By William Fulton
This e-book introduces the $64000 rules of algebraic topology through emphasizing the relation of those rules with different components of arithmetic. instead of selecting one viewpoint of contemporary topology (homotropy conception, axiomatic homology, or differential topology, say) the writer concentrates on concrete difficulties in areas with a number of dimensions, introducing simply as a lot algebraic equipment as valuable for the issues encountered. This makes it attainable to work out a greater variety of significant beneficial properties within the topic than is usual in introductory texts; it's also in concord with the ancient improvement of the topic. The publication is aimed toward scholars who don't unavoidably intend on focusing on algebraic topology.
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Additional info for Algebraic Topology: A First Course (Graduate Texts in Mathematics, Volume 153)
It then suffices to continuously move Re2 , . . , Ren to Ae2 , . . , Aen , respectively, keeping Ae1 fixed. Notice that • Re2 , . . , Ren are all orthogonal to Re1 = Ae1 , because e2 , . . , en are all orthogonal to e1 and R preserves angles. • Ae2 , . . , Aen are all orthogonal to Ae1 , because e2 , . . , en are all orthogonal to e1 and A preserves angles. Thus the required motion can take place in the Rn−1 of vectors orthogonal to Ae1 , where it exists by the assumption that SO(n − 1) is path-connected.
Path-connectedness The most striking difference between SO(n) and O(n) is a topological one: SO(n) is path-connected and O(n) is not. That is, if we view n× n matrices 2 as points of Rn in the natural way—by interpreting the n2 matrix entries a11 , a12 , . . , a1n , a21 , . . , a2n , . . , an1 , . . , ann as the coordinates of a point— then any two points in SO(n) may be connected by a continuous path in SO(n), but the same is not true of O(n). Indeed, there is no continuous path in O(n) from ⎛ ⎞ ⎛ ⎞ 1 −1 ⎜ 1 ⎟ ⎜ ⎟ 1 ⎜ ⎟ ⎜ ⎟ to ⎜ ⎟ ⎜ ⎟ ..
On the geometric side, an octonion projective space (of more than two dimensions) is impossible, because of a theorem of Hilbert from 1899. Hilbert’s theorem essentially states that the coordinates of a projective space satisfy the associative law of multiplication (see Hilbert ). One therefore has only O itself, and the octonion projective plane, OP2 , to work with. Because of this, there are few important Lie groups associated with the octonions. But these are a very select few! They are called the exceptional Lie groups, and they are among the most interesting objects in mathematics.