Topology I, Study notes of Topology

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S. P. Novikov (Ed.)

Topology I

General Survey

With 78 Figures

Springer

Encyclopaedia of

Mathematical Sciences

Volume 12

Editor-in-Chief: RX Gamkrelidze

2

V3.

Contents

Simplicial and cell bundles with a structure group. Obstructions. Universal objects: universal fiber bundles and the universal property of Eilenberg-MacLane complexes. Cohomology operations. The Steenrod algebra. The Adams spectral sequence The classical apparatus of homotopy theory. The Leray spectral sequence. The homology theory of fiber bundles. The Cartan-Serre method. The Postnikov tower. The Adams spectral sequence................................................. Definition and properties of K-theory. The Atiyah-Hirzebruch spectral sequence. Adams operations. Analogues of the Thorn isomorphism and the Riemann-Roth theorem. Elliptic operators and K-theory. Transformation groups. Four-dimensional manifolds............................................... Bordism and cobordism theory as generalized homology and cohomology. Cohomology operations in cobordism. The Adams-Novikov spectral sequence. Formal groups. Actions of cyclic groups and the circle on manifolds..................

Chapter 4. Smooth Manifolds............................... 142 $1. Basic concepts. Smooth fiber bundles. Connexions. Characteristic classes.................................................. $2. The homology theory of smooth manifolds. Complex manifolds. The classical global calculus of variations. H-spaces. Multi-valued functions and functionals....................

53. Smooth manifolds and homotopy theory. Framed manifolds. Bordisms. Thorn spaces. The Hirzebruch formulae. Estimates of the orders of homotopy groups of spheres. Milnor’s example. The integral properties of cobordisms....................... $4. Classification problems in the theory of smooth manifolds. The theory of immersions. Manifolds with the homotopy type of a sphere. Relationships between smooth and PL-manifolds. Integral Pontryagin classes........^....................... $5. The role of the fundamental group in topology. Manifolds of low dimension (n = 2,3). Knots. The boundary of an open manifold. The topological invariance of the rational Pontryagin classes.The classification theory of non-simply-connected manifolds of dimension > 5. Higher signatures. Hermitian K-theory. Geometric topology: the construction of non-smooth homeomorphisms. Milnor’s example. The annulus conjecture. Topological and PL-structures...........................

Concluding Remarks.................... :.............

79

Contents 3

Appendix. Recent Developments in the Topology of 3-manifolds and Knots.................................................

$1. Introduction: Recent developments in Topology.............

52. Knots: the classical and modern approaches to the Alexander polynomial. Jones-type polynomials.....................
53. Vassiliev Invariants............................... $4. New topological invariants for 3-manifolds. Topological Quantum Field Theories..........................

Bibliography ...............................................

Index ......................................................

Chapter 1. The Simplest Topological Properties (^5)

which would return the subject in spirit to the algebraic geometry of the 19th century, when it was regarded as a part of formulaic analysis. This survey constitutes the introduction to a series of essays on topology, in which the development of its various subdisciplines will be expounded in greater detail.

Introduction to the English Translation

This survey was written over the period 1983-84, and published (in Russian) in 1986. The English translation was begun in 1993. In view of the appearance in topology over the past decade of several important new ideas, I have added an appendix summarizing some of these ideas, and several footnotes, in order to bring the survey more up-to-date. I am grateful to several people for valuable contributions to the book: to M. Stanko, who performed a huge editorial task in connexion with the Russian edition; to B. Botvinnik for his painstaking work as scientific editor of the English edition, in particular as regards its modernization; to R. Burns for making a very good English translation at high speed; and to C. Shochet for advice and help with the translation and modernization of the text at the University of Maryland. I am grateful also to other colleagues for their help with modernizing the text.

Sergei P. Novikov, November, 1995

Chapter 1

The Simplest Topological Properties

Topology is the study of topological properties or topological invariants of various kinds of mathematical objects, starting with rather general geometri- cal figures. From the topological point of view the name “geometrical figures” signifies: general polyhedra (polytopes) of various dimensions (complezes); or continuous or smooth “surfaces” of any dimension situated in some Euclidean space or regarded as existing independently (manifolds); or sometimes sub- sets of a more general nature of a Euclidean space or manifold, or even of an infinite-dimensional space of functions. Although it is not possible to give a precise general definition of “topological property” (“topological invariant”) of a geometrical figure (or more general geometrical structure), we may de-

(^6) Chapter 1. The Simplest Topological Properties

scribe such a property intuitively as one which is, generally speaking, “stable” in some well-defined sense, i.e. remains unaltered under small changes or de- formations (homotopies) of the geometrical object, no matter how this is given to us. For instance for a general polytope (complex) the manner in which the polytope is given may be, and often is, changed by means of an operation of subdivision, whereby each face of whatever dimension is subdivided into smaller parts, and so converted into a more complex polyhedron, the subdi- vision being carried out in such a way as to be compatible on that portion of their boundaries shared by each pair of faces. In this way the whole poly- hedron becomes transformed formally into a more complicated one with a larger number of faces of each dimension. The various topological properties, or numerical or algebraic invariants, should be the same for the subdivided complex as for the original.

