03.12.60 · modern-geometry / homotopy

Localisation of nilpotent spaces at a set of primes

shipped3 tiersLean: none

Anchor (Master): Sullivan 1974 *Genetics of homotopy theory and the Adams conjecture* Annals of Mathematics 100 (originator of prime-by-prime localisation and the fracture squares); Bousfield 1975 *The localization of spaces with respect to homology* Topology 14 (the localisation functor); Hilton-Mislin-Roitberg 1975 (the canonical monograph, full nilpotent-group and nilpotent-space development); May-Ponto 2012 Ch. 5-7 (textbook synthesis); Bousfield-Kan 1972 LNM 304 §V (R-localisation of simplicial sets)

Intuition Beginner

Picture the integer $12$ . If you only care about the prime $3$ , the factors of $2$ are a nuisance. You would like to be allowed to divide by $2$ freely, so that $12$ and $3$ become the same up to a unit. Doing this for every prime except $3$ produces a new number system, the ring you get by inverting $2, 5, 7, 11, \dots$ but keeping $3$ . In that system $12 = 4 \times 3$ is just $3$ times an invertible factor, so all the information left is the power of $3$ .

Localising a space at a set of primes $T$ does the same thing one dimension higher. You start with a space $X$ and build a new space $X_{T}$ in which every prime outside $T$ has been made invertible. Concretely, the homotopy groups change from ordinary abelian groups to groups where you are allowed to divide by every prime that is not in $T$ . The part of $X$ that lived at the unwanted primes is washed out, and what survives is exactly the $T$ -primary information.

The two extreme choices are worth naming. If $T$ is the single prime ${p}$ , you keep only the $p$ -part: this is $p$ -localisation, written $X_{(p)}$ . If $T$ is empty, you invert every prime, keeping nothing but the rational skeleton: this is rationalisation $X_{Q}$ . Every other $T$ sits between these.

Visual Beginner

Imagine a dial with a slot for every prime: $2, 3, 5, 7, 11, \dots$ . Choosing a set $T$ is choosing which slots stay switched on. The localisation map $X \to X_{T}$ keeps the switched-on primes and silences the rest. Rationalisation switches every prime off; $p$ -localisation leaves a single switch on.

The picture captures the central feature: localisation does not add anything new, it only forgets the primes outside $T$ . The map $X \to X_{T}$ is always going in one direction, from the full space toward a simpler space that remembers fewer primes.

Worked example Beginner

Take the simplest interesting target group and watch localisation act on it.

Step 1. Start with the abelian group $Z /12$ . Its prime content is one factor of $4$ at the prime $2$ and one factor of $3$ at the prime $3$ , since $12 = 4 \times 3$ . By the structure of finite groups, $Z /12 = Z /4 \times Z /3$ .

Step 2. Localise at $T = {3}$ . The rule is: invert every prime except $3$ . Inverting $2$ kills the factor $Z /4$ entirely, because once $2$ is a unit, the equation $4 x = 0$ forces $x = 0$ . The factor $Z /3$ is untouched, since $3$ is the prime we kept. So the localisation of $Z /12$ at ${3}$ is $Z /3$ .

Step 3. Localise the same group at $T = {2}$ instead. Now $3$ is inverted, which deletes the $Z /3$ factor, and the $Z /4$ survives. The localisation is $Z /4$ .

Step 4. Localise at $T = \emptyset$ (rationalise). Every prime is inverted, so both finite factors vanish. The result is the zero group, matching the fact that a finite group has no rational content.

What this tells us: a space whose only homotopy is $Z /12$ in some degree splits, after localisation, into a clean $3$ -part and a clean $2$ -part, with nothing left over rationally. Localising at $T$ is the space-level version of selecting the $T$ -primary block of every homotopy group.

Check your understanding Beginner

Formal definition Intermediate+

Fix a set $T$ of primes. Let $Z_{T} \subseteq Q$ denote the subring of rationals whose denominators are products of primes outside $T$ ; equivalently $Z_{T} = Z [p^{- 1} : p \in / T]$ , the localisation of $Z$ away from the multiplicative set generated by the primes not in $T$ . An abelian group $A$ is $T$ -local if it is a $Z_{T}$ -module, equivalently if multiplication by every prime $q \in / T$ is an isomorphism of $A$ . The $T$ -localisation of $A$ is $A_{T} := A \otimes_{Z} Z_{T}$ with the evident map $A \to A_{T}$ .

Definition (nilpotent space). A connected based space $X$ is nilpotent if $π_{1} (X)$ is a nilpotent group and acts nilpotently on each higher homotopy group $π_{n} (X)$ , $n \geq 2$ ; this is the standing class, recalled from 03.12.45, on which the constructions below are well behaved. Simply connected spaces are members of this class.

Definition ( $T$ -local space). A nilpotent space $Z$ is $T$ -local if every homotopy group $π_{n} (Z)$ is a $Z_{T}$ -module (for $n = 1$ , a $T$ -local nilpotent group in the sense that each lower-central subquotient is a $Z_{T}$ -module). Equivalently, $Z$ is $T$ -local exactly when its reduced integral homology groups $H_{n} (Z; Z)$ are $Z_{T}$ -modules.

