ATch1 - Algebraic topology can be roughly dened as the...

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Algebraic topology can be roughly defined as the study of techniques for forming algebraic images of topological spaces. Most often these algebraic images are groups, but more elaborate structures such as rings, modules, and algebras also arise. The mechanisms that create these images — the ‘lanterns’ of algebraic topology, one might say — are known formally as functors and have the characteristic feature that they form images not only of spaces but also of maps. Thus, continuous maps between spaces are projected onto homomorphisms between their algebraic images, so topo- logically related spaces have algebraically related images. With suitably constructed lanterns one might hope to be able to form images with enough detail to reconstruct accurately the shapes of all spaces, or at least of large and interesting classes of spaces. This is one of the main goals of algebraic topology, and to a surprising extent this goal is achieved. Of course, the lanterns necessary to do this are somewhat complicated pieces of machinery. But this machinery also has a certain intrinsic beauty. This first chapter introduces one of the simplest and most important functors of algebraic topology, the fundamental group, which creates an algebraic image of a space from the loops in the space, the paths in the space starting and ending at the same point. The Idea of the Fundamental Group To get a feeling for what the fundamental group is about, let us look at a few preliminary examples before giving the formal definitions.
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22 Chapter 1 The Fundamental Group Consider two linked circles A and B in R 3 , as shown in the figure. Our experience with actual links and chains A B tells us that since the two circles are linked, it is impossi- ble to separate B from A by any continuous motion of B , such as pushing, pulling, or twisting. We could even take B to be made of rubber or stretchable string and allow completely general continu- ous deformations of B , staying in the complement of A at all times, and it would still be impossible to pull B off A . At least that is what intuition suggests, and the fundamental group will give a way of making this intuition mathematically rigorous. Instead of having B link with A just once, we could make it link with A two or more times, as in the figures to the right. As a further variation, by assigning an orientation to B A A B - 3 B 2 we can speak of B linking A a positive or a negative number of times, say positive when B comes forward through A and negative for the reverse direction. Thus for each nonzero integer n we have an oriented circle B n linking An times, where by ‘circle’ we mean a curve homeomorphic to a circle. To complete the scheme, we could let B 0 be a circle not linked to A at all.
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This note was uploaded on 07/19/2010 for the course MA 8 taught by Professor Vuletic,m during the Fall '08 term at Caltech.

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ATch1 - Algebraic topology can be roughly dened as the...

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