The simplest examples. 1) Everyone is familiar with the elementary result called “Euler’s Theorem”, which, so we are told, was in fact known prior to Euler: For any closed, convex polyhedron in 3-dimensional Euclidean space IR3, the number of vertices less the number of edges plus the number of (2-dimensional) faces, is 2.

Thus the quantity V - E + F is a topological invariant in that it is the same for any subdivision of a convex polyhedron in lR3.

2. Another elementary observation of a topological nature, also dating back to Euler, is the so-called “problem of the three pipelines and three wells”. Here one is given three points al, ~2, a3 in the plane lR2 (three “houses”) and three other points Al, A2, A3 (“wells”), and it turns out that it is not possible to join each house ai to each well Ai by means of a non-self-intersecting path (“pipeline”) in such a way that no two of the 9 paths intersect in the plane. (Of course, this is possible in IR3.) In topological language this conclusion may be rephrased as follows: Consider the one-dimensional complex (or graph) con- sisting of 6 vertices ai, Aj, and 9 edges xij, i, j = 1,2,3, where the “boundary” of each edge, denoted by dxij, is given by dxij = {ai, Aj}. The conclusion is that this one-dimensional complex cannot be situated in the plane R2 without incurring self-intersections. This represents a topological property of the given complex. 0

These two observations of Euler may be considered as the archetypes of the basic ideas of combinatorial topology, i.e. of the topological theory of polyhe- dra and complexes established much later by Poincare. It is important to bear in mind that the use of combinatorial methods to define and investigate topo- logical properties of geometrical figures represents just one interpretation of such properties, providing a convenient and rigorous approach to the formula- tion of these concepts at the first stage of topology, though of course remaining useful for certain applications. However those same topological properties ad- mit of alternative formulations in various different situations, for instance in

8 Chapter 1. The Simplest Topological Properties

Gauss also discovered certain topological properties of non-self-intersecting (i.e. simple) closed curves in Iw 3. It is well known that a simple, closed, con- tinuous (or if you like smooth, or piecewise smooth, or even piecewise linear) curve separates the plane lR2 into two parts with the property that it is im- possible to get from one part to the other by means of a continuous path avoiding the given curve. The ideally rigorous formulation of this intuitively obvious fact in the context of an explicit system of axioms for geometry and analysis carries the title “The Jordan Curve Theorem” (although of course in fact it is, in somewhat simplified form, already included in the axiom sys- tem; if one is not concerned with economy in the axiom system, then it might just as well be included as one of the axioms). The same conclusion (as for a simple, closed, continuous curve) holds also for any “complete” curve in lR2, i.e. a simple, continuous, unboundedly extended, non-closed curve both of those ends go off to infinity, without nontrivial limit points in the finite plane. This principle generalizes in the obvious way to n-dimensional space: a closed hypersurface in IR” separates it into two parts. In fact a local version of this principle is basic to the general topological definition of dimension (by induction on n). There is however another less obvious generalization of this principle, hav- ing its most familiar manifestation in 3-dimensional space iR3. Consider two continuous (or smooth) simple closed curves (loops) in LR3 which do not in- tersect:

Consider a “singular disc” Di bounded by the curve Tiyi, i.e. a continuous map of the unit disc into lR3: Z$ = $(T,$), i = 1,2, a = 1,2,3, where 0 5 r 5 1, 0 5 4 5 21r, sending the boundary of the unit disc onto yi:

where q5= t for i = 1, and 4 = r for i = 2.

