03.12.E4 · modern-geometry / homotopy

Localization and completion exercise pack (May-Ponto supplement)

shippedIntermediate-onlyLean: nonepending prereqs

Anchor (Master):

Formal definition of the pack Intermediate

May-Ponto's first part takes the homotopy theory of a nilpotent space — a space whose fundamental group is nilpotent and acts nilpotently on the higher homotopy groups — and shows it can be reconstructed from its localizations and completions at the various primes together with its rationalization. Localizing a space $X$ at a set of primes $T$ produces $X_{T}$ whose homotopy and homology groups are the $T$ -local versions of those of $X$ ; completing at a prime $p$ produces $X_{p}^{\land}$ . The book's keystone is the arithmetic (fracture) square, which glues these local pictures back into the integral space as a homotopy pullback.

This pack collects nine exercises drawn from Chapters 5-13 — two easy, four medium, three hard — each with a hint and a full solution. The exercises are grouped: localization of homotopy and homology groups (easy), the universal property of localization and rationalization of spheres (medium), and the fracture square together with the nilpotency hypotheses that make the whole theory work (hard).

The conventions are May-Ponto's: for a set of primes $T$ , $Z_{T}$ is the subring of $Q$ with denominators prime to $T$ ; an abelian group $A$ is $T$ -local if multiplication by each $ℓ \in / T$ is an isomorphism; $X_{T}$ denotes the $T$ -localization and $X_{(p)}$ the localization at the single prime $p$ ; $X_{Q} = X_{0}$ is the rationalization. The completion at $p$ is written $X_{p}^{\land}$ .

Key theorem with full solution Intermediate

Before the pack proper, we work one exercise in full as the model solution. The remaining eight follow the same structure (problem, hint, full answer in <details> blocks).

Lead exercise. State and prove the fracture (arithmetic) square for a simply connected space of finite type.

Solution. Let $X$ be a simply connected space whose homotopy groups are finitely generated. The arithmetic square is the diagram

X ↓ X_{Q} ⟶ ⟶ \prod_{p} X_{p}^{\land} ↓ (\prod_{p} X_{p}^{\land})_{Q}

where the top map collects the completions at all primes, the left map is rationalization, and the bottom map rationalizes the product of completions. The fracture theorem asserts this square is a homotopy pullback.

Proof sketch. On homotopy groups the square realizes the algebraic Hasse square for a finitely generated abelian group $A = π_{n} (X)$ :

A ↓ A \otimes Q \to \to \prod_{p} A_{p}^{\land} = \prod_{p} Z_{p} \otimes A ↓ (\prod_{p} A_{p}^{\land}) \otimes Q = A_{f} \otimes A

This is a pullback of abelian groups: an element of $A$ is determined by its image in $A \otimes Q$ and in each $A_{p}^{\land}$ , compatibly in the finite adeles $A_{f} \otimes A$ . For $A = Z$ this is the classical statement $Z = Q \times_{A_{f}} \prod_{p} Z_{p}$ . Because the square is a pullback on each homotopy group, and a homotopy pullback of nilpotent (here simply connected) spaces is detected on homotopy groups, the space-level square is a homotopy pullback. $□$

This is the master gluing theorem: the integral homotopy type of $X$ is recovered from its rational type and its $p$ -complete types, fitting together over the rationalized adeles. Everything downstream — comparing $X$ to $X_{(p)}$ , building exotic integral spaces from compatible local data — is an application.

Exercises Intermediate

Exercise 1 (easy, numeric). Localization of abelian groups.

For the set of primes $T = {p}$ , compute the $T$ -localizations $Z_{(p)}$ , $(Z / q)_{(p)}$ for a prime $q \neq = p$ , and $(Z / p^{k})_{(p)}$ .

Hint

$A_{(p)} = A \otimes Z_{(p)}$ . Tensoring with $Z_{(p)}$ inverts every prime except $p$ .

Answer

$T$ -localization at $T = {p}$ is $A \mapsto A \otimes Z_{(p)}$ , where $Z_{(p)}$ has all primes except $p$ inverted.