Definition ( $T$ -equivalence). A map $f : X \to Y$ of nilpotent spaces is a $T$ -equivalence if it induces an isomorphism $H_{*} (X; Z_{T}) \to H_{*} (Y; Z_{T})$ on homology with $Z_{T}$ -coefficients; for nilpotent spaces this is equivalent to inducing an isomorphism $π_{*} (X) \otimes Z_{T} \to π_{*} (Y) \otimes Z_{T}$ after localising the homotopy groups.

Definition ( $T$ -localisation of a space). A $T$ -localisation of a nilpotent space $X$ is a map $η : X \to X_{T}$ such that $X_{T}$ is $T$ -local and $η$ is a $T$ -equivalence. As a homological localisation $X_{T} = L_{H Z_{T}} X$ this is the Bousfield localisation at the Moore-spectrum / Eilenberg-MacLane spectrum $H Z_{T}$ , in the framework of 03.12.48; it exists, is functorial, and is determined up to homotopy under $X$ by the universal property below. The named endpoints are $T = {p}$ , giving $p$ -localisation $X_{(p)}$ , and $T = \emptyset$ , giving rationalisation $X_{Q} = X_{(0)}$ of 03.12.06.

Definition (universal property). The map $η : X \to X_{T}$ is initial among maps from $X$ to $T$ -local spaces: for every $T$ -local nilpotent space $Z$ , the induced map $η^{*} : [X_{T}, Z] \to [X, Z]$ on based homotopy classes is a bijection.

Counterexamples to common slips

Localisation is not completion. The $p$ -localisation $X_{(p)}$ has homotopy groups $π_{n} (X) \otimes Z_{(p)}$ , where $Z_{(p)} \subset Q$ is the integers localised at $p$ (a subring of $Q$ ). The $p$ -completion $X_{p}^{\land}$ of 03.12.45 has homotopy groups $π_{n} (X) \otimes Z_{p}$ , the $p$ -adic integers (a profinite ring containing $Z_{(p)}$ ). They agree on torsion at $p$ but differ on the free part: $Z_{(p)}$ is countable, $Z_{p}$ is uncountable.
Non-nilpotent fundamental groups break the homotopy description. When $π_{1} (X)$ is not nilpotent, $π_{n} (X) \otimes Z_{T}$ need not compute the homotopy of any localisation; the homological localisation $L_{H Z_{T}}$ still exists but loses the simple group-by-group description.
Inverting primes is not deleting torsion at those primes only when $T \neq = \emptyset$ . Rationalisation ( $T = \emptyset$ ) deletes all torsion; for nonempty $T$ , the $T$ -primary torsion is retained, and only the primes outside $T$ are cleared.

Key theorem with proof Intermediate+

Theorem (Characterisation of $T$ -localisation; Sullivan 1970/1974, Hilton-Mislin-Roitberg 1975). Let $X$ be a nilpotent space and $η : X \to Y$ a map of nilpotent spaces. The following are equivalent.

(1) $η$ is a $T$ -localisation: $Y$ is $T$ -local and $η$ is a $T$ -equivalence.

(2) For every $n \geq 1$ , the induced map $π_{n} (X) \otimes Z_{T} \to π_{n} (Y)$ is an isomorphism, where the source carries the $Z_{T}$ -localised homotopy group and the target $π_{n} (Y)$ is already $T$ -local.

(3) For every $n$ , the induced map $H_{n} (X; Z_{T}) \to H_{n} (Y; Z_{T})$ is an isomorphism and $H_{n} (Y; Z)$ is a $Z_{T}$ -module.

When these hold, $η$ is the universal map from $X$ to a $T$ -local space, and $Y$ is determined up to homotopy under $X$ .

Proof. We induct up a principal refinement of the Postnikov tower of $X$ , available because $X$ is nilpotent: there is a tower of principal fibrations $X_{n} \to X_{n - 1}$ with fibre an Eilenberg-MacLane space $K (A_{n}, m_{n})$ , classified by a $k$ -invariant in $H^{m_{n} + 1} (X_{n - 1}; A_{n})$ , whose inverse limit is $X$ (this is the structural content recalled from 03.12.45; the nilpotent action is exactly what allows the Postnikov stages to be refined into principal ones).

Step 1. The case of an Eilenberg-MacLane space. For $X = K (A, m)$ with $A$ abelian, localisation of the homotopy group gives the map $K (A, m) \to K (A \otimes Z_{T}, m)$ induced by $A \to A \otimes Z_{T}$ . The target is $T$ -local since its only nonzero homotopy group $A \otimes Z_{T}$ is a $Z_{T}$ -module. By the universal-coefficient computation, $H_{*} (K (A, m); Z_{T})$ depends on $A$ only through $A \otimes Z_{T}$ together with the Tor-term $Tor (A, Z_{T}) = 0$ (because $Z_{T}$ is flat over $Z$ ), so the map is an isomorphism on $Z_{T}$ -homology, hence a $T$ -equivalence. This establishes (1) $\Leftrightarrow$ (2) $\Leftrightarrow$ (3) for a single Eilenberg-MacLane stage.

