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It is strange. But waiting for an expert to fly in and do the work for you is maybe an unreasonable expectation. I've added the homotopy groups for the orthogonal groups, but I don't know what they are in the unitary case.
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The context of Bott periodicity is that the homotopy groups of spheres, which would be expected to play the basic part in algebraic topology by analogy with homology theory, have proved elusive (and the theory is
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I could copy these homotopy groups, but ideally the groups themselves would be be introduced into the article by someone who is an expert in this material.
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It is very strange that, although the periodicity results are expressed clearly in this article, the actual homotopy groups π
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Can someone please make the meaning of this sentence clear? I cannot imagine what it is intended to mean.
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John Baez says (citing a paper by Cartan): "This 'period-8' behavior was discovered by Cartan in 1908 ..."
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What is utterly unclear is "the basic part in algebraic topology by analogy with homology theory".
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This seems relevant. Perhaps someone familiar with the subject matter can add this? —
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Without the Bott periodicity theorems, we would not know that real and complex
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of course part of algebraic topology. So: What is this supposed to mean???
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That "the homotopy groups of sphere" (by which is meant the
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of the homotopy groups of spheres) is "elusive" is clear.
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