In $n>5$, topology = algebra

algebraic-topology general-topology geometric-topology surgery-theory

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.

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