Step 2. Localising a principal fibration. Suppose $X_{n - 1} \to (X_{n - 1})_{T}$ is a $T$ -localisation and $X_{n} \to X_{n - 1}$ is a principal $K (A_{n}, m_{n})$ -fibration with $k$ -invariant $k \in H^{m_{n} + 1} (X_{n - 1}; A_{n})$ . The image of $k$ under $A_{n} \to A_{n} \otimes Z_{T}$ defines a $k$ -invariant over $(X_{n - 1})_{T}$ , hence a principal $K (A_{n} \otimes Z_{T}, m_{n})$ -fibration $(X_{n})_{T} \to (X_{n - 1})_{T}$ and a map $X_{n} \to (X_{n})_{T}$ extending the localisation downstairs. Both the fibre map $K (A_{n}, m_{n}) \to K (A_{n} \otimes Z_{T}, m_{n})$ and the base map $X_{n - 1} \to (X_{n - 1})_{T}$ are $Z_{T}$ -homology isomorphisms by the inductive hypothesis and Step 1. The Zeeman comparison theorem for the Serre spectral sequence with $Z_{T}$ -coefficients then forces the total-space map $X_{n} \to (X_{n})_{T}$ to be a $Z_{T}$ -homology isomorphism, so it is a $T$ -equivalence onto a $T$ -local space.

Step 3. Passage to the limit. The tower of localised stages assembles to a $T$ -localisation $X \to X_{T} = lim (X_{n})_{T}$ . On homotopy groups, the inverse limit produces $π_{n} (X_{T}) = π_{n} (X) \otimes Z_{T}$ with no $lim^{1}$ obstruction, because at each finite stage the homotopy groups are already $Z_{T}$ -modules and the maps are surjective on them; this gives (2). The homological statement (3) follows from (2) for nilpotent spaces by the relative Hurewicz theorem applied stage by stage with $Z_{T}$ -coefficients.

Step 4. Universality. Given any $T$ -local nilpotent space $Z$ and a map $g : X \to Z$ , induct up the tower: at each stage $g$ extends uniquely (up to homotopy) over $(X_{n})_{T}$ because the obstruction and indeterminacy groups $H^{*} (X_{n}; π_{*} (Z))$ are already $Z_{T}$ -modules, so precomposition with the $T$ -equivalence $X_{n} \to (X_{n})_{T}$ is a bijection on the relevant cohomology. Passing to the limit yields a unique factorisation $X \to X_{T} \to Z$ , which is the universal property. $□$

Bridge. This characterisation builds toward the localisation fracture theorem below and appears again in the completion story of 03.12.45, where the same Postnikov-refinement induction runs with $p$ -adic coefficients instead of $Z_{T}$ -coefficients. The foundational reason the argument works is that $Z_{T}$ is a flat subring of $Q$ , so localising homotopy groups is exact and the Tor-terms vanish; this is exactly the homotopical lift of the algebraic statement that $A \mapsto A \otimes Z_{T}$ is an exact functor on abelian groups. The central insight is that nilpotence buys a principal Postnikov refinement, which reduces every space-level statement to a sequence of statements about $K (A, n)$ 's, where localisation is visibly the algebraic localisation of the single coefficient group. Putting these together, the homotopy category of $T$ -local nilpotent spaces is a reflective subcategory of all nilpotent spaces, with reflector $L_{T} = (-)_{T}$ , and this reflection generalises the rationalisation reflector of 03.12.06 from $T = \emptyset$ to arbitrary $T$ . This pattern recurs throughout the subject: every well-behaved localisation of spaces is a reflection onto a local subcategory, and the prime-set case is the arithmetic prototype.

Exercises Intermediate+

Exercise 3 (medium, symbolic).

Compute the homotopy groups of the $p$ -localisation $S_{(p)}^{3}$ of the $3$ -sphere in low degrees, and contrast $π_{3}$ and $π_{6}$ at the prime $p = 2$ against $p = 5$ . Use that $π_{3} (S^{3}) = Z$ and $π_{6} (S^{3}) = Z /12$ .

Hint

$π_{n} (S_{(p)}^{3}) = π_{n} (S^{3}) \otimes Z_{(p)}$ . Localise each finite group at the chosen prime and the free group by tensoring with $Z_{(p)}$ .

Answer

By the characterisation theorem, $π_{n} (S_{(p)}^{3}) = π_{n} (S^{3}) \otimes Z_{(p)}$ .

In degree $3$ : $π_{3} (S^{3}) = Z$ , so $π_{3} (S_{(p)}^{3}) = Z_{(p)}$ for every prime $p$ (the free part survives at every prime).

In degree $6$ : $π_{6} (S^{3}) = Z /12 = Z /4 \times Z /3$ . At $p = 2$ , tensoring with $Z_{(2)}$ inverts $3$ and kills the $Z /3$ , leaving $π_{6} (S_{(2)}^{3}) = Z /4$ . At $p = 5$ , tensoring with $Z_{(5)}$ inverts both $2$ and $3$ , killing the entire group, so $π_{6} (S_{(5)}^{3}) = 0$ .

This is the geometric content of "inverting primes outside $T$ ": at $p = 5$ the sphere $S^{3}$ has no $5$ -primary torsion in this degree, so the localised homotopy group vanishes. Rubric: full credit for $Z_{(p)}$ in degree $3$ , $Z /4$ at $p = 2$ and $0$ at $p = 5$ in degree $6$ .

Exercise 4 (medium, short-answer).

