# The $K$-theory homology of the Eilenberg-MacLane spectrum

Let $$KU$$ be the complex $$K$$-theory spectrum and $$H\mathbb{Z}$$ be the Eilenberg-MacLane spectrum.

For $$n\in \mathbb{Z}$$, it is known what the homology groups $$KU_{n}(H\mathbb{Z})$$ are?

Since for any two spectra $$E,F$$ we have $$E_n(F)=[\mathbb{S}^n,E\wedge F]\cong [\mathbb{S}^n,F\wedge E] = F_n(E),$$ you may as well ask about $$H_n(KU;\mathbb{Z})$$, the integral homology of the complex $$K$$-theory spectrum.

This calculation is carried out e.g. in Chapter 16 of the book

Switzer, Robert M., Algebraic topology – homology and homotopy., Classics in Mathematics. Berlin: Springer. xii, 526 p. (2002). ZBL1003.55002.

In particular, Theorem 16.25 gives the Pontrjagin algebra structure as $$H_*(KU;\mathbb{Z})\cong \mathbb{Q}[u,u^{-1}],$$ the ring of finite Laurent series over $$\mathbb{Q}$$ in an element $$u\in H_2(KU;\mathbb{Z})$$.

We have $$KU_*(H\mathbb Z) = \pi_*(KU\wedge H\mathbb Z)$$; this ring is concentrated in even dimensions and carries an isomorphism between the additive and multiplicative formal group law, hence is rational. Thus the map $$KU\wedge H\mathbb Z\to (KU\wedge H\mathbb Z)\wedge H\mathbb Q\cong KU\wedge H\mathbb Q$$ is an isomorphism. The Chern character $$KU\to \sum_{i\in\mathbb Z} \Sigma^{2i}H\mathbb Q$$ is a rational equivalence, so that $$KU_{2n}(H\mathbb Z) \cong\mathbb Q, KU_{2n+1}(H\mathbb Z) \cong 0\ .$$

Alternatively, the Snaith theorem $$KU\cong \Sigma^{\infty}\mathbb{CP}^{\infty}[\beta^{-1}]$$, where $$\beta\in \pi_2(\mathbb{CP}^{\infty})$$ is the Bott element, shows that $$H_*(KU)\cong H_*(\mathbb{CP}^{\infty})[\beta^{-1}]\ .$$ It's easy to calculate that $$H_*(\mathbb{CP}^{\infty})$$ is a divided power algebra on the generator $$\beta\in H_2(\mathbb{CP}^{\infty})$$. Since $$\beta^n$$ is divisible by $$n!$$, this ring is the Laurent series over $$\mathbb Q$$ generated by $$\beta$$.