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Suppose we have a space which is a direct product of 1-spheres: $\mathbb{S}^1\times\mathbb{S}^1\times...\times\mathbb{S}^1 = \mathbb{T}^N$ (the total number of spheres = $N$), or 2-spheres: $\mathbb{S}^2\times\mathbb{S}^2\times...\times\mathbb{S}^2$. The $\theta_i$ is the polar angle of the point on the sphere number $i$. Let we have also a constraint: $\sum\limits_{i=1}^N \cos(\theta_i) = const$, and $(-N)<const<N$. This constrain significantly changes this topological spaces and the question is how to identify the resulting space. May be some isomorphism to something conventional is possible ?

Philipp
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  • I think things work out so that $\sum \cos (\theta_i)$ is a Morse function. If so, this question is completely answered by Morse theory. The answer depends quite a bit on whether the constant is a critical value. – Qiaochu Yuan Jun 24 '16 at 18:29
  • This constraint which was applied to the direct product of spheres produces a new space which properties we are looking for. For such a space the $\sum \cos( \theta_i)$ is not a Morse function, because it is a constant function - just a identity. – Philipp Jun 25 '16 at 10:18
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    What I'm saying is that if this sum is a Morse function on the original space, then the spaces you care about are its level sets, and then Morse theory completely answers the question of what the level sets of a Morse function look like. – Qiaochu Yuan Jun 26 '16 at 02:18

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