How to think about $\mathbf{Z}(n)_{\mathcal{M}}$

Question

One definition of motivic cohomology for smooth schemes $X$ over a field, is via Friedlander-Suslin complexes.

A refresher (you may skip to the question at the bottom)

One defines

(1) $z_n(X,d) :=$ free abelian group generated by all reduced, irreducible closed $k$-subschemes $W\subset X\times(\mathbf{P}^1_k)^d\times\Delta_k^n$ that, with the projection onto $X$ finite surjective. This is a simplicial abelian group, functorial on smooth $k$-schemes with respect to arbitrary morphisms.

(2) $z_n^{\infty}(X,d) :=$ the sum of $(i_{\infty, j})_*z_n(X,d-1)$ for $j=1,\ldots, d$, where $i_{\infty,j} : (\mathbf{P}^1_k)^{d-1}\to(\mathbf{P}^1_k)^d$ inserts $\infty$ at the $j$-th spot.

(3) $\mathbf{Z}(d)_{X,\mathcal{M}} := z_{\bullet}(X,d)/z_{\bullet}^{\infty}(X,d)$.

It turns out the Zariski, resp. étale hypercohomology of the complex $\mathbf{Z}(d)_{X,\mathcal{M}}$ agrees with $H^*_M(X,\mathbf{Z}(d))$, resp. $H^*_L(X,\mathbf{Z}(d))$.

The construction $\mathbf{Z}(d)_{X,\mathcal{M}}$ sort of consists of singular cohomology "twisted" by higher spheres $S^{2d}$, where one uses the "smash product" $(\mathbf{P}^1_k)^{\wedge d}$ as a replacement for $S^{2d}$. If this smash product existed, the Suslin complex of $X\times (\mathbf{P}^1_k)^d$ would be $\mathbf{Z}(d)_{X,\mathcal{M}}$.

QUESTION What is the precise topological analog of the Friedlander-Suslin complex $\mathbf{Z}(d)_{X,\mathcal{M}}$? That is, if $X$ is a topological manifold, then $\mathbf{Z}(0)_{X,\mathcal{M}}$ is the usual singular cochain complex. What should $\mathbf{Z}(d)_{X,\mathcal{M}}$ be? Some singular cohomology of a pair? (if so, what pair?)

EDIT: followup question here

Answer 1

[All cohomology will be reduced cohomology for ease of notation].

There is no analog for classical homotopy theory. This is related to the fact that the Picard group of the category of spectra is $\mathbb{Z}$ (so the only twists are shifts in degree).

But not all is lost.

Let us enter the more exotic, but still quite familiar, world of $C_2$-equivariant homotopy theory. Its objects are spaces equipped with a $C_2$-action, morphisms are $C_2$-equivariant morphisms and homotopies are $C_2$-equivariant homotopies. In particular, if we have a homotopy equivalence it induces an equivalence both on the underlying space and on the fixed points. The analog of $\mathbb{Z}(0)$ here is given by the cohomology for the constant Mackey functor $\underline{\mathbb{Z}}$ which can, if the action is nice [think $C_2$-manifolds], be defined as $$H^*(X;\underline{\mathbb{Z}}):=H^*(X/C_2;\mathbb{Z})\,.$$ (note for example that if $X$ has trivial $C_2$-action, this is just ordinary cohomology). In the equivariant world we are equipped with the classical shifts $$H^*(X\wedge S^n;\underline{\mathbb{Z}})\cong H^{*-n}(X;\underline{\mathbb{Z}})\,,$$ but no one prevents us to use more exotic shifts. Let $S^\rho$ the 1-point compactification of the regular representation of $C_2$ (and $S^{n\rho}=(S^\rho)^{\wedge n}$). Then we can define $$H^*(X;\underline{\mathbb{Z}}(n)):=H^*(X\wedge S^{n\rho};\underline{\mathbb{Z}})\,.$$ (This can in fact be defined also for negative $n$, but I don't want to enter into the details here).

The analogy is more taut than one could at first think. In fact, there is a "realization" functor from smooth schemes over $\mathbb{R}$ to $C_2$-equivariant spaces, sending $X$ to $X(\mathbb{C})$ with the conjugation action. This sends $\mathbb{P}^1$ to $S^\rho$, and you obtain a map $$H^*_{mot}(X;\mathbb{Z}(n))\to H^*(X(\mathbb{C})\wedge S^{n\rho};\underline{\mathbb{Z}})$$ In fact, $H\underline{\mathbb{Z}}$ is the initial $C_2$-cohomology theory receiving such a map.

References

For the definition and the basic properties of $H^*(-;\underline{\mathbb{Z}})$ I like section 3 (Mackey functors, homology and homotopy) of

Hill, M.A.; Hopkins, M.J.; Ravenel, D.C., On the nonexistence of elements of Kervaire invariant one, Ann. Math. (2) 184, No. 1, 1-262 (2016). ZBL1366.55007.

For the connection with real realization, real realization is defined (although it is not the most modern or elegant presentation) in section 3.3.3 of

Morel, Fabien; Voevodsky, Vladimir, $\mathbb{A}^1$-homotopy theory of schemes, Publ. Math., Inst. Hautes Étud. Sci. 90, 45-143 (1999). ZBL0983.14007.

Finally, the fact that the real realization sends the motivic cohomology spectrum to $H\underline{\mathbb{Z}}$, is theorem 4.17 of (thanks to Drew Heard for this one!)

Heller, J.; Ormsby, K., Galois equivariance and stable motivic homotopy theory, Trans. Am. Math. Soc. 368, No. 11, 8047-8077 (2016). ZBL1346.14049.

Moreover theorem 4.18 gives you a range in which the natural map $$H^p(X;\mathbb{Z}/n(q))\to H^p(X(\mathbb{C})_+\wedge S^{q\rho};\underline{\mathbb{Z}/n})$$ is an isomorphism (beware that their indexing conventions for cohomology differ from the ones in this answer)