Definition 0.1 Two curves yr and 72 in lR3 are said to be nontrivially linked if the curve 72 meets every singular disc D1 with boundary yi (or, equivalently, if the curve yi meets every singular disc Dz with boundary 72). Simple examples are shown in Figure 1.1. In n-dimensional space IRn certain pairs of closed surfaces may be linked, namely submanifolds of dimensions p and q where p + q = n - 1. In particular a closed curve in IR2 may be linked with a pair of points ( a “zero-dimensional surface”) - this is just the original principle that a simple closed curve separates the plane. Gauss introduced an invariant of a link consisting of two simple closed curves yi, 72 in lR3, namely the signed number of turns of one of the curves around the other, the linking coeficient {n,n} of the link. His formula for this is

Chapter 1. The Simplest Topological Properties 9

(a) Unlinked curves (b) Linking coefficient 1

Fig. 1.

(c) Linking coefficient 4

N = {n,rz> = ; !.f

@%w-+Y2(~)1 ,Yl - 72) Ir1(t> - r2(t)13 ’^

(0.3) 71 72

where [ , ] denotes the vector (or cross) product of vectors in lR3 and ( , ) the Euclidean scalar product. Thus this integral always has an integer value N. If we take one of the curves to be the z-axis in lR3 and the other to lie in the (5, y)-plane, then the formula (0.3) gives the net number of turns of the plane curve around the z-axis. It is interesting to note that the linking coefficient (0.3) may be zero even though the curves are nontrivially linked (see Figure 1.2). Thus its having non-zero value represents only a sufficient condition for nontrivial linkage of the loops. Elementary topological properties of paths and homotopies between them played an important role in complex analysis right from the very beginning of that subject in the 19th century. They without doubt represent one of

Fig. 1.2. The linking coefficient = 0, yet the curves are non-trivially linked

Chapter 1. The Simplest Topological Properties 11

Fig. 1.

nontrivial “monodromy” , i.e. that we arrive at one of the other solutions at 20: w,(zo) # q(zo), s # j. Proceeding more systematically, consider all possible loops r(t), a 2 t 2 b, in the region U = lR2 \ { zi, ... ,zm}, with $a) = y(b) = za. Each such loop determines a permutation of the branches of the function w(z): if we start at the branch given by wj(z) and continue around the loop from a to b, then we arrive when t = b at the branch defined by w,, so that the loop y(t) determines a permutation j -+ s of the branches (or sheets) above ZO:

Y + o-r> u,(j)^ =^ s.

The inverse path y-l (i.e. the path traced backwards from b to a) yields the inverse permutation cry-l : s + j, and the superposition yi .ys of two paths yi (traced out from time a to time b) and 72 (from b to c), i.e. the path ob- tained by following yi by 79, corresponds to the product of the corresponding permutations: ~71% = ~-72Ou-Y17 uy-1 = (up.^ (0.7) In the general, non-degenerate, situation the permutations of the form CJ? generate the full symmetric group of permutations of n symbols. (This is the underlying reason for the general insolubility by radicals of the algebraic equa- tion (0.6) for n 25.) To seethis, note that the “basic” path 35, j = 1,... , m, which starts from za, encircles the single branch point zj, and then proceeds back to zo along the same initial segment (see Figure 1.3) corresponds, in the typical situation of maximally non-degenerate branch points, to the inter- change of two sheets (i.e. (T-,, is just a transposition of two indices). The claim then follows from the fact that the transpositions generate all permutations. It is noteworthy that the permutation cry is unaffected if the loop y is subjected to a continuous homotopy within U, throughout which its begin- ning and end remain fixed at za. This is analogous to the preservation of the Cauchy integral under homotopies (see (0.4) above), but is algebraically

12 Chapter 1. The Simplest Topological Properties

more complicated: the dependence of the permutation gy on the path y is non-commutative, in contrast with the Cauchy integral:

This sort of consideration leads naturally to a group with elements the ho- motopy classes of continuous loops y(t) beginning and ending at a particular point .zo E U, for any region, or indeed any manifold, complex or topological space U. This group is called the fundamental group of U (with base point ze) and is denoted by ~1 (U, 20). The Riemann surface defined by F(z, w) = 0 thus gives rise to a homomorphism - monodromy - from the fundamental group to the group of permutations of its “sheets”, i.e. the branches of the function w(z) in a neighborhood of z = ~0:

f7 : m(U, zo) + &, (^) (0.9)

where S, denote the symmetric group on n symbols, and U is as before - a region of E2. For transcendental functions F, on the other hand, the equation F(z, 20) = 0 may determine a many-valued function w(z) with infinitely many sheets (n = oo). Here the simplest example is

F(z,w)=expw-z=O, U=R2\0, w=lnz.

