map
ϕ (^) α and is a homeomorphism from the interior of D nα onto e nα. Namely, we can take
Φ α to be the composition D nα ↩ X n −^1
α D^ nα → X^ n^ ↩^ X^ where the middle map is
the quotient map defining X n^. For example, in the canonical cell structure on S n described in Example 0.3, a characteristic map for the n cell is the quotient map
D n → S n^ collapsing ∂D n^ to a point. For RP n^ a characteristic map for the cell e i^ is
the quotient map D i →RP i^ ⊂ RP n^ identifying antipodal points of ∂D i^ , and similarly
for CP n^. A subcomplex of a cell complex X is a closed subspace A ⊂ X that is a union of cells of X. Since A is closed, the characteristic map of each cell in A has image contained in A , and in particular the image of the attaching map of each cell in A is contained in A , so A is a cell complex in its own right. A pair (X, A) consisting of a cell complex X and a subcomplex A will be called a CW pair. For example, each skeleton X n^ of a cell complex X is a subcomplex. Particular cases of this are the subcomplexes RP k^ ⊂ RP n^ and CP k^ ⊂ CP n^ for k ≤ n. These are in fact the only subcomplexes of RP n^ and CP n^. There are natural inclusions S^0 ⊂ S^1 ⊂ ··· ⊂ S n^ , but these subspheres are not subcomplexes of S n^ in its usual cell structure with just two cells. However, we can give S n^ a different cell structure in which each of the subspheres S k^ is a subcomplex, by regarding each S k^ as being obtained inductively from the equatorial S k −^1 by attaching two k cells, the components of S k − S k −^1. The infinite-dimensional sphere S ∞^ =
n S^ n
then becomes a cell complex as well. Note that the two-to-one quotient map S ∞^ →RP∞
that identifies antipodal points of S ∞^ identifies the two n cells of S ∞^ to the single n cell of RP∞^.
8 Chapter 0 Some Underlying Geometric Notions
In the examples of cell complexes given so far, the closure of each cell is a sub- complex, and more generally the closure of any collection of cells is a subcomplex. Most naturally arising cell structures have this property, but it need not hold in gen- eral. For example, if we start with S^1 with its minimal cell structure and attach to this
a 2 cell by a map S^1 → S^1 whose image is a nontrivial subarc of S^1 , then the closure
of the 2 cell is not a subcomplex since it contains only a part of the 1 cell.
Operations on Spaces
Cell complexes have a very nice mixture of rigidity and flexibility, with enough rigidity to allow many arguments to proceed in a combinatorial cell-by-cell fashion and enough flexibility to allow many natural constructions to be performed on them. Here are some of those constructions.
Products. If X and Y are cell complexes, then X × Y has the structure of a cell complex with cells the products e mα × e nβ where e mα ranges over the cells of X and e nβ ranges over the cells of Y. For example, the cell structure on the torus S^1 × S^1 described at the beginning of this section is obtained in this way from the standard cell structure on S^1. For completely general CW complexes X and Y there is one small complication: The topology on X × Y as a cell complex is sometimes finer than the product topology, with more open sets than the product topology has, though the two topologies coincide if either X or Y has only finitely many cells, or if both X and Y have countably many cells. This is explained in the Appendix. In practice this subtle issue of point-set topology rarely causes problems, however.
Quotients. If (X, A) is a CW pair consisting of a cell complex X and a subcomplex A , then the quotient space X/A inherits a natural cell complex structure from X. The cells of X/A are the cells of X − A plus one new 0 cell, the image of A in X/A. For a
cell e nα of X − A attached by ϕα : S n −^1 → X n −^1 , the attaching map for the correspond-
ing cell in X/A is the composition S n −^1 → X n −^1 → X n −^1 /An −^1.
For example, if we give S n −^1 any cell structure and build D n^ from S n −^1 by attach- ing an n cell, then the quotient D n^ /S n −^1 is S n^ with its usual cell structure. As another example, take X to be a closed orientable surface with the cell structure described at the beginning of this section, with a single 2 cell, and let A be the complement of this 2 cell, the 1 skeleton of X. Then X/A has a cell structure consisting of a 0 cell with a 2 cell attached, and there is only one way to attach a cell to a 0 cell, by the constant map, so X/A is S^2.
Suspension. For a space X , the suspension SX is the quotient of X × I obtained by collapsing X × { 0 } to one point and X × { 1 } to an- other point. The motivating example is X = S n^ , when SX = S n +^1 with the two ‘suspension points’ at the north and south poles of S n +^1 , the points ( 0 , ··· , 0 , ± 1 ). One can regard SX as a double cone
10 Chapter 0 Some Underlying Geometric Notions
If X and Y are CW complexes, then there is a natural CW structure on X ∗ Y having the subspaces X and Y as subcomplexes, with the remaining cells being the product cells of X × Y × ( 0 , 1 ). As usual with products, the CW topology on X ∗ Y may be weaker than the quotient of the product topology on X × Y × I.
