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Algebraic Topology Notes, Part I: Homology ALGEBRAIC TOPOLOGY NOTES, PART I: HOMOLOGY JONATHAN A. HILLMAN Abstract. The teaching material that forms this web site is copyright. Other than for the purposes of and subject to the conditions prescribed under the Copyright Act, no part of it may in any form or by any means (electronic, mechanical, microcopying, photocopying, recording or otherwise) be altered, reproduced, stored in a retrieval system or transmitted without prior written permission from the University of Sydney. COPYRIGHT. The University of Sydney 2003 (revised in June 2011 and August 2014). 1 2 JONATHAN A. HILLMAN 1. Introduction Algebraic topology advanced more rapidly than any other branch of mathematics during the twentieth century. Its influence on other branches, such as algebra, algebraic geometry, analysis, differential geometry and number theory has been enormous. The typical problems of topology such as whether Rm is homeomorphic to Rn or whether the projective plane can be embedded in R3 or whether we can choose a continuous branch of the complex logarithm on the whole of Cnf0g may all be interpreted as asking whether there is a suitable continuous map. The goal of Algebraic Topology is to construct invariants by means of which such problems may be translated into algebraic terms. The homotopy groups πn(X) and homology groups Hn(X) of a space X are two important families of such invariants. The homotopy groups are easy to define but in general are hard to compute; the converse holds for the homology groups. In Part I of these notes we consider homology, beginning with simplicial homol- ogy theory. Then we define singular homology theory, and develop the properties which are summarized in the Eilenberg-Steenrod axioms. (These give an axiomatic characterization of homology for reasonable spaces.) We then apply homology to various examples, and conclude with two or three lectures on cohomology and dif- ferential forms on open subsets of Rn. Although we shall assume no prior knowledge of Category Theory, we shall introduce and use categorical terminology where ap- propriate. (Indeed Category Theory was largely founded by algebraic topologists.) Part II of these notes is an introduction to the fundamental group and combi- natorial group theory. We do not consider some topics often met in a first course on Algebraic Topology, such as reduced homology, or the Mayer-Vietoris Theorem, which is a very useful consequence of the Excision property. Other topics not considered here, but which are central to the further study and application of algebraic topology include the Jordan-Brouwer separation theorems, orientation for manifolds, cohomology and Poincar´eduality. References Algebraic Topology: A First Course by M.J.Greenberg (2nd edition: and J.Harper), Benjamin/Cummings (1981). Algebraic Topology, by A. Hatcher, Cambridge University Press (2002). Also available through the WWW (\www.math.cornell.edu/∼hatcher"). Further reading: Lectures on Algebraic Topology by A. Dold, Springer-Verlag (1972). A Concise Course in Algebraic Topology, by J.P.May, Chicago Lectures in Math- ematics, Chicago UP (1999). Algebraic Topology, by E.H.Spanier, McGraw-Hill (1966). ALGEBRAIC TOPOLOGY NOTES, PART I: HOMOLOGY 3 2. some spaces The most interesting spaces for geometrically minded mathematicians are man- ifolds, cell-complexes and polyhedra. Notation. X =∼ Y means X is homeomorphic to Y , if X; Y are spaces. n n n n R+ = f(x1; : : : ; xn) 2 R j xn ≥ 0g. @R+ = f(x1; : : : ; xn) 2 R j xn = 0g. n n 2 D = f(x1; : : : ; xn) 2 R j Σxi ≤ 1g. n−1 n n 2 S = @D = f(x1; : : : ; xn) 2 R j Σxi = 1g. n n n n D+ = f(x0; : : : xn) 2 S j xn ≥ 0g and D− = f(x0; : : : xn) 2 S j xn ≤ 0g. intDn = Dn n @Dn = Dn n Sn−1. O = (0;:::; 0). Exercise 1. (a) Show that Dn n fOg =∼ Sn−1 × (0; 1] and Rn n fOg =∼ Sn−1 × R. (b) Show that Dn =∼ [0; 1]n. Definition. An n-manifold is a space M whose topology arises from a metric and such that for all m 2 M there is an open neighbourhood U and a homeomorphism n h : U ! h(U) onto an open subset of R+. The boundary @M is the set of points m n for which there is such a homeomorphism h with h(m) 2 @R+. n n We shall show later that the dimension n is well-defined, and that @(R+) = @R+, so @M is an (n − 1)-manifold and @@M = ;. (See Exercise 17 below.) FACT. The metric condition is equivalent to requiring that M be Hausdorff (T2) and that each connected component of M have a countable base of open sets. Open n subspaces of R+ clearly satisfy these conditions, but there are bizarre examples which demonstrate that these conditions are not locally determined. Examples. (n = 2): disc, sphere, torus (T ), annulus, M¨obiusband (Mb). 