## 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 $$\alpha\in MGL^{0,0}(\mathbb{1})$$?

## Beilinson and Deligne’s Motivic Polylogarithm and Zagier Conjecture

Where can I find the preprint Motivic Polylogarithm and Zagier Conjecture by Beilinson and Deligne? I see it referenced in a lot of papers but no one seems to host a copy.