State the difference between $S_{(p)}^{3}$ ( $p$ -localisation) and $S_{p}^{3 \land}$ ( $p$ -completion) at the level of $π_{3}$ , and explain why they nevertheless induce the same map on mod- $p$ homology.

Hint

$π_{3} = Z$ . Localisation tensors with $Z_{(p)} \subset Q$ ; completion tensors with $Z_{p}$ . Compare both with $Z / p$ .

Answer

On $π_{3} = Z$ , localisation gives $π_{3} (S_{(p)}^{3}) = Z_{(p)}$ , the subring of $Q$ with denominators prime to $p$ — a countable subring of $Q$ . Completion gives $π_{3} (S_{p}^{3 \land}) = Z_{p}$ , the $p$ -adic integers — an uncountable profinite ring. The natural map $Z_{(p)} \to Z_{p}$ is the completion, injective but far from surjective.

They induce the same map on mod- $p$ homology because $Z_{(p)} \otimes Z / p = Z / p = Z_{p} \otimes Z / p$ : both rings have the same residue field, so after reducing mod $p$ the difference between the subring of $Q$ and the $p$ -adic integers disappears. This is precisely why localisation and completion agree on torsion and mod- $p$ data but differ on the free part. Rubric: full credit for the $Z_{(p)}$ versus $Z_{p}$ contrast plus the shared residue field $Z / p$ .

Exercise 5 (medium, short-answer).

Prove that the $T$ -localisation map $η : X \to X_{T}$ is itself a $T$ -local space's identity in the sense that $(X_{T})_{T} ≃ X_{T}$ , i.e. that $L_{T}$ is idempotent.

Hint

Use that $X_{T}$ is already $T$ -local and apply the universal property to the identity map $X_{T} \to X_{T}$ .

Answer

$X_{T}$ is $T$ -local by construction. Applying the universal property of $T$ -localisation to $X_{T}$ , the map $η_{X_{T}} : X_{T} \to (X_{T})_{T}$ is initial among maps from $X_{T}$ to $T$ -local spaces. But the identity $id : X_{T} \to X_{T}$ is a map to a $T$ -local space, so it factors uniquely as $X_{T} η_{X_{T}} (X_{T})_{T} r X_{T}$ with $r \circ η_{X_{T}} = id$ .

Conversely $η_{X_{T}}$ is a $T$ -equivalence between $T$ -local spaces; a $T$ -equivalence of $T$ -local spaces is a homotopy equivalence, because on each homotopy group it is an isomorphism of $Z_{T}$ -modules (the localisation $(-) \otimes Z_{T}$ acts as the identity on a $Z_{T}$ -module). Hence $η_{X_{T}}$ is an equivalence and $(X_{T})_{T} ≃ X_{T}$ . This is the idempotence of the reflector $L_{T}$ . Rubric: full credit for the universal-property factorisation plus the observation that a $T$ -equivalence of $T$ -local spaces is an equivalence.

Exercise 7 (hard, short-answer).

State and prove the localisation fracture (pullback) theorem in the two-set form: for disjoint sets of primes $T_{1}, T_{2}$ with $T = T_{1} \cup T_{2}$ , the space $X_{T}$ is the homotopy pullback of $X_{T_{1}}$ and $X_{T_{2}}$ over $X_{T_{1} \cap T_{2}} = X_{T_{1}}_{Q -glued}$ .

Hint

Reduce to homotopy groups via the principal Postnikov refinement, and use the algebraic pullback square for $A \otimes Z_{T}$ in terms of $A \otimes Z_{T_{1}}$ , $A \otimes Z_{T_{2}}$ , and $A \otimes Z_{T_{1} \cap T_{2}}$ .

Answer

Statement. For disjoint $T_{1}, T_{2}$ and $T = T_{1} \cup T_{2}$ , the commutative square $$ \begin{array}{ccc} X_T & \to & X_{T_1} \ \downarrow & & \downarrow \ X_{T_2} & \to & X_{T_1 \cap T_2} \end{array} $$ (where the right and bottom maps further localise away the extra primes, and $T_{1} \cap T_{2} = \emptyset$ gives the rational corner $X_{Q}$ ) is a homotopy pullback of nilpotent spaces.

Proof. By the principal Postnikov refinement it suffices to prove the statement on a single Eilenberg-MacLane stage $K (A, n)$ , i.e. to prove that for a finitely generated abelian group $A$ the square of localised groups $$ A_T \to A_{T_1}, \qquad A_{T_2} \to A_{T_1 \cap T_2} $$ is a pullback of abelian groups. Concretely $A_{T} = A \otimes Z_{T}$ , and the claim is $$ A \otimes \mathbb{Z}T ;\cong; (A \otimes \mathbb{Z}{T_1}) \times_{A \otimes \mathbb{Z}{T_1 \cap T_2}} (A \otimes \mathbb{Z}{T_2}). $$ This is the tensor product of $A$ with the ring-level pullback $Z_{T} = Z_{T_{1}} \times_{Z_{T_{1} \cap T_{2}}} Z_{T_{2}}$ , which holds because a fraction with denominator a product of primes outside $T$ is precisely one that lies in $Z_{T_{1}}$ and in $Z_{T_{2}}$ simultaneously, with the two views agreeing in the common further-localisation $Z_{T_{1} \cap T_{2}}$ . Flatness of each $Z_{T_{i}}$ over $Z$ preserves the pullback under $A \otimes (-)$ , and since $Z_{T_{1} \cap T_{2}}$ is flat the higher derived terms vanish, so no $lim^{1}$ obstruction enters. The homotopy-group pullback then lifts to a homotopy-pullback of spaces by the Postnikov induction, exactly as in the Zeeman comparison argument of the main proof. Rubric: full credit for the ring-level pullback $Z_{T} = Z_{T_{1}} \times_{Z_{T_{1} \cap T_{2}}} Z_{T_{2}}$ , the flatness remark, and the lift to spaces.