$Z_{(p)} \otimes Z_{(p)} = Z_{(p)}$ (localization is idempotent).
$(Z / q) \otimes Z_{(p)} = 0$ for $q \neq = p$ : since $q$ is invertible in $Z_{(p)}$ , multiplication by $q$ is an isomorphism on $Z / q$ , but it is also the zero map there, forcing the group to be $0$ .
$(Z / p^{k}) \otimes Z_{(p)} = Z / p^{k}$ : $p$ is not inverted, and every other prime acts invertibly already on a $p$ -group, so the group is unchanged.

So localizing at $p$ keeps the $p$ -primary torsion and the $p$ -local part of free groups, and kills all torsion prime to $p$ . This is the algebraic heart of space-level $p$ -localization. May-Ponto Ch. 4-6.

Exercise 2 (easy, proof). Localization commutes with homology.

State the theorem that for a nilpotent space $X$ and set of primes $T$ , the localization map $X \to X_{T}$ induces $H_{*} (X) \otimes Z_{T} ≅ H_{*} (X_{T})$ , and explain why this characterizes $X_{T}$ .

Hint

A $T$ -localization is characterized by being $T$ -local with the universal map from $X$ ; the homology criterion is the working definition.

Answer

Theorem (May-Ponto Ch. 6). For a nilpotent space $X$ , the $T$ -localization map $ϕ : X \to X_{T}$ induces isomorphisms

π_{*} (X) \otimes Z_{T} ≅ π_{*} (X_{T}), H_{*} (X; Z) \otimes Z_{T} ≅ H_{*} (X_{T}; Z) .

Equivalently, $ϕ$ is a $Z_{T}$ -homology isomorphism and $X_{T}$ is $T$ -local (its homotopy groups are $Z_{T}$ -modules).

This characterizes $X_{T}$ because the pair (a $T$ -local space, a map from $X$ inducing $H_{*} (-; Z_{T})$ -isomorphism) is unique up to homotopy: any two such receive a comparison equivalence by the relative Hurewicz/Whitehead machinery for nilpotent spaces. The homology criterion is the practical definition — one builds $X_{T}$ by a Postnikov-tower induction, localizing each Eilenberg-MacLane stage $K (A_{n}, n)$ to $K (A_{n} \otimes Z_{T}, n)$ and reassembling. Nilpotence is what allows the Postnikov tower to be built with abelian-group data the localization can act on. May-Ponto Ch. 5-6.

Exercise 3 (medium, proof). Universal property of localization.

Prove that $ϕ : X \to X_{T}$ is initial among maps from $X$ to $T$ -local spaces: for any $T$ -local nilpotent space $Z$ , $ϕ^{*} : [X_{T}, Z] \to [X, Z]$ is a bijection.

Hint

A map $X \to Z$ into a $T$ -local target inverts the primes outside $T$ on homotopy groups, so it factors through the localization.

Answer

Let $Z$ be a $T$ -local nilpotent space, so $π_{*} (Z)$ are $Z_{T}$ -modules. We show $ϕ^{*} : [X_{T}, Z] \to [X, Z]$ is a bijection.

Surjectivity. Given $f : X \to Z$ , induct up the Postnikov tower of $Z$ . At each Eilenberg-MacLane stage $K (π_{n} Z, n)$ , the obstruction and lifting groups are cohomology groups of $X$ with $T$ -local coefficients $π_{n} Z$ . Because the coefficients are $T$ -local, $H^{*} (X; π_{n} Z) = H^{*} (X; π_{n} Z) \otimes Z_{T} = H^{*} (X_{T}; π_{n} Z)$ via the homology-localization isomorphism (Exercise 2) and universal coefficients. So $f$ extends compatibly over $X_{T}$ , producing $\tilde{f} : X_{T} \to Z$ with $\tilde{f} \circ ϕ ≃ f$ .

Injectivity. Two extensions $\tilde{f}, \tilde{g} : X_{T} \to Z$ with $\tilde{f} ϕ ≃ \tilde{g} ϕ$ agree after restriction along $ϕ$ ; the same obstruction-theoretic argument, applied to the difference, shows the homotopy between them defined on $X$ extends to $X_{T}$ . Hence $\tilde{f} ≃ \tilde{g}$ .

Therefore $ϕ^{*}$ is a bijection, and $X_{T}$ is the universal $T$ -local space under $X$ . This universal property, not the homology formula, is what makes localization a functor and a left adjoint (a Bousfield localization). May-Ponto Ch. 6.

