Math 2.0

Algebraic topology Definition

Whitehead tower

Builds on Path-connected spaceHomotopy groupn-connected spaceEilenberg–MacLane spaceUniversal coverPostnikov tower

Definition

Let XX be a path-connected space with base point, and let πk\pi_k denote the kk-th homotopy group. A Whitehead tower of XX is a sequence of pointed spaces X⟨n⟩X\langle n \rangle, n≥0n \ge 0, with X⟨0⟩=XX\langle 0 \rangle = X. It comes with pointed maps qn ⁣:X⟨n⟩→X⟨n−1⟩q_n \colon X\langle n \rangle \to X\langle n - 1 \rangle for n≥1n \ge 1. For each n≥0n \ge 0, the composite rn=q1∘⋯∘qn ⁣:X⟨n⟩→Xr_n = q_1 \circ \cdots \circ q_n \colon X\langle n \rangle \to X must satisfy:

  1. X⟨n⟩X\langle n \rangle is nn-connected, that is, πk(X⟨n⟩)=0\pi_k(X\langle n \rangle) = 0 for all k≤nk \le n.
  2. rnr_n induces an isomorphism πk(X⟨n⟩)→πk(X)\pi_k(X\langle n \rangle) \to \pi_k(X) for all k>nk > n.

The space X⟨n⟩X\langle n \rangle is the nn-connected cover of XX.

Remarks

For a CW complex XX a Whitehead tower exists and each X⟨n⟩X\langle n \rangle is unique up to weak homotopy equivalence over XX. Up to homotopy, qnq_n can be taken to be a fibration, and then its fibre is an Eilenberg–MacLane space K(πn(X),n−1)K(\pi_n(X), n - 1). For XX a CW complex, X⟨1⟩→XX\langle 1 \rangle \to X is the universal cover. The tower is dual to the Postnikov tower: X⟨n⟩X\langle n \rangle is the homotopy fibre of X→XnX \to X_n.

Example: for X=S3X = S^3, X⟨1⟩=X⟨2⟩=S3X\langle 1 \rangle = X\langle 2 \rangle = S^3, since S3S^3 is 22-connected. The cover S3⟨3⟩→S3S^3\langle 3 \rangle \to S^3 is a fibration with fibre K(ℤ,2)=ℂP∞K(\Z, 2) = \C P^\infty.

Non-example: for X=S1X = S^1, the identity S1→S1S^1 \to S^1 as X⟨1⟩X\langle 1 \rangle fails clause 1, since π1(S1)≅ℤ≠0\pi_1(S^1) \cong \Z \ne 0. The universal cover ℝ→S1\R \to S^1 satisfies both clauses.

Reading path

To understand this page from the ground up, read these 7 pages in order. The foundations come first.

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Eilenberg–MacLane space, concept, no evidence levelEilenberg–MacLane spa…conceptHomotopy group, concept, no evidence levelHomotopy groupconceptPath-connected space, concept, no evidence levelPath-connected spaceconceptPostnikov tower, concept, no evidence levelPostnikov towerconceptUniversal cover, concept, no evidence levelUniversal coverconceptn-connected space, concept, no evidence leveln-connected spaceconceptWhitehead tower, definition, no evidence levelWhitehead towerdefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Eilenberg–MacLane space concept
  2. Homotopy group concept
  3. Path-connected space concept
  4. Postnikov tower concept
  5. Universal cover concept
  6. n-connected space concept
  7. Whitehead tower definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Postnikov_system
About this page
TagZYCX4PN96G
URIm2://ZYCX4PN96G
Kinddefinition
WritingW0 stub
MSC55S45, 55P20
DefinesWhitehead tower
UsesPath-connected space, Homotopy group, n-connected space, Eilenberg–MacLane space, Universal cover, Postnikov tower
Changed2 November 2026

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