In this example the sheets are numbered in a natural way by means of the integers: taking zo = 1, we have wk = lnzs = 2rik, where k ranges over the integers. The path y(t) with \yJ = 1, y(O) = y(27r) = 1, going round the point z = 0 in the clockwise direction exactly once, yields the monodromy y -+ cry, a,(k) = k - 1. An interesting topological theory where the non-abelianness of the funda- mental group r(U, ze) plays an important role is that of knots, i.e. smooth (or, if preferred, piecewise smooth, or piecewise linear) simple, closed curves

-Y(t) c R3, Y(t + 27r) = 7(t), or,^ more^ generally,^ the^ theory^ of^ links,^ as intro- duced above, a link being a finite collection of simple, closed, non-intersecting curves yi,. , yk C Iw3. For k > 1, one has the matrix with entries the linking coefficients {ri, rj}, i # j, given by the formula (0.3), which however does not determine all of the topological invariants of the link. In the case k = 1, that of a knot, there is no such coefficient available. Let y be a knot and U the complementary region of Iw3:

U=R3\y. (0.10)

It turns out that the fundamental group rr(U, za), where za is any point of U, is abelian precisely when the given knot y can be deformed by means of a smooth homotopy-of-knots (i.e. by an “isotopy” , as it is called) into the trivial knot, i.e. into the unknotted circle S1 c IK2 c Iw3, where the circle S1 lies in

14 Chapter 1. The Simplest Topological Properties

Fig. 1.

be confused with the kernel of an integral operator!), and finite-dimensional

“cokernel”, i.e. kernel of the adjoint operator A* : Hz --+ HI. It turns out

that the index i(A) of such an operator A, defined by

i(A) = dim(Ker A) - dim(Ker A*), (0.11)

i.e. the difference in the respective numbers of linearly independent solutions of the equations A(h) = 0 and A(h) = 0, is a homotopy invariant. This means simply that the index i(A) remains unchanged under continuous deformations of the operator A, although the individual dimensions on the right-hand side of the equation (0.11) may change.

In the simplest case of nonsingular kernels K(z, y) of integral operators .6?, the “Fredholm alternative” was discovered at the beginning of this century. In the language of functional analysis this is in egect just t&e assertion that i(A) = 0 for operators A of the form A = 7 + K, where K is a “compact operator”, i.e. g(M) is a compact subset of Hz for every bounded subset M of HI, and the operator 7 is an isomorphism of the Hilbert spaces HI and Hz. In fact the addition of a compact operator to any Noetherian operator A preserves the Noetherian property, so that the simple$ deformation in the class of Noetherian operators has the form At = A0 + tK (with A0 = 7 in the classical Fredholm situatian). For singular integral operators, on the other hand, the index is a rather more complicated topological characteristic. Much classical work of the 1920s and 1930s was devoted to explicit calculation of the index via the kernel of the operator. Far-reaching generalizations of this theory to higher-dimensional manifolds, culminating in the Atiyah-Singer Index Theorem, have come to be of exceptional significance for topology and its applications.

$1. Observations from generaltopology. Terminology 15

This example shows how topological properties arise not only in connexion with geometrical figures in the naive sense,but also in mathematical contexts of a quite different nature.

Chapter 2

Topological Spaces. Fibrations. Homotopies

31. Observations from general topology.

Terminology

Although topological properties are sometimes hidden behind a combinato- rial - algebraic mask, they nonetheless all partake organically of the concept of continuity. The most general definition of a continuous map or function between sets requires very little in the way of structure on the sets. As intro- duced by Frechet, this structure or topology on a set X, making it a topological space, consists merely in the designation of certain subsets U of X (among all subsets of X) as the open sets of X, subject only to the requirements that the empty set and the whole space X be open, and that the collection of all the open sets be closed under the following operations: the intersection of any finite subcollection of open sets should again be open, and the union of any collection, infinite or finite, of open sets should likewise be open. The complement X \ U of any open set U is called a closed set. The closure of any subset V c X, denoted by v, is the smallest closed set containing V. A continuous map (or, briefly, map) f : X +^ Y between topological^ spaces X and Y, is then one for which the complete inverse image f-‘(U) of every open set U of Y is open in X. (The complete inverse image f-‘(D) of any subset D c Y, is just the set of all z E X such that f(x) E 0.) A compact space is a topological space X with the property that, given any covering of X by open sets U,, i.e. (^) U U,^ = X,^ there always exist a finite^ set of indices fflr..., (YN such that the open sets U,,,.. , U,, already cover X, i.e. there is always a finite subcover. It can be shown that in a compact space X every sequenceof points xi, i = 1,2,.. ., has a limit point z, in X, i.e. a point such that every open set containing it contains also terms xi of the sequence for infinitely many i. A Hausdorfl topological space is one with the property that for every two distinct points xi, x2 there are disjoint open sets U1, U2 containing them:

