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 $\mathbf{Z}(d)_{X,\mathcal{M}}$.

QUESTIONWhat 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**