Wedge Sum. This is a rather trivial but still quite useful operation. Given spaces X and Y with chosen points x 0 ∈ X and y 0 ∈ Y , then the wedge sum X ∨ Y is the quotient of the disjoint union X q Y obtained by identifying x 0 and y 0 to a single point. For example, S^1 ∨ S^1 is homeomorphic to the figure ‘8,’ two circles touching at a point. More generally one could form the wedge sum ∨ α Xα of an arbitrary collection of spaces X (^) α by starting with the disjoint union
α Xα and identifying points^ xα ∈^ Xα to a single point. In case the spaces Xα are cell complexes and the points xα are 0 cells, then
α X^ α is a cell complex since it is obtained from the cell complex^
α Xα by collapsing a subcomplex to a point. For any cell complex X , the quotient X n/X n −^1 is a wedge sum of n spheres ∨ α S (^) αn , with one sphere for each n cell of X.
Smash Product. Like suspension, this is another construction whose importance be- comes evident only later. Inside a product space X × Y there are copies of X and Y , namely X × { y 0 } and { x 0 }× Y for points x 0 ∈ X and y 0 ∈ Y. These two copies of X and Y in X × Y intersect only at the point (x 0 , y 0 ) , so their union can be identified with the wedge sum X ∨ Y. The smash product X ∧ Y is then defined to be the quo- tient X × Y /X ∨ Y. One can think of X ∧ Y as a reduced version of X × Y obtained by collapsing away the parts that are not genuinely a product, the separate factors X and Y. The smash product X ∧ Y is a cell complex if X and Y are cell complexes with x 0 and y 0 0 cells, assuming that we give X × Y the cell-complex topology rather than the product topology in cases when these two topologies differ. For example, S m^ ∧ S n^ has a cell structure with just two cells, of dimensions 0 and m + n , hence S m^ ∧ S n^ = S m + n^. In particular, when m = n = 1 we see that collapsing longitude and meridian circles of a torus to a point produces a 2 sphere.
Two Criteria for Homotopy Equivalence
Earlier in this chapter the main tool we used for constructing homotopy equiva- lences was the fact that a mapping cylinder deformation retracts onto its ‘target’ end. By repeated application of this fact one can often produce homotopy equivalences be- tween rather different-looking spaces. However, this process can be a bit cumbersome in practice, so it is useful to have other techniques available as well. We will describe two commonly used methods here. The first involves collapsing certain subspaces to points, and the second involves varying the way in which the parts of a space are put together.
Two Criteria for Homotopy Equivalence Chapter 0 11
Collapsing Subspaces
The operation of collapsing a subspace to a point usually has a drastic effect on homotopy type, but one might hope that if the subspace being collapsed already has the homotopy type of a point, then collapsing it to a point might not change the homotopy type of the whole space. Here is a positive result in this direction:
If (X, A) is a CW pair consisting of a CW complex X and a contractible subcomplex A ,
then the quotient map X → X/A is a homotopy equivalence.
A proof will be given later in Proposition 0.17, but for now let us look at some examples showing how this result can be applied.
E xample 0.7: Graphs. The three graphs are homotopy equivalent since
each is a deformation retract of a disk with two holes, but we can also deduce this from the collapsing criterion above since collapsing the middle edge of the first and third graphs produces the second graph. More generally, suppose X is any graph with finitely many vertices and edges. If the two endpoints of any edge of X are distinct, we can collapse this edge to a point, producing a homotopy equivalent graph with one fewer edge. This simplification can be repeated until all edges of X are loops, and then each component of X is either an isolated vertex or a wedge sum of circles. This raises the question of whether two such graphs, having only one vertex in each component, can be homotopy equivalent if they are not in fact just isomorphic graphs. Exercise 12 at the end of the chapter reduces the question to the case of connected graphs. Then the task is to prove that a wedge sum ∨ m S^1 of m circles is not homotopy equivalent to
n S^1 if^ m^ ≠^ n^. This sort of thing is hard to do directly. What one would like is some sort of algebraic object associated to spaces, depending only on their homotopy type, and taking different values for
m S^1 and^
n S^1 if^ m^ ≠^ n^. In fact the Euler characteristic does this since
m S^1 has Euler characteristic 1− m^. But it is a rather nontrivial theorem that the Euler characteristic of a space depends only on its homotopy type. A different algebraic invariant that works equally well for graphs, and whose rigorous development requires less effort than the Euler characteristic, is the fundamental group of a space, the subject of Chapter 1.
E xample 0.8. Consider the space X obtained
from S^2 by attaching the two ends of an arc A to two distinct points on the sphere, say the north and south poles. Let B be an arc in S^2 joining the two points where A attaches. Then X can be given a CW complex structure with the two endpoints of A and B as 0 cells, the interiors of A and B as 1 cells, and the rest of S^2 as a 2 cell. Since A and B are contractible,