Exercise 4 (medium, numeric). Rationalization of spheres.

Compute the rational homotopy groups $π_{*} (S^{n}) \otimes Q$ for $n$ odd and $n$ even, i.e. describe the rationalization $S_{Q}^{n}$ in each parity.

Hint

This is the Serre computation. An odd sphere is rationally a $K (Q, n)$ ; an even sphere has one extra class from the Whitehead square.

Answer

By Serre's theorem (the rational homotopy of spheres):

$n$ odd. $π_{k} (S^{n}) \otimes Q = Q$ for $k = n$ and $0$ for all other $k$ . So $S_{Q}^{n} ≃ K (Q, n)$ : rationally an odd sphere is an Eilenberg-MacLane space. Its minimal Sullivan model is the free graded-commutative algebra on a single generator $x$ in degree $n$ with $d x = 0$ (an exterior generator, since $n$ is odd).

$n$ even. $π_{k} (S^{n}) \otimes Q = Q$ for $k = n$ and for $k = 2 n - 1$ , and $0$ otherwise. The extra class in degree $2 n - 1$ is the Whitehead square $[ι_{n}, ι_{n}]$ . The minimal model is the free algebra on a degree- $n$ generator $x$ (polynomial, since $n$ even) and a degree- $(2 n - 1)$ generator $y$ with $d y = x^{2}$ , which kills the would-be class $x^{2}$ and creates the $2 n - 1$ homotopy class.

So rationally, odd spheres are homotopy-simple ( $K (Q, n)$ ) and even spheres carry exactly one extra rational homotopy class. This parity split is the prototype computation of rational homotopy theory and feeds the localization fracture. May-Ponto Ch. 9, Sullivan.

Exercise 5 (medium, proof). Completion of $Z$ at $p$ and the homotopy of $S^{1}$ .

Describe the $p$ -completion of the integers and explain why the circle $S^{1}$ is its own $p$ -completion only after care — i.e. why $S^{1}$ is not $p$ -complete in the naive sense, contrasting nilpotent versus non-nilpotent behavior.

Hint

$Z_{p}^{\land} = lim Z / p^{k}$ . The circle has $π_{1} = Z$ , which is not a finite $p$ -group, and $S^{1}$ is not simply connected.

Answer

The $p$ -completion of $Z$ is the $p$ -adic integers $Z_{p}^{\land} = lim_{k} Z / p^{k}$ , an uncountable torsion-free $Z_{p}$ -module.

For a simply connected space, $p$ -completion replaces each homotopy group $A$ by its derived $p$ -completion. The circle is not simply connected: $π_{1} (S^{1}) = Z$ , which is abelian (so $S^{1}$ is nilpotent) but its completion is subtle. The Bousfield-Kan $p$ -completion of $S^{1} = K (Z, 1)$ is $K (Z_{p}^{\land}, 1)$ , not $S^{1}$ itself — completing the fundamental group changes the space. So $S^{1}$ is not $p$ -complete; its completion $(S^{1})_{p}^{\land} ≃ K (Z_{p}^{\land}, 1)$ has uncountable fundamental group.

The contrast: for simply connected finite complexes the completion is well-behaved and the fracture square reassembles $X$ from ${X_{p}^{\land}}$ and $X_{Q}$ . For non-simply-connected spaces like $S^{1}$ one must check nilpotence of the $π_{1}$ -action and accept that completion genuinely enlarges $π_{1}$ . May-Ponto's whole framework is stated for nilpotent spaces precisely so that these completions exist and the fracture theorem holds. May-Ponto Ch. 10-11.

Exercise 6 (medium, proof). Localization at $T$ from localizations at single primes.

Show that for a nilpotent space $X$ of finite type and a set of primes $T$ , the localization $X_{T}$ is recovered as a homotopy pullback of the single-prime localizations $X_{(p)}$ ( $p \in T$ ) over the rationalization $X_{Q}$ .

Hint

This is the local fracture square: $Z_{T} = ⋂_{p \in T} Z_{(p)}$ inside $Q$ , realized as a pullback.

