Notations for Whitehead tower in Anderson duality

Question

In appendix B of Hopkins and Singer's paper, Lemma B.15., the authors claimed that we can deduce the following isomorphisms $$ [X, \Sigma^n \tilde{I}]\rightarrow [X \langle n-1, \infty\rangle , \Sigma^n \tilde{I}]\leftarrow [X \langle n-1, n\rangle , \Sigma^n \tilde{I}]$$ $$[X \langle n-1, n\rangle , \Sigma^n \tilde{I}\langle n-1, \infty\rangle]\rightarrow[X \langle n-1, 0\rangle , \Sigma^n \tilde{I}]$$ from the short exact sequence: $$0\rightarrow \mathrm{Ext}(\pi_{n-1}X, \mathbb{Z})\rightarrow\tilde{I}(X)\rightarrow\mathrm{Hom}(\pi_{n}X, \mathbb{Z})\rightarrow 0,$$ where $X$ and $\tilde{I}$ are two spectra. (Actually $\tilde{I}$ is the Anderson dual of the sphere defined in appendix B.2., but I think it is suffice to use this short exact sequence instead of the definition for this lemma.)

Then they also claimed that $$\Sigma^n \tilde{I}\langle n-1, ...,\infty\rangle\approx \Sigma^nH\mathbb{Z}.$$

What confuses me is the notation $\langle ...\rangle$. I believe this relates to the Whitehead tower of a space $X$, where $X\langle n\rangle$ is $n$-connected and has other homotopy groups same as $X$. I would like to know what does this notation mean if there are multiple numbers in the bracket and how to get these two claims from the definitions.

Thank you in advance for your comments and answers!

Edit: On page 75 of the paper, there is a footnote saying that $X\langle n,...,m\rangle$ indicates the Postnikov section of $X$ having homotopy groups only in dimension $n,...,m$. Then the first isomorphism of the lemma is straightforward from the splitting of the exact sequence. But it seems to me there is a typo in the second isomorphism, maybe it should be $$[X \langle n-1, n\rangle , \Sigma^n \tilde{I}\langle n-1, \infty\rangle]\rightarrow[X \langle n-1, n\rangle , \Sigma^n \tilde{I}].$$

Answer 1

You've probably figured this out by now, but you're right that there's a typo in the second isomorphism. For example, if $n-1 > 0$, then $X\langle n-1,0\rangle = 0$, and the statements on that page would imply that $H^n(X\langle n-1,n\rangle;\mathbb{Z}) = 0$ for any $X$, which isn't true.

On the other hand, it is true that $[X\langle n-1,n\rangle,\Sigma^n\widetilde{I}\langle n-1,\infty\rangle] = [X\langle n-1,n\rangle,\Sigma^n\widetilde{I}]$. In fact you always have $[X\langle m,\infty\rangle,Y] = [X\langle m,\infty\rangle,Y\langle m,\infty\rangle]$ for spectra $X$ and $Y$.

For $\Sigma^n \widetilde{I}\langle n-1,\ldots,\infty\rangle\simeq \Sigma^n H\mathbb{Z}$, it suffices to check the following three conditions:

1. $\pi_0 \widetilde{I} = \mathbb{Z}$.
2. $\pi_{-1} \widetilde{I} = 0$.
3. $\pi_k \widetilde{I} = 0$ for $k > 0$.

These can be read off the long exact sequence $$ \cdots\rightarrow \pi_{k+1}H\mathbb{Q}\rightarrow \text{Hom}(\pi_{-(k+1)}S,\mathbb{Q}/\mathbb{Z})\rightarrow \pi_k\widetilde{I}\rightarrow \pi_k H\mathbb{Q}\rightarrow \text{Hom}(\pi_{-k}S,\mathbb{Q}/\mathbb{Z})\rightarrow\cdots $$ coming from the definition of $\widetilde{I}$.