In topology the spheres $S^n$ are the "simplest" closed manifolds, and they are like "Dirac's delta at $n$" for (reduced) cohomology groups. Furthermore they are boundaries of the simplest compact manifolds-with-boundary, i.e. the disks $D^{n+1}$, which are contractible. And $S^{n}$ is obtained by glueing two copies of $D^{n}$ along their boundary $S^{n-1}$.
My question is:



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> Are there some objects of algebraic geometric nature that somehow reproduce the same pattern, or that are considerable as the equivalent of spheres from topology?
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> More generally, are there "homology spheres" for some homology theory like -say- Chow groups? What about an "algebraic Poincaré conjecture"?
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If they do exist, I don't expect them to be standard varieties or schemes, otherwise they probably would have made their appearence "classically".



What is the algebraic geometry version of the spheres?
To expand on Tom Goodwillie's answer:


a precise definition of "motivic sphere" would be
$$S^{p,q} = \big( \Delta^{p-q} / \partial \Delta^{p-q} \big) \wedge \big( \bigwedge^q \mathbb{G}\_m\big)$$
which you can interpret as the $q$-fold smash product of the multiplicative group $\mathbb{G}\_m$ smashed with a $(p-q)$-dimensional simplicial sphere. This smash product makes sense in a category of simplicial presheaves on smooth schemes (which is what you work with in A¹-homotopy theory).


The indices can be explained from looking at the motive $M(S^{p,q}) = \mathbb{Z}(q)[p]$ or at the realizations, where the complex realization gives $S^p$ and the real realization gives $S^{p-q}$. More about this can be read in the Morel-Voevodsky paper.


Now these are not algebraic varieties, by definition, so a good question is



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> For integers $p$ and $q$, does an algebraic variety $X$ exist which is A¹-weakly equivalent to $S^{p,q}$?
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We'll also say "$X$ is a $S^{p,q}$".
From looking at realizations you can already exclude $p,q$ negative.
One positive example is (by definition) $\mathbb{G}\_m$, which is a $S^{1,1}$.


As Tom Goodwillie explained, $\mathbb{A}^n / (\mathbb{A}^n \setminus 0)$ is a $S^{2n,n}$ (which some people shorten to "motivic $2n$-sphere") and $\mathbb{A}^n \setminus 0$ is a $S^{2n-1,n}$.


There is not much known beyond that, I suppose.




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There are even more varieties that could count as algebraic versions of spheres. An interesting feature of algebraic geometry is that you can look at many fields of definition. Projective quadrics over the complex numbers all look the same (in each dimension), as they are isomorphic (given the same dimension) to the split quadrics $Q\_{2n}^{split} = \{\sum\_{i=0}^n x\_iy\_i = 0\}$ or $Q\_{2n+1}^{split} = \{\sum\_{i=0}^n x\_iy\_i = z^2\}$ as well as to the anisotropic quadrics $Q\_{m} = \{\sum\_{i=0}^m z\_i^2\}$ (by change of basis). Over smaller, non-quadratically closed fields, these quadrics are no longer isomorphic.


Now look at the affine quadrics inside the projective quadrics (by removing a suitable hyperplane section). Conjecturally, the split ones are motivic spheres (at least they have the right motive), while the anisotropic ones aren't. You can consider these forms of motivic spheres.


