The creation of algebraic topology is a major accomplishment of 20th-century mathematics. The goal of this book is to show how geometric and algebraic ideas met and grew together into an important branch of mathematics in the recent past. The book also conveys the fun and adventure that can be part of a mathematical investigation.

Combinatorial topology has a wealth of applications, many of which result from connections with the theory of differential equations. As the author points out, "Combinatorial topology is uniquely the subject where students of mathematics below graduate level can see the three major divisions of mathematics — analysis, geometry, and algebra — working together amicably on important problems."

To facilitate understanding, Professor Henle has deliberately restricted the subject matter of this volume, focusing especially on surfaces because the theorems can be easily visualized there, encouraging geometric intuition. In addition, this area presents many interesting applications arising from systems of differential equations. To illuminate the interaction of geometry and algebra, a single important algebraic tool — homology — is developed in detail.

Written for upper-level undergraduate and graduate students, this book requires no previous acquaintance with topology or algebra. Point set topology and group theory are developed as they are needed. In addition, a supplement surveying point set topology is included for the interested student and for the instructor who wishes to teach a mixture of point set and algebraic topology. A rich selection of problems, some with solutions, are integrated into the text.

vertexes 0> = {Pn}, 1 = {Q„}, and 0t = {Rn} consisting only of points with northeast-, northwest-, and south-pointing vectors, respectively. By compactness there exists a point P of Q) near ^ . Since the sides of the triangles tend to zero, it follows that P is also near J and near ^ . By continuity the vector V(P) is near the sequence of vectors V{^) as well as the sequences V(£>) and V{?A\ The vectors V(0>) all point northeast. Since the set of northeast-pointing vectors, namely the first

about each point, and the stability or instability of the critical points. These properties are part of a purely topological theory of differential equations developed by Poincare. For if we imagine the portion of the plane represented in Figure 7.1 cut out of a sheet of rubber as in §1 and subjected to a continuous transformation, then, although the shape and length of the paths could change, the number and nature of the critical points cannot change; nodes remain nodes (stable or unstable);

neighborhood contains a point S of B because & <- B. Now R can be connected with S by a polygonal chain lying in the given neighborhood of R and hence lying in G, and it can also be connected with P by a polygonal chain lying in G. Therefore P can be connected with S by such a chain, contradicting the fact that S is in B. A similar contradiction results if instead B contains a point near A. Q.E.D. Consider the application of this theorem to the proof of the Jordan curve theorem. If f is a Jordan

case, the removal decreases both F and E by one, while in the second case, F and V decrease by one and E decreases by two. In any event, the quantity F — E + V is Figure 1.7 Step two: the topless cube triangulated. 8 BASIC CONCEPTS Figure 1.8 Step three: removal of triangles. unchanged by the removal. Eventually we are left with just one triangle for which F = 1, E = 3, and V = 3. Obviously here F — E + V = 1. Since Steps 2 and 3 did not alter the sum F — E + V, this proves Euler's formula.

depends on the set {yl9 y2,. .., yn} but does not depend on the set {j>i, )'i, • • •, 3;n-i}> then yn depends on the set {yu y2,..., yn_u x). Proofs of the first two properties are easy to supply (Exercise 1). The third property is called the exchange axiom and is proved as follows. Let x be dependent on {yl9 y2,..., yn}. Then x is a linear combination of the elements Vi, >'2> • • •, );n> s o that x = at)\ + a2y2 + • • • + anyn9 where each coefficient al9 a2,..., an is a zero or a one. If the