Exercise 8 (hard, short-answer).

Explain how the localisation fracture square of Exercise 7, taken to the extreme one-prime-at-a-time decomposition, relates to the arithmetic-square completion fracture of 03.12.45. Identify precisely where the two diagrams differ.

Hint

Both reassemble $X$ from $p$ -by- $p$ pieces and a rational corner. One uses $X_{(p)}$ (localisation, $Z_{(p)} \subset Q$ ); the other uses $X_{p}^{\land}$ (completion, $Z_{p}$ ). Compare the gluing corner.

Answer

Iterating the localisation fracture (Exercise 7) down to single primes assembles $X = X_{(all primes)}$ as a homotopy limit of its $p$ -localisations $X_{(p)}$ over the rationalisation $X_{Q}$ : the homotopy groups satisfy $π_{n} (X) = ⋂_{p} π_{n} (X_{(p)})$ inside $π_{n} (X_{Q}) = π_{n} (X) \otimes Q$ , the algebraic statement that $Z = ⋂_{p} Z_{(p)}$ inside $Q$ .

The arithmetic-square completion fracture of 03.12.45 reassembles $X$ instead from its $p$ -completions $X_{p}^{\land}$ and rationalisation, with gluing corner the rationalised product $X_{Q} (\prod_{p} X_{p}^{\land})$ , governed by the adelic short exact sequence $0 \to Z \to Q \oplus \hat{Z} \to A_{f} \to 0$ .

The precise difference: localisation fracture uses $Z_{(p)} \subset Q$ and glues over the rationals directly, with $⋂_{p} Z_{(p)} = Z$ and no finite-adele correction; completion fracture uses the $p$ -adic $Z_{p}$ and must glue over the finite adeles $A_{f} = \hat{Z} \otimes Q$ , which carries a genuine $lim^{1}$ bookkeeping requirement and a finite-type hypothesis. The localisation square needs no profinite gluing corner; the completion square does. Rubric: full credit for the $⋂_{p} Z_{(p)} = Z$ localisation statement, the adelic completion statement, and the identification of the gluing corner as the precise point of difference.

Advanced results Master

The localisation functor and its homological description

Theorem (Bousfield 1975; existence of $T$ -localisation). For every set of primes $T$ , the assignment $X \mapsto X_{T} = L_{H Z_{T}} X$ is an idempotent functor $\mathrm{Ho}(\mathbf{Top}_){\mathrm{nil}} \to \mathrm{Ho}(\mathbf{Top})_{\mathrm{nil}} $o n t h e h o m o t o p y c a t e g or y o f ni l p o t e n t s p a ces, w i t hna t u r a l t r an s f or ma t i o n$ \eta : \mathrm{id} \to L_T $e x hibi t in g$ X_T $a s t h er e f l ec t i o n o f$ X $in t o t h e f u l l s u b c a t e g or y o f$ T$-local spaces.

The functor is the Bousfield localisation at the homology theory $H Z_{T}$ , constructed by the small-object argument inside a localised model structure on $sSet_{*}$ whose weak equivalences are the $Z_{T}$ -homology isomorphisms, in the sense of 03.12.48. On the nilpotent class the abstract localisation coincides with the explicit Postnikov-induction construction of the Key theorem, and the homotopy groups are computed by $π_{n} (X_{T}) = π_{n} (X) \otimes Z_{T}$ with the $π_{1}$ -action localised correspondingly. The flatness of $Z_{T}$ over $Z$ is what makes the homology theory $H Z_{T}$ behave like a localisation rather than a completion: $Z_{T}$ -homology is ordinary homology with coefficients in a subring of $Q$ , so all Tor-terms vanish and the localisation map is a homology isomorphism in $Z_{T}$ -coefficients by construction.

Localisation of nilpotent groups

Theorem (Hilton-Mislin-Roitberg 1975; localisation of nilpotent groups). Let $G$ be a nilpotent group with lower central series $G = Γ_{1} \supseteq Γ_{2} \supseteq \dots \supseteq Γ_{c + 1} = 1$ . There is a $T$ -localisation $G \to G_{T}$ , a homomorphism into a $T$ -local nilpotent group, universal among maps from $G$ to $T$ -local nilpotent groups, and it is computed by localising each subquotient: $Γ_{i} (G_{T}) / Γ_{i + 1} (G_{T}) = (Γ_{i} (G) / Γ_{i + 1} (G)) \otimes Z_{T}$ . The localisation $G \to G_{T}$ is a $T$ -isomorphism: its kernel and cokernel (as a map of nilpotent groups) are $T^{'}$ -torsion for $T^{'}$ the complementary set of primes.

