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From my study of physics I have arrived at the question of how to classify all maps $\mathbb{T}^2\to S^4$ where $\mathbb{T}^2$ is the two-torus. The classification should be up to homotopies.

The reason I am stuck is that my usual mechanism to classify such maps (the fundamental group) cannot work here because the domain is $\mathbb{T}^2$ and not $S^2$.

I "guess" that the answer is that there are only two classes up to homotopies so that the fundamental group (sort of abuse of terminology as the fundamental group has to classify maps where $S^n$ is the domain, if I understand correctly) is $\mathbb{Z}/2\mathbb{Z}$.

To be concrete, let us parametrize $\mathbb{T}^2\equiv\mathbb{R}^2/\mathbb{Z}^2$ so that we may write $(x,y)\in\mathbb{T}^2$ Thus we are given a continuous map $\mathbb{T}^2\stackrel{d}{\to} S^4;(x,y)\stackrel{d}{\mapsto}d(x,y)\in\mathbb{R}^5$ such that $||d(x,y)||=1$. Given such a map $d$, I would ideally like some formula or clear criterion to put it in either class.

EDIT: After thinking about it some more I realized I want to classify a subset of the maps above, and not the whole space. The subspace is defined by all maps for which $(0.5+x,0.5+y)$ and $(0.5-x,0.5-y)$ for all $(x,y)\in[0,0.5]^2$ go to the same point on the sphere.

SECOND EDIT: After thinking yet some more, let $S\subset\{1,2,3,4,5\}$ such that $|S|\neq0$ be given. Now we assume that $$d_i(0.5-x,0.5-y) = - d_i(0.5+x,0.5+y) \forall i\in S$$ for all $(x,y)\in[0,0.5]^2$ and $$d_i(0.5-x,0.5-y) = + d_i(0.5+x,0.5+y) \forall i\notin S$$ for all $(x,y)\in[0,0.5]^2$

What is the homotopic classification of this set of maps?

PPR
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1 Answers1

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Considering these spaces as CW complexes, every map is homotopic to a cellular map. This implies that all maps from the torus to $S^4$ are nulhomotopic.

Matt Samuel
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  • Hi, thanks for your answer. Is there any kind of other classification that will distinguish such maps from one another? Perhaps something to do with an involution on $T^2$?? – PPR Feb 25 '15 at 16:29
  • Trivially, the maps can be distinguished from one another by not imposing any additional equivalence relation. Beyond that it really depends what you're trying to do. – Matt Samuel Feb 25 '15 at 16:32
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    @PPR nontrivially, the maps can likely be distinguished by whether they can be sent to one another via a homeomorphism of $S^4$. This is related to knot theory, and the answer is probably very complicated. – Matt Samuel Feb 25 '15 at 16:36
  • I am trying to realize something like this: http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.61.1329 – PPR Feb 25 '15 at 16:39
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    Sorry, I don't understand the abstract well enough to see how it's related, and a subscription is needed to access the text. I would recommend posting another question that is more specific. – Matt Samuel Feb 25 '15 at 16:44
  • Hi, I edited the question with some new information to reflect what I meant yesterday.. – PPR Feb 26 '15 at 17:43
  • @PPR my answer still applies. This is essentially the same as considering maps from a certain quotient space of the torus which itself can be realized as a two dimensional CW complex. – Matt Samuel Feb 26 '15 at 18:04
  • Hi Matt, I made a second edit, perhaps now I got something? – PPR Feb 27 '15 at 15:06
  • The point is that everything is trivial unless you have something that is at least 4 dimensional – Matt Samuel Feb 27 '15 at 15:08