## If there are nontrivial maps $H\mathbb{Z}^{top}\to MU$ then there are nontrivial maps $H\mathbb{Z}\to MGL$?

Let $$H\mathbb{Z}^{top}$$ be the Eilenberg-MacLane spectrum and $$MU$$ be the complex cobordism spectrum.

Consider now the motivic counterparts of the above spectral, namely $$MGL$$ (the motivic cobordism) and $$H\mathbb{Z}$$ (the motivic Eilenberg Mac-Lane spectra).

There is an adjunction $$(Re_{B},c):SH(\mathbb{C})\to SH$$, where $$Re_{B}$$ is the Betti realization functor and $$c$$ is the functor induced by sending a space to the constant presheaf of spaces on smooth varieties over $$\mathbb{C}$$. It is known that the functor $$c$$ is fully faithful.

My question is the following:

If there are nontrivial maps $$H\mathbb{Z}^{top}\to MU$$, can I deduce that there are nontrivial maps $$H\mathbb{Z}\to MGL$$?

## Are there non trivial maps from $H\mathbb{Z}$ to $MGL$?

Let $$k$$ be a field of characteristic $$0$$. Let us denote by $$\mathbf{1}_{k}$$ the sphere spectrum. Let $$MGL$$ be the algebraic cobordism spectrum.

We have the following diagram

$$H\mathbb{Z}\leftarrow \mathbf{1}_{k}\rightarrow MGL$$

My question is the following:

Are there non trivial maps $$H\mathbb{Z}\to MGL$$ such that the above triangle commutes?