The non-abelian content is that localisation respects the nilpotent filtration but does not in general distribute over a presentation; the construction proceeds by induction on the nilpotency class $c$ , localising the central extension $1 \to Z (G) \to G \to G / Z (G) \to 1$ one stage at a time. This is the group-theoretic input that drives the $π_{1}$ -level of the space-level localisation: a nilpotent space localises because its fundamental group is a nilpotent group, for which the localisation above exists, and its higher homotopy groups are modules over $π_{1}$ on which the action localises compatibly.

The localisation fracture theorem

Theorem (Sullivan 1974; Hilton-Mislin-Roitberg 1975, fracture for localisation). Let $X$ be a nilpotent space and let ${T_{i}}$ be a family of sets of primes whose union is the set of all primes and whose pairwise intersections all equal a common subset $S$ . Then the natural map $$ X_{\bigcup_i T_i} ;\longrightarrow; \operatorname{holim}i , X{T_i} $$ is a homotopy equivalence, where the homotopy limit is taken over the diagram of further-localisation maps $X_{T_{i}} \to X_{S}$ . In the basic two-set case with $S = \emptyset$ , this is the pullback square $X = X_{(p)} \times_{X_{Q}}^{h} X_{(p^{'})}$ reassembling $X$ from a prime and its complement over the rationalisation.

This is the Part 2 fracture theorem of May-Ponto, distinct from the Part 3 arithmetic-square completion fracture of 03.12.45: here the corners are localisations $X_{T_{i}}$ (homotopy groups $\otimes Z_{T_{i}}$ , subrings of $Q$ ) and the gluing corner is a further localisation, with no profinite completion and no finite-adele correction. The full prime-by-prime form recovers $X$ as the homotopy limit of all its $p$ -localisations over the rationalisation, the homotopical lift of $Z = ⋂_{p} Z_{(p)}$ inside $Q$ .

Synthesis. The localisation $X \to X_{T}$ is the foundational reason the homotopy theory of nilpotent spaces splits into independent prime-local pieces, and this is exactly the homotopical lift of the arithmetic fact that an abelian group is assembled from its localisations $A \otimes Z_{(p)}$ glued over $A \otimes Q$ . The construction builds toward the completion theory of 03.12.45 and appears again in 03.12.06, where rationalisation is the endpoint $T = \emptyset$ of the present family. The central insight is that nilpotence provides a principal Postnikov refinement, which reduces every localisation statement to the algebraic localisation of a single coefficient group; putting these together, the localisation fracture theorem is dual to the completion fracture in the precise sense that one glues over the rationals $Q$ while the other glues over the finite adeles $A_{f}$ , and the comparison map $X_{(p)} \to X_{p}^{\land}$ from localisation to completion generalises the inclusion $Z_{(p)} ↪ Z_{p}$ of a subring of $Q$ into a profinite ring. The bridge is that both fracture squares are instances of a single principle, that a nilpotent finite-type space is the homotopy limit of its arithmetic localisations over their common rational shadow, and the two theorems differ only in whether the local data is localised (subring of $Q$ ) or completed (profinite).

Full proof set Master

Proposition (homotopy groups of a $T$ -localisation). Let $X$ be a nilpotent space and $X \to X_{T}$ its $T$ -localisation. Then for every $n \geq 1$ the map $π_{n} (X) \to π_{n} (X_{T})$ induces an isomorphism $π_{n} (X) \otimes Z_{T} ≅ π_{n} (X_{T})$ , where for $n = 1$ the tensor is the localisation of the nilpotent group $π_{1} (X)$ on its lower-central subquotients.

Proof. Run the principal Postnikov refinement of $X$ , a tower $\dots \to X_{n} \to X_{n - 1} \to \dots$ with principal fibres $K (A_{n}, m_{n})$ and $lim X_{n} ≃ X$ , available by nilpotence (recalled from 03.12.45). The base case $K (A, m) \to K (A \otimes Z_{T}, m)$ has the asserted homotopy-group statement by definition, since $A \otimes Z_{T}$ is the localised coefficient group and $Z_{T}$ is flat, so $Tor (A, Z_{T}) = 0$ and no correction term appears.

For the inductive step, the fibration sequence $K (A_{n} \otimes Z_{T}, m_{n}) \to (X_{n})_{T} \to (X_{n - 1})_{T}$ produces a long exact sequence of homotopy groups mapping from that of $K (A_{n}, m_{n}) \to X_{n} \to X_{n - 1}$ . By the inductive hypothesis the base and fibre maps are localisations on homotopy groups; since $(-) \otimes Z_{T}$ is exact (flatness of $Z_{T}$ ), the five-lemma applied to the localised long exact sequence gives that $π_{k} (X_{n}) \otimes Z_{T} \to π_{k} ((X_{n})_{T})$ is an isomorphism for all $k$ .

