By Saunders Mac Lane (auth.)

In proposing this therapy of homological algebra, it's a excitement to recognize the assistance and encouragement which i've got had from both sides. Homological algebra arose from many resources in algebra and topology. Decisive examples got here from the learn of workforce extensions and their issue units, an issue I realized in joint paintings with OTTO SCHIL LING. yet another improvement of homological rules, which will their topological purposes, got here in my lengthy collaboration with SAMUEL ElLENBERG; to either collaborators, especial thank you. for a few years the Air strength place of work of clinical examine supported my learn tasks on a number of matters now summarized right here; it's a excitement to recognize their full of life realizing of easy technology. either REINHOLD BAER and JOSEF SCHMID learn and commented on my whole manuscript; their recommendation has resulted in many advancements. ANDERS KOCK and JACQUES RIGUET have learn the total galley evidence and stuck many slips and obscurities. one of the others whose sug gestions have served me good, I be aware FRANK ADAMS, LOUIS AUSLANDER, WILFRED COCKCROFT, ALBRECHT DOLD, GEOFFREY HORROCKS, FRIED wealthy KASCH, JOHANN LEICHT, ARUNAS LIULEVICIUS, JOHN MOORE, DIE TER PUPPE, JOSEPH YAO, and a few my present scholars on the collage of Chicago - to not m~ntion the auditors of my lectures at Chicago, Heidelberg, Bonn, Frankfurt, and Aarhus. My spouse, DOROTHY, has cheerfully typed extra types of extra chapters than she wish to count number. Messrs.

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2) Two cycles c and c' in the same homology class are said to be homologous; in symbols c'"'-'c'. As first examples we shall give a number of specific differential groups with their homology. Most of these examples will be found by dissecting a simple geometric figure into cells and taking d to be the operator which q P assigns to each cell the sum of its boundary cells, each affected with a suitable sign. Example 1. Take two points p and q on a circle 51 which divide the circle into two b semicircular arcs a and b.

Hence, for this D the only homology classes are those yielded by the constant functions, and H(C) is the additive group of real numbers. The same conclusion holds if D is the interior of a circle, but fails if D is, say, the interior of a circle with the origin deleted. In this latter case an exact differential need not be the differential of a function f. For example (-y dx+ x dy)/(x 2+ y2) is not such. Example 7. A circular cylinder may be regarded as the cartesian product SIxI of a circle SI and a unit interval I.

IX and XII. Exercises In the category of topological spaces, show that the disjoint union of two spaces provides a direct sum diagram, and that the cartesian product XxY of two spaces, with its usual topology and with the natural projections on X and Y, provides a direct product diagram. 1. 2. Show that any two objects in the category of groups may be ends of a direct product diagram and of a direct sum diagram. ) 3. II of elements IX, fJ, y in which a product fJIXEvlt is sometimes defined. II if "fJ= fJ whenever "fJ is defined and IX"= IX whenever IX" is defined.