U1 n U2 = 8, xi E U1, x2 E U2. A topological space X is called a metric space

if there is a real-valued “distance” p(xr, x2) defined for each pair of points xi, 22 E X, continuous in xi, x2 (with respect to the “product topology” on X x X - see below), satisfying

5 1. Observations from general topology. Terminology (^17)

x v Y = XOY/xo x y(). (^) (1.1)

More generally, given any closed subset A c X and map f : A ---+ Y, one may by means of identification form the following analogue of the bouquet:

Xv(A,f)Y=XtiY /x=f(z), zcA.

An important case of this is the mapping cylinder Cf of a map f : Z -+ Y. Consider the product of 2 with an interval I = [a, b], and form the identifica- tion space Cf = (2 x I)LJY/(z,b) M f(z), z E 2. (^) (1.2) (Here 2 x I plays the role of X and 2 x {b} that of A.) The topology on the space Cf is defined in the natural way: a subset of Cf is taken as open precisely if its complete inverse images under the natural maps 2 x I + Cf and Y -+ Cf , are both open. On any subset A of a topological space X the subspaceor induced topology is defined by taking the open sets of A to be simply the intersections with A of the open sets of X. A sequence of points xi of a topological space X is said to converge in X, if it has a limit in X , i.e. a point xW of X with the property that every open set U containing 5, contains the xi for almost all i (that is for all but finitely many i). The topology on X may be recovered from the knowledge of its convergent sequences. A homeomorphism between topological spaces X and Y is a continuous, one-to-one surjection f : X + Y, such that the inverse f-l : Y --+ X is also continuous. Here the continuity of the inverse function f-l does not in general follow from the other defining conditions; it does follow, however, if X is compact and Y Hausdorff. Functional analysis provides many examples of continuous bijections with discontinuous inverses. In particular, for spacesof real-valued smooth func- tions there are various natural kinds of convergence definable in terms of different numbers of derivatives, so that the existence of continuous’bijections with discontinuous inverses is to be expected even in such relatively concrete contexts. One often encounters topological spaceswhich carry at the same time some algebraic structure, compatible with the topological structure in the sensethat the various algebraic operations are continuous when considered as maps; thus one has topological groups, topological vector spaces, topological rings, etc. From the purely abstract point of view, it is very natural to consider topo- logical spaces which have the property of being locally Euclidean, although in fact most naturally occurring examples of such entities come with some additional smooth or piecewise linear structure (PL-structure).

Definition 1.1 A topological manifold (of dimension n) is a Hausdorff topological space X with the property that each of its points x has an open neighbourhood U (i.e. open set, or “region”, containing x) which

18 Chapter 2. Topological spaces. Fibrations. Homotopies

is homeomorphic to an open set of n-dimensional Euclidean space W (for some fixed n).

Thus an n-manifold is covered by open sets U, each homeomorphic to II%“, and therefore each having induced on it via some specific homeomorphism cp: U, + RF, local co-ordinates x:,.. , x”, On each region of overlap U, n Uo there will then be defined two systems (or more) of local co-ordinates, and hence a co-ordinate transformation from each of these to the other:

X

$2. Homotopies. Homotopy type

A continuous homotopy (or briefly homotopy or deformation) of a map

f : X ---t Y, is a continuous map of the cylinder X x I to Y:

F=F(z,t):XxI-+Y, XEX, a<t<b,

(I an interval [a, b]) for which

F(x, a) = f(x) for all x E X.

Two maps f, g : X ---+ Y are homotopic if there is a continuous homotopy F such that F(x, a) = f(x), F(x, b) = g(x), II: E X.

One often needs to consider in this context pointed spacesX, Y, i.e. with particular points x0 E X, yo E Y specified. For such spacesmaps f : X -+ Y are usually also required to be “pointed”, i.e. to satisfy f(xo) = yj~, and homotopies between “pointed” maps are then also normally “pointed”, in the sensethat one requires F(xo, t) = yo for all t. Each equivalence class of homotopic maps f : X --+ Y constitutes a path- component of the function space Yx, and is called a homotopy classof maps X --) Y (or of pointed maps, as the case may be). Thus the set ro(Yx) is comprised of homotopy classes. Sometimes one has to deal with pairs of spacesA c X, B c Y, where the appropriate maps f : X --+ Y are those for which f(A) c B. Such a map of pairs is denoted by