Passing to the inverse limit, $π_{n} (X_{T}) = π_{n} (lim (X_{n})_{T})$ . The relevant $lim^{1}$ term vanishes: at each finite stage the homotopy groups are finitely generated nilpotent (so finitely generated $Z_{T}$ -modules after localisation) and the tower maps are surjective on $π_{n}$ in the eventual range, satisfying the Mittag-Leffler criterion. Hence $π_{n} (X_{T}) = lim π_{n} ((X_{n})_{T}) = lim (π_{n} (X_{n}) \otimes Z_{T}) = (lim π_{n} (X_{n})) \otimes Z_{T} = π_{n} (X) \otimes Z_{T}$ , where the third equality uses that localisation commutes with the eventually-constant inverse system. For $n = 1$ the same argument runs on the lower-central subquotients of the nilpotent group $π_{1} (X)$ , using the localisation of nilpotent groups of the Advanced-results theorem. $□$

Proposition (homology characterisation of $T$ -local spaces). A nilpotent space $Z$ has $Z_{T}$ -module homotopy groups if and only if its reduced integral homology groups $H_{n} (Z; Z)$ are $Z_{T}$ -modules for all $n$ .

Proof. Suppose the homotopy groups are $Z_{T}$ -modules. Induct up the Postnikov tower: $H_{*} (K (A, m); Z)$ for a $Z_{T}$ -module $A$ consists of $Z_{T}$ -modules, because the integral homology of $K (A, m)$ is built from $A$ by iterated Tor and tensor operations (the homology of Eilenberg-MacLane spaces), all of which preserve the property of being a $Z_{T}$ -module when $A$ is one and $Z_{T}$ is a localisation of $Z$ . The Serre spectral sequence of each principal fibration has $E^{2}$ -page a bigraded $Z_{T}$ -module by the inductive hypothesis on base and the just-established statement on fibre; the property passes to $E^{\infty}$ and hence to the homology of the total space, since an extension of $Z_{T}$ -modules is a $Z_{T}$ -module.

Conversely, suppose the reduced homology groups are $Z_{T}$ -modules. For a simply connected $Z$ the Hurewicz theorem gives $π_{2} (Z) = H_{2} (Z; Z)$ , a $Z_{T}$ -module; inducting with the relative Hurewicz theorem up the Postnikov tower shows each $π_{n} (Z)$ is a $Z_{T}$ -module. For the nilpotent non-simply-connected case the same induction runs on the principal refinement, with the action of $π_{1}$ on the higher groups being $Z_{T}$ -linear because $π_{1}$ itself is $T$ -local on its subquotients. $□$

Connections Master

The completion counterpart of this unit is the arithmetic square and integral fracture theorem 03.12.45, which reassembles a nilpotent finite-type space from its $p$ -completions $X_{p}^{\land}$ and rationalisation. The present unit builds directly atop it: it inherits the nilpotent-space definition, the rationalisation endpoint, and the homotopy-pullback fracture pattern, and replaces the profinite $p$ -adic completion $Z_{p}$ by the subring-of- $Q$ localisation $Z_{T}$ , gluing over $Q$ rather than over the finite adeles.
The rationalisation $X_{Q}$ studied through Sullivan minimal models 03.12.06 is the endpoint $T = \emptyset$ of the localisation family developed here. Every statement about $X_{T}$ specialises, when all primes are inverted, to a statement about the rational homotopy type, and the present unit shows rationalisation is one member of a one-parameter family indexed by sets of primes.
The construction realises $X_{T}$ as a Bousfield localisation of a model category 03.12.48, with localising homology theory $H Z_{T}$ ; the abstract reflector $L_{H Z_{T}}$ of that framework is identified, on nilpotent spaces, with the explicit Postnikov-induction localisation of this unit, giving a worked instance of the general localisation machinery.
The nilpotent-group and nilpotent-space theory underlying the construction — the lower central series, the principal Postnikov refinement, and the recognition criteria — is the natural prerequisite developed in the co-produced unit on nilpotent groups and spaces 03.12.61, which supplies the structural input that the localisation here consumes group-by-group along the central series.

Historical & philosophical context Master

The localisation of spaces at a set of primes originates with Dennis Sullivan's 1970 MIT notes and his 1974 Annals of Mathematics paper ^{[Sullivan 1974]}, which introduced the prime-by-prime study of homotopy types and the fracture squares assembling a space from its localisations and rationalisation; the same paper used the profinite completion and its Galois action to prove the Adams conjecture. Aldridge Bousfield's 1975 Topology paper ^{[Bousfield 1975]} recast localisation as an idempotent functor on the homotopy category determined by a homology theory, making the existence and functoriality of $X_{T}$ a special case of homological localisation and freeing it from the case-by-case Postnikov constructions. The systematic group-theoretic and space-level development, including the careful treatment of nilpotent fundamental groups and the characterisation theorems, was carried out by Peter Hilton, Guido Mislin, and Joseph Roitberg in their 1975 monograph ^{[Hilton-Mislin-Roitberg 1975]}, which remains the canonical reference for the nilpotent case and fixes the convention, retained here, that $Z_{T}$ denotes the integers with the primes outside $T$ inverted.

The conceptual debt to number theory is explicit in Sullivan's own framing: the assembly of a space from its $p$ -local pieces over the rationals mirrors the assembly of an integer from its images in the local rings $Z_{(p)}$ , and the distinction between localisation and completion at a prime is the topological shadow of the inclusion $Z_{(p)} ↪ Z_{p}$ of a subring of $Q$ into a profinite ring. May and Ponto ^{[May-Ponto 2012]} organise the modern textbook account by separating the localisation theory (their Part 2) from the completion theory (their Part 3), the division this unit and 03.12.45 respect.

