# In $n>5$, topology = algebra

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?

• 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 – Henry T. Horton Aug 15 '13 at 21:22
• This is quite beautifully explained in the first chapter of Scorpan's nice The Wild World of 4-Manifolds. I recommend it strongly. – PseudoNeo Aug 26 '14 at 18:12
• 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. – anomaly Jun 12 '16 at 5:03