Can we reconstruct positive weight invariants in algebraic topology using algebraic geometry?

Question

I can't really say that I understand what a weight is, but the qualitative distinction between weight zero and positive weight has come up a couple times in MathOverflow questions:

1. The étale fundamental group of a pointed connected complex scheme has a canonical map to the profinite completion of the topological fundamental group, and for regular varieties, this seems to be an isomorphism. However, in the case of a nodal rational curve (see this question), one finds that the étale fundamental group is not profinite, and has an honest isomorphism with the topological fundamental group. Similarly, the degree 1 étale cohoomology of the nodal curve with coefficients in $\mathbb{Z}$ is just $\mathbb{Z}$ as expected from topology, where one typically expects étale cohomology with torsion-free coefficients to break badly in positive degree. Emerton explained in this blog comment that the good behavior of étale cohomology and the étale fundamental group in these cases is due to the fact that the contribution resides in motivic weight zero, and the singularity is responsible for promoting it to cohomological degree 1.
2. Peter McNamara asked this question about how well formal loops detect topological loops, and Bhargav suggested in a rather fantastic answer that the formal loop functor only detects weight zero loops (arising from removing a divisor). In particular, he pointed out that maps from $\operatorname{Spec}\mathbb{C}((t))$ only detect the part of the fundamental group of a smooth complex curve of positive genus that comes from the missing points.

I have a pre-question, namely, how does one tell the weight of a geometric structure, such as a contribution to cohomology, or the fact that removing a divisor yields a weight zero loop?

My main question is: Are there algebraic (e.g., not using the complex topology) tools that always yield the correct invariants in positive weight, such as cohomology with coefficients in $\mathbb{Z}$ and the fundamental group of a pointed connected complex scheme?

I've heard a claim that motivic cohomology has a Betti realization that yields the right cohomology, but I don't know enough about that to understand how. Any hints/references?

With regard to the second example above, I've seen some other types of loops in algebraic geometry, but I don't really know enough to assess them well. First, there are derived loops, which you get by generalizing to Top-valued functors on schemes, defining $S^1$ to be the sheaf associated to the constant circle-valued functor (in some derived-étale topology), and considering the topological space $X(S^1)$ or the output of a Hom functor. As far as I can tell, derived loops are only good at detecting infinitesimal things (e.g., for $E$ an elliptic curve, $LE$ is just $E \times \operatorname{Spec}\operatorname{Sym}\mathbb{C}[-1]$, which has the same complex points as $E$). Second, there is also some kind of formal desuspension operation in stable motivic homotopy that I don't understand at all. One kind of loop has something to do with gluing 0 to 1 in the affine line, and the other involves the line minus a point. I'm having some trouble seeing a good fundamental group come out of either of these constructions, but perhaps there is some miracle that pops out of all of the localizing.

Answer 1

I don't think I understand the question, but for the pre-question, I recommend Deligne's ICM address on weights. The point is that the cohomology of algebraic varieties "knows its place." If the variety is smooth and complete, the weight matches (by definition) the degree of the cohomology. By resolution of singularities, it is possible to arrange for all the cohomology to come from smooth complete varieties. But this process of relating them goes through long exact sequences / cones which can shift the degree that the vector spaces end up in.

For example, if X is a smooth complete curve, then its 2nd cohomology is weight 2 and is called Z(1). If you delete a point, its top cohomology is now zero. You have "subtracted" Z(1) from degree 2. If you delete another point, the same thing should happen, but you can't subtract again, and the proper generalization is coning, resulting in Z(1) appearing in degree 1. But Z(1) is still weight 2, so it is now out of place in cohomological degree 1. Dually, if you glue two points together, you should kill off the difference of their homology classes. But they were already homologous, so instead you create a Z in degree 1. This is weight 0 sitting in degree 1. Singularities raise the degree; holes lower the degree.