By Massey

This e-book is meant to function a textbook for a path in algebraic topology in the beginning graduate point. the most themes lined are the class of compact 2-manifolds, the basic workforce, protecting areas, singular homology concept, and singular cohomology concept. those subject matters are constructed systematically, warding off all unecessary definitions, terminology, and technical equipment. anywhere attainable, the geometric motivation at the back of many of the techniques is emphasised. The textual content includes fabric from the 1st 5 chapters of the author's prior publication, ALGEBRAIC TOPOLOGY: AN advent (GTM 56), including just about all of the now out-of- print SINGULAR HOMOLOGY thought (GTM 70). the fabric from the sooner books has been conscientiously revised, corrected, and taken brand new.

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**Example text**

S/. In fact, the converse of this statement is also true: Theorem 17 ([54, Thm. 1]). S/ that has hD; Di D 0. Then there exists an n such that nD is the class of a fibre in a genus one fibration on S. S/ with hD; Di D 0. Then, possibly after negating, we may assume that D is in the closure of the positive cone CS . 34 A. Harder and A. S/. D/ is the class of a fibre in a genus one fibration. Thus we have: Corollary 3 ([54, Cor. 3]). S/ admits a nonzero element u with hu; ui D 0. Example 17 (Anticanonical hypersurfaces in P1 P2 ).

Harder was supported by an NSERC PGS D scholarship and a University of Alberta Doctoral Recruitment Scholarship. A. Thompson was supported by a Fields-Ontario-PIMS postdoctoral fellowship with funding provided by NSERC, the Ontario Ministry of Training, Colleges and Universities, and an Alberta Advanced Education and Technology Grant. Appendix: Lattice Theory In this appendix we present a short description of the lattice theory that is used in the preceding article. The main reference for this section will be [49].

2) 77, 563–626 (1963) 32. : On compact analytic surfaces, III. Ann. Math. (2) 78, 1–40 (1963) 33. : Threefolds and deformations of surface singularities. Invent. Math. 91(2), 299–338 (1988) 34. : Algebraic K3 surfaces with finite automorphism groups. Nagoya Math. J. 116, 1–15 (1989) 35. : Abelian varieties attached to polarized K3 -surfaces. Math. Ann. 169, 239–242 (1967) 36. : Degenerations of K3 surfaces and Enriques surfaces. Math. USSR Izv. 11(5), 957–989 (1977) 37. : On modifications of degenerations of surfaces with Ä D 0.