Differential Topology, Foliations, and Group Actions: by Paul A. Schweitzer, Steven Hurder, Nathan Moreira DOS Santos

By Paul A. Schweitzer, Steven Hurder, Nathan Moreira DOS Santos

This quantity includes the court cases of the Workshop on Topology held on the Pontif?cia Universidade Cat?lica in Rio de Janeiro in January 1992. Bringing jointly approximately one hundred mathematicians from Brazil and all over the world, the workshop lined numerous issues in differential and algebraic topology, together with team activities, foliations, low-dimensional topology, and connections to differential geometry. the most focus was once on foliation idea, yet there has been a full of life alternate on different present subject matters in topology. the quantity comprises a great record of open difficulties in foliation study, ready with the participation of a few of the pinnacle international specialists during this region. additionally provided listed here are surveys on workforce actions---finite staff activities and tension thought for Anosov actions---as good as an easy survey of Thurston's geometric topology in dimensions 2 and three that may be obtainable to complicated undergraduates and graduate scholars.

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The map ϕ : SU(2) → SO(3) defined by ϕ (t) = {±t} is a 2-to-1 homomorphism, because the two elements t and −t of SU(2) go to the single pair ±t in SO(3). Thus SO(3) looks “simpler” than SU(2) because SO(3) has only one element where SU(2) has two. Indeed, SO(3) is “simpler” because SU(2) is not simple—it has the normal subgroup {±1}—and SO(3) is. We now prove this famous property of SO(3) by showing that SO(3) has no nontrivial normal subgroup. Simplicity of SO(3). The only nontrivial subgroup of SO(3) closed under conjugation is SO(3) itself.

This discovery motivates much of Lie theory. There are infinitely many simple Lie groups, and most of them are generalizations of rotation groups in some sense. However, deep ideas are involved in identifying the simple groups and in showing that we have enumerated them all. To show why it is not easy to identify all the simple Lie groups we make a special study of SO(4), the rotation group of R4 . Like SO(3), SO(4) can be described with the help of quaternions. But a rotation of R4 generally depends on two quaternions, and this gives SO(4) a special structure, related to the direct product of S3 with itself.

He still had not noticed that there is no three square identity, but he suspected that multiplying triples of the form a + bi + c j requires a new object k = i j. Also, he began to realize that there is no hope for the commutative law of multiplication. Desperate to salvage something from his 13 years of work, he made the leap to the fourth dimension. He took k = i j to be a vector perpendicular to 1, i, and j, and sacrificed the commutative law by allowing i j = − ji, jk = −k j, and ki = −ik. On October 16, 1843 he had his famous epiphany that i, j, and k must satisfy i2 = j2 = k2 = i jk = −1.

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