By William S. Massey

"This publication is meant to function a textbook for a path in algebraic topology first and foremost graduate point. the most subject matters coated are the class of compact 2-manifolds, the basic workforce, overlaying areas, singular homology concept, and singular cohomology conception. those themes are constructed systematically, keeping off all pointless definitions, terminology, and technical equipment. at any place attainable, the geometric motivation in the back of a few of the recommendations is emphasised. The textual content comprises fabric from the 1st 5 chapters of the author's previous booklet, ALGEBRAIC TOPOLOGY: AN creation (GTM 56), including just about all of the now out-of-print SINGULAR HOMOLOGY idea (GTM 70). the cloth from the sooner books has been rigorously revised, corrected, and taken as much as date."

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**Sample text**

3). • Sl — reflection in a line l (see sec. 2), We see that involutive movements correspond to geometric elements of two kinds: points and lines. This correspondence is in fact one-to-one, because different points and different lines produce different involutions. Therefore, the passage from geometrical objects to involutions preserves all information, and every fact about points and lines can be reformulated in terms of the corresponding involutions. Problem 24. Find the property of a pair of reflections S l , Sm which is equivalent to the fact that the lines l and m are mutually perpendicular.

Through the intersection point of two circles draw a line on which these circles cut equal chords. Exercise 36. There is a round table and an unlimited number of equal round coins. Two players take turns at placing the coins on the table in such a way that they do not touch each other. What is the winning strategy for the first player? 4. Functions of a complex variable We return once again to Problem 16 (see page 42). Apart from the geometric solution given above, this problem also has an algebraic solution.

A glide reflection with axis l and vector v is a movement that consists in a reflection with respect to a line l and a translation by the vector v, which is supposed to be parallel to the line l (see Fig. 19). Denoting the glide reflection by Ulv , we can write the definition as Ulv = Tv ◦ Sl = Sl ◦ Tv . e. the two products taken in different order are indeed equal, because the figure AA1 A A2 is always a rectangle. Glide reflections, like all other types of plane movements, can be successfully used for solving geometrical problems.