Answer

On an abelian group $A$ of finite type, the algebraic identity is

A \otimes Z_{T} = A \otimes p \in T ⋂ Z_{(p)},

and $Z_{T}$ sits in the pullback square

Z_{T} ↓ Q \to \to \prod_{p \in T} Z_{(p)} ↓ \prod_{p \in T} Q

(an element of $⋂ Z_{(p)}$ is a rational with denominator prime to every $p \in T$ , i.e. a $T$ -local integer). Tensoring with the finite-type $A$ preserves this pullback.

Realizing degreewise on the homotopy groups of the nilpotent space $X$ , the square

X_{T} ↓ X_{Q} \to \to \prod_{p \in T} X_{(p)} ↓ \prod_{p \in T} X_{Q}

is a homotopy pullback: it is a pullback on each $π_{n}$ , and homotopy pullbacks of nilpotent finite-type spaces are detected on homotopy groups. So $X_{T}$ is reconstructed from the single-prime localizations glued over the rational type. This is the local fracture square, the building block from which the global arithmetic square of the lead exercise is assembled. May-Ponto Ch. 8.

Exercise 7 (hard, proof). Nilpotent action is necessary.

Give an example of a non-nilpotent space for which naive localization at a prime fails to be detected on homology, illustrating why the theory restricts to nilpotent spaces.

Hint

Take a space with a non-nilpotent action of $π_{1}$ on a higher homotopy group, e.g. a non-orientable circle bundle or $R P^{2} \lor$ -type example; consider the action of $π_{1}$ on $π_{n}$ .

Answer

Localization theory needs $π_{1}$ to be nilpotent and to act nilpotently on each $π_{n}$ — meaning the lower central series of the action terminates, so $π_{n}$ has a finite filtration by $π_{1}$ -submodules on whose quotients $π_{1}$ acts as the identity.

Counterexample to drop nilpotence. Take $X$ with $π_{1} = Z / p$ acting non-nilpotently (and not as the identity) on $π_{n} = A$ for some $n \geq 2$ — for instance a space realizing a $Z / p$ -representation $A$ on which the generator acts by an automorphism of order prime to $p$ . Then the homotopy groups of any candidate "localization" cannot be built from the $π_{1}$ -modules $π_{n}$ by the Postnikov induction, because the action does not reduce to the identity on associated-graded pieces and the localized coefficient system is not a $π_{1}$ -module in a controllable way. Concretely, the localization map need not induce $H_{*} (X) \otimes Z_{(p)} ≅ H_{*} (X_{(p)})$ : the homology of the total space mixes the group homology of $π_{1}$ with the (now non-nilpotently twisted) higher homotopy, and tensoring with $Z_{(p)}$ does not commute with the twisted homology in the needed way.

For a clean failure, $X =$ a $K (G, 1)$ with $G$ a non-nilpotent group (e.g. a nonabelian free group or $S_{3}$ ): $G$ has no localization $G_{(p)}$ that is a group localizing homology, since localization of groups requires nilpotence (Malcev/Bousfield-Kan). The Postnikov-tower construction of $X_{(p)}$ breaks at the very first stage.

This is why May-Ponto's Part 1 is stated for nilpotent spaces: nilpotence is exactly the hypothesis that lets the Postnikov tower be localized stage-by-stage, with each stage an abelian-group localization. May-Ponto Ch. 4-5.

Exercise 8 (hard, proof). Fracture for the abelian-group level.

Prove the Hasse-square pullback for a finitely generated abelian group $A$ : that $A$ is the pullback of $A_{Q} \to (A^{\land})_{Q} \leftarrow A^{\land}$ , where $A^{\land} = \prod_{p} A_{p}^{\land}$ .

Hint

Reduce to $A = Z$ and $A = Z / p^{k}$ by the structure theorem; check the pullback in each case.

Answer

By the structure theorem, $A = Z^{r} \oplus (torsion)$ , and pullbacks commute with finite direct sums, so it suffices to check $A = Z$ and $A = Z / p^{k}$ .

