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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeApr 24th 2018

changed page name to singular

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeApr 24th 2018

…or rather I meant to. Something went wrong. I’d still want to, but I refrain from doing it now in order not to generate a messy flood of empty threads here

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeOct 27th 2021

made explicit (here) statement and proof (via Euler characteristics) that the only finite group with a free action on any even-dimensional sphere is $\mathbb{Z}/2$.

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeOct 27th 2021

I have expanded and re-organized the material on free actions of finite groups on $n$-spheres a little, merging its subsection (now here) with the previously puny subsection on spherical space forms.

In particular, and for what it’s worth, I have made explicit (here) the example of the free action of finite subgroups of $Sp(1)$ on all $S^{4n+3}$-spheres.

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeOct 27th 2021

added also pointer to the Lefschetz fixed point theorem and the implication (here) that therefore the only free action of a finite group on an even-dimensional sphere must be by $\mathbb{Z}/2$ and must be orientation reversing.

This almost shows that it must be the antipodal action, but still needs an argument for why there can be no other orientation-reversing action of $\mathbb{Z}/2$.

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeOct 27th 2021

• CommentRowNumber7.
• CommentAuthorrafayaashary01
• CommentTimeOct 28th 2021
• (edited Oct 28th 2021)

For what it’s worth, the hinted proof of Remark 2.6 should also suffice as an independent proof of Proposition 2.4:

Every free continuous $G$ action on $S^{2n}$ induces a group morphism $deg \colon G \to End_{Ab}(H_{2n}(S^{2n}))^{\times} \simeq \mathbb{Z}/2$ (where $End_{Ab}(H_{2n}(S^{2n}))^{\times}$ is the group of units of the endomorphism ring of $H_{2n}(S^{2n})$), and Lefschetz says that any element of the kernel of $deg$ has a fixed point and is thus the identity, i.e., that $ker(deg) = 1$.

As for that the nontrivial map of any such $\mathbb{Z}/2$ action is homotopic to the antipodal action, is there any simple approach that doesn’t boil down to that $\pi_{2n}(S^{2n}) \simeq H_{2n}(S^{2n})$ naturally by Hurewicz?

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeOct 28th 2021

Thanks. True, the $\mathbb{Z}/2$-action is necessarily unique up to homotopy, but when comparing continuous actions, we’d want to classify them up to homeomorphism. But yeah, I suppose that follows. Will make an edit…

• CommentRowNumber9.
• CommentAuthorrafayaashary01
• CommentTimeOct 28th 2021
• (edited Oct 29th 2021)

Edit: Ignore the following.

Per this paper, there exist continuous involutions of $S^{4n}$ with quotient not homeomorphic to $\mathbf{P}(\mathbb{R}, 4n)$, making such a classification apparently difficult.

• CommentRowNumber10.
• CommentAuthorUrs
• CommentTimeOct 28th 2021
• (edited Oct 28th 2021)

Thanks, interesting. That would explain why I am getting stuck here…

But does that article give a pair of non-homomorphic involutions on an actual even-dim sphere, or just on a homotopy sphere?

• CommentRowNumber11.
• CommentAuthorUrs
• CommentTimeOct 28th 2021

Ah, their other article here is very explicit at least about non smoothly-homomorphic involutions on $S^{2n}$s. Will dig into this, thanks again for the pointer.

• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeOct 28th 2021
• (edited Oct 28th 2021)

I have now added here a list of numbers of isomorphism classes of examples of non-standard smooth free involutions on $n$-spheres for low $n$, with pointers to the literature.

My understanding is that these all refer to the standard smooth structure on the spheres, with just the involution being non-standard. But particularly for the reference Lopez de Medrano 1971 this detail remains somewhat hidden (to me) behind the notation used there.

