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K-theory Definition

Wall's finiteness obstruction

Builds on CW complexFundamental groupUniversal coverGroup ringChain complexChain homotopyFinitely generated projective moduleReduced K0Homotopic maps

Definition

Let XX be a connected CW complex with fundamental group π\pi. Assume XX is finitely dominated: there are a finite CW complex KK and maps i ⁣:X→Ki \colon X \to K and r ⁣:K→Xr \colon K \to X with r∘ir \circ i homotopic to the identity of XX. Consider the cellular chain complex C∗(X~)C_*(\tilde X) of the universal cover, a complex of free modules over the group ring ℤ[π]\Z[\pi]. It is chain homotopy equivalent to a complex P∗P_* with only finitely many nonzero terms. Each PiP_i is a finitely generated projective ