Chromatic Homotopy and Telescopic Localization

4 Telescopic Localization and the Bousfield Kuhn functor

Now we will look the telescopic localization functors more in depth and see their relation to unstable homotopy theory. We will set \(n\geq 1\), and let \(V_n\) denote a type n

space with a \(v_n\)-self map \(v_n:\Sigma ^dV_n\to V_n\).

Definition 4.1. For a space/spectrum \(X\), the \(v_n\)-periodic homotopy groups with coefficients in \(V_n\), denoted \(v_n^{-1}\pi _*(X;V_n)\) are defined as \(v_n^{-1}\pi _*(\Map _*(V_n,X))\).

It isn’t hard to see that \(v_n^{-1}\pi _*(X;V_n)\) is the homotopy groups of a \(d\)-periodic spectrum called \(\Phi _{V_n}(X)\), given by the formula \(\colim _k \Sigma ^{\infty -kd}\Map _*(V_n,X)\) where the colimit is uses the map \(\Map _*(V_n,X) \to \Map _*(\Sigma ^dV_n,X)=\Omega ^{d}\Map _*(V_n,X)\) induced by \(v_n\).

Note that if \(X\) is a spectrum, then \(\Phi _{V_n}(X)\) is the spectrum \(X\otimes DV[v_n^{-1}] = X\otimes T(n)\).

Definition 4.2. If \(f:X \to Y\) is a map of spaces or spectra, \(f\) is a \(v_n\)-periodic equivalence if \(\Phi _{V_n}X \to \Phi _{V_n}Y\) is an equivalence.

Essentially by the thick subcategory theorem, the notion of \(v_n\)-periodic equivalence only depends on \(n\).

Lemma 4.3. Let \(n\geq 1\), and \(X\) be a spectrum. Then
- 1. \(\Phi _{V_n}X = \Phi _{V_n}\Omega ^\infty X\).
- 2. The map \(\tau _{\geq k}X \to X\) is a \(v_n\)-periodic equivalence.
- 3. The map \(X \to X^{\wedge }_p\) is a \(v_n\)-periodic equivalence.

Proof. \((1)\): This follows from the adjunction between \(\Sigma ^\infty \) and \(\Omega ^\infty \).

\((2)\): The fibre is bounded above, and \(v_n^{-1}\pi _*(Y;V_n) = 0\) whenever \(Y\) is bounded above because \(|v_n|>0\), and the homotopy groups of the mapping space are bounded above.

\((3)\): The fibre \(F \to X \to X^{\wedge }_p\) is \(\SP /p\)-acyclic. This means that it is killed by tensoring with \(\SP /p^m\) for all \(m\). But for any type \(n\) spectrum, some power of \(p\) acts by \(0\), so the same is true for \(T(n)\). Thus \(F\otimes \SP /p^k\otimes T(n) = F\otimes (T(n)\oplus \Sigma T(n)) = 0\) for \(k\gg 0\), so \(F\) is \(T(n)\)-acyclic. □

The above lemma, along with the fact that \(\Phi _{V_n}X= X\otimes T(n)\) implies that for \(n\geq 1\), \(L_{T(n)}X\) only depends on \(\Omega ^\infty X\) as an \(\EE _{\infty }\)-space. A wonderful insight of Bousfield and Kuhn is that in fact it only depends on \(\Omega ^\infty X\) as a space!

To see this, we start by thinking about the construction taking a pair of a type \(n\) space and \(v_n\)-self map \((V_n,v_n)\) to the functor \(\Phi _{V_n}:S_* \to \Sp \). If we replace \(v_n\) by an iterate, it is easy to see that it doesn’t change \(\Phi _{V_n}\), so since \(v_n\)-self maps are unique, the data of \(v_n\) is not important in the construciton of \(\Phi _{V_n}\).

Secondly, if we replace \(V_n\) by \(\Sigma V_n\), \(\Phi _{V_n}\) changes to \(\Phi _{\Sigma V_n} = \Sigma ^{-1}\Phi _{V_n}\). Thus \(\Phi _{V_n}\) only depends on the spectrum \(\Sigma ^\infty V_n\).

These observations can be souped up to construct a functor

\[ \Sp _{\geq n} \to \Fun (S_*,\Sp )\]

that sends a type \(n\) spectrum \(V\) to \(\Phi _V\).

Definition 4.4. The Bousfield-Kuhn functor \(\Phi \) is a functor \(S_* \to \Sp \) given by \(\Phi := \lim _{V \to \SP }\Phi _V\).

Another way to describe it is that you right Kan extend the functor \(\Sp _{\geq n} \to \Fun (S_*,\Sp )\) along the inclusion to \(\Sp \), and evaluate on \(\SP \). An important property of \(\Phi \) is that it realizes the factorization of \(L_{T(n)}\) through \(\Omega ^\infty \) as a space:

Proposition 4.5. \(\Phi \Omega ^{\infty }X = L_{T(n)}X\).

