A question about the (motivic) integral cohomology of the Eilenberg-MacLane spectrum

Let $ H\mathbb{Z}$ be the Eilenberg-MacLane spectrum. Let $ n\geq 0$ be any integer.

Is it known the structure of the group $ [H\mathbb{Z},\Sigma^{n}H\mathbb{Z}]$ ?

Is there any reference in this direction?

What about of its motivic version, i.e. if $ M\mathbb{Z}$ denotes the motivic Eilenberg-MacLane spectrum,

What $ [M\mathbb{Z},\Sigma^{2n,n}M\mathbb{Z}]$ is?

I would appreciate any help.

Motivic Knot Embedding

I’ve been trying to be more diligent about reading motivic homotopy theory, and have been reading Levine’s ‘An Overview of Motivic Homotopy Theory.’ I think the subject is fascinating, and I’ve developed a question during my reading.

Levine defines $ \mathcal{X}:\mathbf{Sm}/S^{\text{op}}\rightarrow\mathbf{Spc}$ , where the source is the category of smooth, separated $ S$ -schemes of finite type.

There are three suspension functors, $ $ \Sigma_{S^1}\mathcal{X}\simeq\mathcal{X}\wedge S^1\quad\Sigma_{\mathbb{G}_m}\mathcal{X}\simeq\mathcal{X}\wedge\mathbb{G}_m\quad\Sigma_{\mathbb{P^1}}\mathcal{X}\simeq\mathcal{X}\wedge\mathbb{P}^1 $ $

Assume two smooth embeddings (“motivic knots”) $ f,f^*$ $ $ S^{a+b,b}\xrightarrow{f\>\wedge\text{ id}}\Sigma_{S^1}^2S^{a+b,b} \quad S^{a+b,b}\xrightarrow{\text{id}\>\wedge\>f^*}\Sigma_{\mathbb{G}_m}^2S^{a+b,b} $ $ but Levine states that $ \mathbb{P}^1$ is the preferred object to stabilize the category of motivic spectra about, which suggests to me as a ‘more natural’ setting for the preferred motivic knot $ $ S^{a+b,b}\xrightarrow{f\wedge f^*}\Sigma_{\mathbb{P}^1}^2S^{a+b,b} $ $

Question: Is there any research on a “motivic knots” (in any broad sense encompassing interesting behavior around $ f,f^*$ or some combination)?

A question about the unit map of the motivic cobordism spectrum

Consider the motivic sphere spectrum $ \mathbb{1}$ and the motivic cobordism spectrum $ MGL$ .

Let $ 1_{MGL}:\mathbb{1}\to MGL\in MGL^{0,0}(\mathbb{1})$ be the unit for $ MGL$ . We have, by a combination of results of Levine (that use the Hoyois-Morel-Hopkins spectral sequence) and Quillen that $ MGL^{0,0}(\mathbb{1})\cong MU^{0}(*)\cong \mathbb{Z}$ . In this form, there are maps other than $ 1_{MGL}$ . My question is:

What relation exist, if any, between $ 1_{MGL}$ and $ \alpha\in MGL^{0,0}(\mathbb{1})$ ?