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On a manifold $M=S^{2}\times S^{1}$ I'm trying to find the homogenous model/Klein space geometry, and define a reduction of it's structure group. Now since $S^{2}=\frac{SO(3)}{SO(2)}$ we have that:

$$\frac{G}{H}=\frac{SO(3)\times SO(2)}{SO(2)}$$

Where $G=SO(3)\times SO(2)$ is transitive on $M$ and $H=SO(2)$ is the point stabilizer. As far as I can tell everythings all right here; however, in reading about Thurston's Geometerization Conjecture, this isn't one of the eight Thurston geometries for a closed, oriented, 3-manifold. For this manifold it should be of the form:

$$\frac{G}{H}=\frac{SO(3)\times\mathbb{R}\times\mathbb{Z}_{2}}{SO(2)\times\mathbb{Z}_{2}}$$

Which is pretty close. I'm guessing these spaces have to be isomorphisms of one another. I can equivalently define the first equation in terms of double covers:

$$\frac{G}{H}=\frac{SU(2)\times U(1)}{U(1)}$$

($U(1)=SO(2)$ double covers itself) and use a hand-wavy argument that $SO(n)\times\mathbb{Z}_{2}=Doublecover(SO(n))$. That still doesn't explain $\mathbb{R}$ versus $SO(2)$ in our transitive group. The former is the universal cover of the latter so maybe that comes into play? Or perhaps ${\mathbb{R} \over \mathbb{Z}_{2}} =SO(2)$?

I'm specifically interested in a reduction of the G-bundle to an H-bundle so how this breaks down is important for me.

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Recall that a geometric structure on $M$ is a diffeomorphism $\varphi:X/\Gamma\to M$ where $G$ is a Lie group, $H\subseteq G$ is a compact subgroup, $X=G/H$ is a model geometry, and $\Gamma$ is a discrete subgroup of $G$ acting on $X$ in the canonical way.

The model geometry of $M=S^2\times S^1$ is $X=\frac{O(3)\times\mathbb{R}\times\mathbb{Z}_2}{O(2)\times\mathbb{Z}_2}$, but this does not mean that $M\cong X$. Instead, we have $M\cong X/\Gamma$, and $\Gamma$ is nontrivial in this case (notice that $\Gamma$ is always isomorphic to $\pi_1(M)$). Instead, we have $\Gamma=\mathbb{Z}$, regarded as a subgroup of the $\mathbb{R}$ factor in $O(3)\times\mathbb{R}\times\mathbb{Z}_2$.

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  • $\begingroup$ Aha! I was missing the connection with the fundamental group $\pi_{1}$. So we have a reduction to a $U(1)\times \mathbb{Z}_2 $ (H)-structure on $M$. Any idea what kind of invariant object this corresponds to? If it was just $U(1)$ I'd say it's an invariant 2-form similar to the electromagnetic field two-form, but with the $\mathbb{Z}_2$ I'm not sure $\endgroup$
    – R. Rankin
    Commented Aug 14, 2021 at 15:22
  • $\begingroup$ @R.Rankin A geometric structure need not correspond to any particular invariant object, and for a general geometry $X=G/H$ it is possible to come up with all kinds of objects whose stabilizer is $G$. For instance, for $S^2\times\mathbb{R}\cong\frac{SO(3)\times\mathbb{R}\times\mathbb{Z}_2}{SO(2)\times\mathbb{Z}_2}$, one can regard the geometry as the stabilizer of $(H,V,g,[\omega])$, where $H$ and $V$ are the horizontal and vertical distributions on the product, $g$ is the standard product metric, and $[\omega]$ is an orientation of the $S^2$ fibers. $\endgroup$
    – Kajelad
    Commented Aug 14, 2021 at 15:59
  • $\begingroup$ I had no idea that you took of quotient of the manifold with the fundamental group to get a model geometry, where can I read an intro to this? Is this at all related to the Postnikov/whitehead tower, say then for a spin geometry we'd divide by the second homotopy group to get our manifold locally? Or am I way off base here? $\endgroup$
    – R. Rankin
    Commented Aug 21, 2021 at 5:01
  • $\begingroup$ You seem to have things backwards: the manifold $M$ is a quotient of the model geometry $X=G/H$ by a discrete group $\Gamma\subseteq G$ acting freely on $X$. The quotient map $X\to M$ is thus a covering map, and since $X$ is by definition simply connected, we have $\Gamma\cong\pi_1(M)$. This has nothing to do with Postnikov towers or spin geometry; it's just basic covering space theory. $\endgroup$
    – Kajelad
    Commented Aug 21, 2021 at 5:30
  • $\begingroup$ Apologies, I'm new to this, coming from a physics background here. Essentially I'm dealing with a space that's a connected sum of several of the above manifolds. My understanding is that such a space (any oriented 3 manifold) can be represented by a three sphere with a knot (specifically unknot) for each connected sum with Dehn surgery performed on the individual links. I was hoping then to find a model space in terms of the knot complement. (So a three sphere with some extra structure) this is a large area of math for a "newbie" to chew on admittedly $\endgroup$
    – R. Rankin
    Commented Aug 21, 2021 at 6:12

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