• CommentRowNumber13.
• CommentAuthorUrs
• CommentTimeOct 28th 2021

I have expanded the section “General obstructions and existence” (here) adding mentioning now also of Milnor’s $2p$-condition and of Zassenhausen’s $p q$-conditions for orthogonal actions (though that really belongs to a section on isometric actions which is as yet missing)

• CommentRowNumber14.
• CommentAuthorUrs
• CommentTimeOct 29th 2021

For completeness, I have added as an example (here) how the ADE-classification confirms that the finite subgroups of $SU(2)$ satisfy Smith’s $p^2$-condition and Milnor’s $2 p$-condition, while finite subgroups of $SO(3)$ may violate the latter.

• CommentRowNumber15.
• CommentAuthorrafayaashary01
• CommentTimeOct 29th 2021

Re comments 10 and 11, I agree and now doubt whether my claim in comment 9 is even true. I asked a few friends to no avail and subsequently submitted the question to MO. Hopefully someone will know!

• CommentRowNumber16.
• CommentAuthorUrs
• CommentTimeOct 30th 2021

Thanks again.

The reply by I. Belegradek here seems to again be concerned with double covers by homotopy-spheres instead of actual spheres, though I haven’t yet followed the references given.

On the other hand, comparing to the the book by Lopez de Medrano, which is mostly concerned with classification of $\mathbb{Z}/2$-actions on $S^{n}$ in the piecewise-linear category (with some remarks for the smooth category but apparently no comments on the continuous category), the counting turns out at least similar to what Belegradek gives (which is maybe not surprising, just saying it for the record):

By the classification result that Lopez de Medrano previews on p. 2 (11 of 114) there are always at least four involutions up to piecewise-linear homomorphisms on the $n$-sphere for $n \gt 4$, and there is a finite number of them unless $n-3$ is divisible by 4.

At least for $n \in \{5,6,7\}$ this pattern also applies to involutions up to smooth homomorphism, as LdM discusses in Sec. V.6.1. By the discussion there, the counting up to diffeomorphism agrees with that up to pl-homeo for $S^5$ and $S^6$, but where the 7-sphere has $\mathbb{Z}/4 \oplus \mathbb{Z}$ worth of involutions up to pl-homeo, it has $\mathbb{Z}/2 \oplus \mathbb{Z}/28 \oplus \mathbb{Z}$ worth up to diffeos. (I am unsure whether this refers to non-standard involutions on the standard smooth 7-sphere, or if it also allows exotic smooth structure on the 7-sphere. On first reading/scanning of the book I thought the former, now I think maybe the latter. I wish the author had made this a little more transparent. )

It’s somewhat strange that, as far as i can see, LdM does not at least comment on the situation up-to-homeomorphism.

• CommentRowNumber17.
• CommentAuthorrafayaashary01
• CommentTimeOct 30th 2021

After racking my brain for a long while, I think I’ve found the missing ingredient: the Poincaré conjecture for topological manifolds of the appropriate dimension! After all, a topological manifold homotopy equivalent to $\mathbf{P}(\mathbb{R}, 2n)$ will have universal cover a topological manifold homotopy equivalent to $S^{2n}$, whence homeomorphic to $S^{2n}$. (The best citable reference that I could find in the $n\geq 5$ case is Theorem 7 of this paper.)

• CommentRowNumber18.
• CommentAuthorUrs
• CommentTimeOct 30th 2021

I see. That’s of course a good point.

Just to record this, I have made a brief note in the entry on what we have now regarding continuous involutions up to homeomorphism: here.

This is not meant to do justice to the topic, but just a note not to forget. Please feel invited to edit.

• CommentRowNumber19.
• CommentAuthorUrs
• CommentTimeNov 22nd 2021
• (edited Nov 22nd 2021)

I am wondering about the following:

Is this example an instance of a general pattern?

Namely, the plain dihedral groups do not have free actions on spheres, but the binary dihedral groups do. Could it be true in general that for $G$ any group which does not act freely on any sphere, there is a $\widehat G \to G$ such that $\widehat G$ does have free actions on some spheres?