By William Fulton

To the trainer. This publication is designed to introduce a pupil to a few of the real principles of algebraic topology through emphasizing the re­ lations of those rules with different parts of arithmetic. instead of making a choice on one standpoint of modem topology (homotopy conception, simplicial complexes, singular idea, axiomatic homology, vary­ ential topology, etc.), we focus our cognizance on concrete prob­ lems in low dimensions, introducing merely as a lot algebraic machin­ ery as important for the issues we meet. This makes it attainable to work out a greater variety of significant good points of the topic than is common in a starting textual content. The booklet is designed for college kids of arithmetic or technology who're now not aiming to turn into working towards algebraic topol­ ogists-without, we are hoping, discouraging budding topologists. We additionally consider that this strategy is in higher concord with the ancient devel­ opment of the topic. What may we adore a pupil to understand after a primary path in to­ pology (assuming we reject the reply: half what one would prefer the coed to grasp after a moment path in topology)? Our solutions to this have guided the alternative of fabric, together with: less than­ status the relation among homology and integration, first on aircraft domain names, afterward Riemann surfaces and in larger dimensions; wind­ ing numbers and levels of mappings, fixed-point theorems; appli­ cations equivalent to the Jordan curve theorem, invariance of area; in­ dices of vector fields and Euler features; basic groups

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