Non-embedding results via Spanier Whitehead Duality

Ishan Levy

February 2, 2023

What is the minimal dimension such that \(\CC \PP ^2\) or \(\HH \PP ^2\) can be embedded in Euclidean space? It turns out one gets the optimal answer by answering the corresponding stable problem, namely what is the minimal dimension such that a space with the stable homotopy type of \(\CC \PP ^2\) or \(\HH \PP ^2\) can be embedded in Euclidean space?

We can answer a generalization of this question as follows (due to Hilton and Spanier):

Theorem 0.1. Let \(f\) be a map \(S^{m-1} \to S^n\), and let \(C_f\) denote the cofibre. Suppose that \(f\) cannot be stably desuspended. Then the minimum dimension embedding of \(C_f\) is \(m+n+1\).

In particular, \(\CC \PP ^2\) and \(\HH \PP ^2\) are cofibres of Hopf maps, which cannot stably be desuspended (a stable desuspension is another map of spheres of lower dimension agreeing stably with \(f\)). Note that \(C_f\) can be embedded in \(S^{m+n+1}\) because the mapping cylinder of \(f\) embeds into the join of \(S^{m-1}\) and \(S^n\), which is \(S^{m+n}\), and the cylinder end can be coned off in \(S^{m+n+1}\). So it suffices to show one cannot embed into anything smaller.

First we consider the simplest cases. If \(m<n+1\), then \(f\) is trivial, so, we must have \(n=0\), in which case the assertion is obvious. If \(m=n+1\), then \(n=1\). Then, one can use Alexander duality to observe that were there an embedding into \(S^3\), then the complement would have zero dimensional homology that is not free.

Thus we can assume that \(n>1, m>n+1\).

Lemma 0.2. If \(n>1,m>n+1\), \(f\) can be stably desuspended iff \(C_f\) can be stably desuspended.

Proof. Clearly the stable homotopy type of \(C_f\) depends only on that of \(f\), proving one direction. On the other hand, if \(C_f\) can be stably desuspended to a space \(X\), a homology decomposition of \(X\) will be the cofibre of a map \(g\) between spheres. After suspending enough, these will be of the same dimension, and since the map between the middle skeleton has to extend to a homotopy equivalence between the two spaces, the attaching maps differ by a unit (i.e an integer multiple), so \(f\) can be stably desuspended. □

The essential input of working stably is the following observation: the Spanier-Whitehead dual of the cofibre of \(C_f\) (denoted \(DC_f\)) is \(\Sigma ^{-m-n}C_{\pm f}\), where the sign (unimportant) I think is \((-1)^{mn}\). To see this, the dual of a map \(f\) between spheres is \(\pm f\), suspended to have the right degrees. We have a cofibre sequence, \(f:S^{m-1} \to S^n \to C_f\), which taking duals gives a cofibre sequence \(DC_f \to S^{-n} \to S^{1-m}\). Rearranging this shows that \(DC_f = \Sigma ^{-m-n} C_{\pm f}\). Now the complement of \(C_f\) inside \(S^{m+n}\) would be \(\Sigma ^{m+n-1}DC_f = \Sigma ^{-1} C_f\)! This completes the proof via the lemma.

The same argument gives the slightly stronger version:

Theorem 0.3. Let \(f\) be a map \(S^{m-1} \to S^n\), and let \(C_f\) denote the cofibre. Let \(k\) be the maximum number of times that \(f\) can be stably desuspended. Then the minimum dimension embedding of \(C_f\) is \(m+n+1-k\).