Case $A = Z$ . Here $A_{Q} = Q$ , $A^{\land} = \prod_{p} Z_{p}$ , and $(A^{\land})_{Q} = Q \otimes \prod_{p} Z_{p} = A_{f}$ (the finite adeles). The pullback

P = Q \times_{A_{f}} \prod_{p} Z_{p} = {(q, (z_{p})) : q = (z_{p}) in A_{f}}

consists of rationals $q$ that are $p$ -integral for every $p$ — that is, $q \in Z$ . So $P = Z$ . This is the classical "an integer is a rational that is everywhere $p$ -integral."

Case $A = Z / p^{k}$ . The rationalization $A_{Q} = (Z / p^{k}) \otimes Q = 0$ (torsion dies rationally). The completion: $A_{q}^{\land} = 0$ for $q \neq = p$ and $A_{p}^{\land} = Z / p^{k}$ (already $p$ -complete), so $A^{\land} = Z / p^{k}$ , and $(A^{\land})_{Q} = 0$ . The pullback $0 \times_{0} Z / p^{k} = Z / p^{k} = A$ .

In both cases $P = A$ , so the Hasse square is a pullback for every finitely generated $A$ . Realizing this degreewise on $π_{*} (X)$ for a simply connected finite-type $X$ gives the space-level arithmetic square of the lead exercise. May-Ponto Ch. 13.

Exercise 9 (hard, proof). Localization as a Bousfield localization.

Explain how $T$ -localization of spaces is an instance of a Bousfield localization of the model category $sSet$ (or $Top$ ), identifying the relevant homology theory and the local objects.

Hint

Localize at the homology theory $H_{*} (-; Z_{T})$ . Local objects are the $T$ -local spaces; local equivalences are $Z_{T}$ -homology isomorphisms.

Answer

Fix the homology theory $E_{*} = H_{*} (-; Z_{T})$ . The Bousfield localization $L_{E}$ of the model category of spaces is the left Bousfield localization at the class of $E_{*}$ -isomorphisms: a new model structure with the same cofibrations, more weak equivalences (the $E_{*}$ -isomorphisms), and accordingly fewer fibrant objects.

Local equivalences are maps inducing isomorphisms on $H_{*} (-; Z_{T})$ .
Local objects ( $E$ -local, the fibrant objects of the localized structure) are exactly the $T$ -local nilpotent spaces: those $Z$ for which $[-, Z]$ inverts every $E_{*}$ -isomorphism. Equivalently, $π_{*} (Z)$ are $Z_{T}$ -modules.
The localization functor $L_{E}$ is fibrant replacement in this structure, and on nilpotent spaces $L_{E} X ≃ X_{T}$ — the universal $T$ -local space under $X$ (Exercise 3's universal property is exactly the defining property of a Bousfield localization).

The existence of $L_{E}$ follows from Bousfield's theorem (the localization exists for any homology theory $E$ on spaces). This reframes the hands-on Postnikov-tower construction of $X_{T}$ as an instance of the general machine: localization at $Z_{T}$ -homology. The same machine, run at $E =$ Morava $K$ -theory, gives chromatic localization, linking Part 1 of May-Ponto to the periodicity theorems. May-Ponto Ch. 19, Bousfield.

Exercise pack EP4. May-Ponto supplement: $p$ -localization, rationalization, completion, the arithmetic (fracture) square, and nilpotent spaces across Part 1, Ch. 5-13.

Prerequisites

03.12.60 pending
03.12.61 pending
03.12.45 pending
03.12.46 pending
03.12.48 pending
03.12.47 pending
03.12.19 pending

Tier anchors

beginner
intermediate: May-Ponto, More Concise Algebraic Topology, Part 1 (Ch. 5-13): localization at a set of primes, completion, the arithmetic (fracture) square, nilpotent spaces, the Hasse/local-to-global square
master

References

May-Ponto — More Concise Algebraic Topology: Localization, Completion, and Model Categories · Ch. 5 (nilpotent spaces and Postnikov towers), Ch. 6 (localization of nilpotent spaces at a set of primes $T$), Ch. 10 (completions at a prime $p$), Ch. 8 / 13 (the fracture / arithmetic square gluing local and rational data)
Sullivan — Geometric Topology: Localization, Periodicity, and Galois Symmetry · the original localization and arithmetic-square program
Bousfield-Kan — Homotopy Limits, Completions and Localizations · Ch. I-VI — $R$-completion and the fracture squares

Estimated time

intermediate: 6h