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One of the answers to this MO question implies that loop spaces of $S^n$ for $n>1$ have non-zero homology in arbitrarily high degree.

Is there any simple (or, better yet, geometric) way to prove this?

And if the general result is too strong, is there any simple way to at least show an example of a loop space of a sphere with some high degree non-vanishing homology?

I'm curious because the only sphere loop space I can actually imagine ($\mathbb{Z} \approx \Omega S^1$) does not exhibit this kind of behaviour.

Edit: This is not exactly what I meant. It is clear to me that $H^{n-1}(\Omega S^n) \approx \mathbb{Z}$ because it follows from Hurewicz theorem. I am looking for non-trivial homology in higher degree than $n-1$. I'm sorry if my question wasn't clear, in this case geometric basically means "by hands" or "with little theory", ie. without spectral sequences or cohomology operations.

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Hi Piotr, a while back I wrote a short run-through of Milnor's book "Morse Theory", in which he incidentally computes the homology of $\Omega S^n$ in all degrees. I don't think I explicitly mentioned this result since my goal was to get to Bott periodicity as quickly as possible, but all the necessary machinery to understand this fact geometrically (as Ryan outlines below) is certainly there:… – Aaron Mazel-Gee Sep 25 '11 at 21:27
However, if you're willing to learn about the Serre spectral sequence (and you should -- it's awesome!), the proof is all but trivial. And you can even recover the ring structure on cohomology, which IIRC you can't do (easily, at least) using Morse-theoretic techniques alone. – Aaron Mazel-Gee Sep 25 '11 at 21:29
This will be very useful, thanks! – Piotr Pstrągowski Sep 25 '11 at 21:54
up vote 2 down vote accepted

For the purpose of your question, what do you mean by "geometric way"?

Consider $\Omega S^n$. Think about the subspace of $\Omega S^n$ consisting of great circles that pass-through the base-point of $S^n$, parametrized to be constant speed. This is a subspace of $\Omega S^n$ that is homeomorphic to $S^{n-1}$. It's also the generator of $H_{n-1} \Omega S^n$, provided $n>1$.

So the generator consists of loops that have been "pulled tight". Usually when people talk about geometry in the case of homology of loop-spaces, they usually mean Bott-style Morse theory. So you might want to consider the geodesic DE on the smooth free loop space $L(S^n)$, its critical points, the index of such critical points, etc.

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