Bibliography Master

@article{Sullivan1974,
  author    = {Sullivan, Dennis},
  title     = {Genetics of homotopy theory and the {Adams} conjecture},
  journal   = {Annals of Mathematics},
  volume    = {100},
  year      = {1974},
  pages     = {1--79}
}

@book{Sullivan2005,
  author    = {Sullivan, Dennis P.},
  title     = {Geometric Topology: Localization, Periodicity and {Galois} Symmetry},
  series    = {K-Monographs in Mathematics},
  volume    = {8},
  publisher = {Springer},
  year      = {2005},
  note      = {Revised and edited version of the 1970 MIT lecture notes}
}

@article{Bousfield1975,
  author    = {Bousfield, Aldridge K.},
  title     = {The localization of spaces with respect to homology},
  journal   = {Topology},
  volume    = {14},
  year      = {1975},
  pages     = {133--150}
}

@incollection{HiltonMislinRoitberg1975,
  author    = {Hilton, Peter and Mislin, Guido and Roitberg, Joseph},
  title     = {Localization of Nilpotent Groups and Spaces},
  series    = {North-Holland Mathematics Studies},
  volume    = {15},
  publisher = {North-Holland},
  year      = {1975}
}

@book{BousfieldKan1972,
  author    = {Bousfield, Aldridge K. and Kan, Daniel M.},
  title     = {Homotopy Limits, Completions and Localizations},
  series    = {Lecture Notes in Mathematics},
  volume    = {304},
  publisher = {Springer-Verlag},
  year      = {1972}
}

@book{MayPonto2012,
  author    = {May, J. Peter and Ponto, Kate},
  title     = {More Concise Algebraic Topology: Localization, Completion, and Model Categories},
  publisher = {University of Chicago Press},
  year      = {2012},
  series    = {Chicago Lectures in Mathematics}
}

@book{Hatcher2002,
  author    = {Hatcher, Allen},
  title     = {Algebraic Topology},
  publisher = {Cambridge University Press},
  year      = {2002}
}

Prerequisites

03.12.45
03.12.48
03.12.06

Tier anchors

beginner: May-Ponto *More Concise Algebraic Topology* (Chicago 2012) Ch. 5 introduction; Hatcher §4.J informal; Sullivan 1974 *Annals* 100 motivational §1 (inverting primes)
intermediate: May-Ponto 2012 Ch. 5-7; Hilton-Mislin-Roitberg 1975 *Localization of Nilpotent Groups and Spaces* Ch. I-V; Bousfield-Kan 1972 LNM 304 §V
master: Sullivan 1974 *Genetics of homotopy theory and the Adams conjecture* Annals of Mathematics 100 (originator of prime-by-prime localisation and the fracture squares); Bousfield 1975 *The localization of spaces with respect to homology* Topology 14 (the localisation functor); Hilton-Mislin-Roitberg 1975 (the canonical monograph, full nilpotent-group and nilpotent-space development); May-Ponto 2012 Ch. 5-7 (textbook synthesis); Bousfield-Kan 1972 LNM 304 §V (R-localisation of simplicial sets)

References

Sullivan — Genetics of homotopy theory and the Adams conjecture · Annals of Mathematics 100 (1974), pp. 1-79; the prime-by-prime localisation of a space, the fracture squares assembling X from its p-localisations and rationalisation, and the Galois action on the profinite completion; the foundational source for localisation at a set of primes as distinct from completion
Bousfield — The localization of spaces with respect to homology · Topology 14 (1975), pp. 133-150; the homological localisation functor L_E; the special case E = H Z_T giving T-localisation; functoriality and idempotence
Hilton-Mislin-Roitberg — Localization of Nilpotent Groups and Spaces · North-Holland Mathematics Studies 15 (1975); Ch. I-III localisation of nilpotent groups at a set of primes T (the Z_T-localisation of the lower central quotients), Ch. IV-V localisation of nilpotent spaces via the Postnikov tower, the characterisation of T-local spaces, and the localisation fracture (pullback) theorem
Bousfield-Kan — Homotopy Limits, Completions and Localizations · Springer Lecture Notes in Mathematics 304 (1972); §V R-localisation of simplicial sets for a subring R of the rationals, §III-§IV the completion tower, the comparison of R-localisation with p-completion
May-Ponto — More Concise Algebraic Topology · University of Chicago Press 2012; Ch. 5 localisations of nilpotent groups and spaces at a set of primes T, Ch. 6 characterisation and computation of T-local spaces, Ch. 7 the fracture theorems for localisation, the global-to-local reassembly distinct from the completion fracture of Ch. 10-13
Sullivan — Geometric Topology, Localization, Periodicity and Galois Symmetry · MIT notes 1970; revised as K-Monographs in Mathematics 8 (Springer 2005); §1-§3 the localisation of a simply connected space at a set of primes, with the localised homotopy and homology groups computed as Z_T-modules
Hatcher — Algebraic Topology · Cambridge 2002; §3.A and §4.J localisation of abelian groups and the rational/local homotopy of spaces as motivation

Estimated time

beginner: 20m
intermediate: 48m
master: 95m