2 2 ∼ 2 The projective plane P (R) = S =(x ∼ −x) = Mb [ D . Let D+ = f(x; y; z) 2 2 1 2 1 2 1 S j z ≥ 2 g, D− = f(x; y; z) 2 S j z ≥ 2 g and E = f(x; y; z) 2 S j jzj ≤ 2 g. 2 Then S = D+ [ E [ D−. Since the antipodal map interchanges D+ and D− and identifying antipodal points of E gives a M¨obiusband we see that P 2(R) =∼ Mb[D2 is the union of a M¨obiusband with D2. All surfaces without boundary are locally homeomorphic to each other. We need a global invariant to distinguish them. Homology provides such invariants. (It is not so successful in higher dimensions.) 3. projective spaces Let F = R, C or H (the quaternion algebra). Then F is a skew field and has finite dimension as a real vector space. Let d = dimRF (= 1, 2 or 4). n+1 Given a point X = (x0; : : : ; xn) 2 F n fOg, let [x0 : ··· : xn] be the line through O and X in F n+1. Let P n(F ) (or FP n) be the set of all such lines through O in F n+1. Two nonzero points determine the same line if and only if they are proportional, i.e., [x0 : ··· : xn] = [y0 : ··· : yn] if and only if there is a λ 2 × n n+1 × F = F nfOg such that yi = λxi for 0 ≤ i ≤ n. Hence P (F ) = (F nfOg)=F . Each line through O in F n+1 =∼ R(n+1)d passes through the unit sphere S(n+1)d−1, and two points on the unit sphere determine the same line if and only if one is a multiple of the other by an element of Sd−1 (= {±1g, S1 or S3), the subgroup of elements of F × of absolute value 1. Thus there is a canonical surjection from S(n+1)d−1 to P n(F ), and P n(F ) = S(n+1)d−1=Sd−1 is the orbit space of a group action. In particular, P n(R) is obtained from the n-sphere by identifying antipodal points. 4 JONATHAN A. HILLMAN The group GL(n + 1;F ) acts transitively on the lines through O, and so P n(F ) may be identified with the quotient of GL(n + 1;F ) by the subgroup which maps the line (x; 0;:::; 0) 2 F n+1 to itself. n Let Ui = f[x0 : ··· : xn] 2 P (F ) j xi 6= 0g, for each 0 ≤ i ≤ n. Then n i=n n P (F ) = [i=0 Ui. There are bijections φi : Ui ! F , given by φi([x0 : ··· : xn]) = (x0=xi;:::; x[i=xi; : : : ; xn=xi); for all [x0 : ··· : xn] 2 Ui; −1 and with inverse φi (y1; : : : ; yn) = [y1 : ··· : yi : 1 : yi+1 : ··· : yn]. Moreover there n−1 n is an obvious bijection from P (F ) to P (F ) n Ui. As F n has a natural metric topology, we may topologize P n(F ) by declaring the subsets Ui to be open and the bijections φi to be homeomorphisms. We may identify Dnd with the unit ball in F n. Then the map h : Dnd ! P n(F ) given by n h(x1; : : : ; xn) = [x1 : ··· : xn : 1 − jxj] is a continuous surjection. Hence P (F ) nd nd nd−1 is compact. Moreover, h maps intD = D n S homeomorphically onto Un, nd−1 nd n−1 n while it maps S = @D onto P (F ) = P (F ) n Un. Exercise. Show that P n(F ) is Hausdorff and separable. A compact Hausdorff space is completely regular. If moreover it is separable then it is metrizable, by the Urysohn embedding theorem. (One can also define a metric on P n(F ) directly.) Thus these projective spaces are manifolds. 1 d Special cases. P (F ) = F [f1g = S , where 1 = [1 : 0], and points [x : y] 2 U1 are identified with the ratio x=y 2 F . In particular, P 1(C) is the extended complex plane. The map h : S3 ! S2 = P 1(C) given by h(u; v) = [u : v] for all (u; v) 2 C2 such that juj2 + jvj2 = 1 is known as the Hopf fibration. Remark. The above construction of the set P n(F ) works equally well for any skew field F , in particular for F a finite field! 4. cell complexes n−1 n Definition. Let X be a space and f : S ! X a map. Then X [f e = X q Dn=(y ∼ f(y); 8y 2 Sn−1) is the space obtained by adjoining an n-cell to X n along f. (The topology on X [f e is the finest such that the quotient function n n q : X q D ! X [f e is continuous.) n n We may identify X with a closed subset of X [f e .
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