Proof. We have from the definition and our previous observations \(\Phi \Omega ^{\infty }X = \lim _{V \to \SP }\Phi _V X = \lim _{V \to \Sp } DV[v_n^{-1}]\otimes X\).

Each term in the limit is \(T(n)\)-local, so it agrees with \(L_{T(n)}(DV[v^{-1}]\otimes X) = L_{T(n)}(DV\otimes X) = L_{T(n)}(X^V)\).

Putting this together, we have

\[\Phi \Omega ^{\infty }X = \lim _{V \to \SP }L_{T(n)}X^V\]

Now I claim that \(T(n)\)-locally \(\SP \) is a filtered colimit of type \(n\) spectra. This claim completes the proof, because it identifies \(\lim _{V \to \SP }L_{T(n)}X^V\) with \(L_{T(n)}X^\SP = L_{T(n)}X\).

To see the claim we recall that we had a cofibre sequence

\[\Sigma ^{-n}\SP /p^\infty ,\dots ,v_{n-1}^\infty \to \SP \to L_{n-1}^f\SP \]

\(L_{n-1}^f\SP \) is \(T(n)\)-acyclic since \(T(n)\) is a filtered colimit of type \(n\) spectra, which \(L_{n-1}^f\) kills. Thus applying \(L_{T(n)}\) to the cofibre sequence above, we get a formula for \(L_{T(n)}\SP \) as a filtered colimit of (\(T(n)\)-localizations of) type \(n\) spectra. □

Some other important facts about the Bousfield-Kuhn functor are:

• It inverts \(v_n\)-periodic equivalences, and takes values in \(T(n)\)-local spectra. This is indicated by the factorization below, where \(S_*^{v_n}\) is the localization of pointed spaces at the \(v_n\)-periodic equivalences. The factored map is also denoted \(\Phi \).
• \(\Phi :S_*^{v_n} \to \Sp _{T(n)}\) preserves limits.

A consequence of the factorization in the proposition above is:

Corollary 4.6. Let \(f:X \to Y \in \Sp \) be a map. If \(\Sigma ^\infty \Omega ^\infty f\) is a \(T(n)\)-equivalence, so is \(f\).

Proof. By assumption, \(L_{T(n)}\Sigma ^{\infty }\Omega ^\infty f\) is an equivalence. But this is equal to \(\Phi \Omega ^\infty \Sigma ^\infty \Omega ^\infty f\), and by the triangle identity for the adjunction between \(\Sigma ^\infty \) and \(\Omega ^\infty \), the map \(\Phi \Omega ^\infty f\) is a retract of \(\Phi \Omega ^\infty \Sigma ^\infty \Omega ^\infty f\). Thus \(\Phi \Omega ^{\infty }f = L_{T(n)}f\) is also an equivalence. □

The functor \(\Sigma ^\infty \Omega ^\infty \) doesn’t preserve \(T(n)\)-local equivalences in general.

Example 4.7. \(H\ZZ \) is \(T(n)\)-acyclic, but \(\Sigma ^\infty \Omega ^\infty H\ZZ \) is a sum of spheres, so is not.

Nevertheless, for sufficiently connected maps, \(\Sigma ^\infty \Omega ^\infty \) does preserve \(T(n)\)-local equivalences. Here is a version of that statement for the finite localizations.

Proposition 4.8. Let \(n\geq 1\). There is an \(m\geq 2\) such that:
- 1. If F is an \(m\)-connected pointed space such that \(v_i^{-1}\pi _*(F; V_i) = 0\) for \(0 \leq i \leq n\), then \(F\) is \(L_n^f\)-acyclic.
- 2. If \(f:X \to Y\) is an \(m\)-connected map that is a \(v_i\)-periodic equivalence for \(0 \leq i \leq n\), then \(\Sigma ^\infty f\) is an \(L_n^f\) equivalence.
- 3. \(\Sigma ^\infty \Omega ^\infty \) preserves \(m\)-connected \(L_n^f\) equivalences.

Proof. \((1)\): Omitted. This relies on results of Bousfield on unstable localization.

\((2)\): The fibre \(F\) satisfies the hypotheses of \((1)\). Then \(f\) can be identified with \(\colim _YF \to \colim _Y*\), which is a \(T(n)\) equivalence since \(F\) is \(L_n^f\)-acyclic.

\((3)\): Apply \(\Omega ^\infty \) and \((2)\).

□

Remark 4.9. In fact in the above proposition, \(m\) can be taken to be \(n+1\). This is a consequence of ambidexterity of the \(T(n)\)-local category, which was proven by Carmeli, Schlank, and Yanovski.