During the study of the surgery theory I faced following sentence:
Surgery theory works best for $n > 5$, when "topology = algebra".
I don't know what is the meaning of topology=algebra. Can someone clear the sentence to me?
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asked Aug 15, 2013 at 21:08
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19$\begingroup$ There is a completely algebraic theory of surgery developed by Ranicki. See the following notes (especially section 5): maths.ed.ac.uk/~aar/surgery/ranicki.pdf $\endgroup$– Henry T. HortonAug 15, 2013 at 21:22
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12$\begingroup$ This is quite beautifully explained in the first chapter of Scorpan's nice The Wild World of 4-Manifolds. I recommend it strongly. $\endgroup$– PseudoNeoAug 26, 2014 at 18:12
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$\begingroup$ The meaning is simply that the $h$- and $s$-cobordism theorems only hold in dimensions $n\geq 5$, where they make algebraic machinery effective. For an explanation of what happens below dimension $5$ (particularly dimension $4$), I can't recommend Scorpan's book highly enough. $\endgroup$– anomalyJun 12, 2016 at 5:03
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In the interest of recording the answers in the comments above and removing the question from the "unanswered" queue, I note the following.
Henry Horton:
There is a completely algebraic theory of surgery developed by Ranicki. See the following notes (especially section 5): http://www.maths.ed.ac.uk/~aar/surgery/ranicki.pdf
PseudoNeo:
This is quite beautifully explained in the first chapter of Scorpan's nice The Wild World of 4-Manifolds. I recommend it strongly.