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I've been learning about Morse homology, and I find it easy to compute the homology group of surfaces embedded in R3 by defining a Morse function on it, seeking the critical points and the stable und unstable manifolds etc, but what if I am to compute the homology of manifolds of a greater dimension? for example, if a take the 3-torus (or the n-torus in general), how would I proceed to calculate Hn?

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    Are you particularly wondering how to define a Morse function on these tori? Are you aware that these homology groups can be easily computed by other means? Either way this question would probably be more appropriate for M.SE. – Noah Stein Jul 11 '13 at 23:39
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    On $(S^1)^n$ a nice Morse function (that gives a minimal CW-complex) is just the sum of the squares of any linear height function on each $S^1$ factor. This gives you $2^n$ cells, corresponding to the $n$-tuples of critical points of the linear functions on the factors. – Ryan Budney Jul 11 '13 at 23:46
  • I believe the OP is asking how to compute the homology of $T^3$ embedded in $R^4$ using Morse theory. –  Jul 12 '13 at 00:04

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I believe the question is: what is a nice formula for embedding a torus in $\mathbb{R}^4.$ Once you have that, you can compute the critical points, etc. The nicest is presumably the Clifford Torus. The OP can compute the critical points of his favorite height function.

Igor Rivin
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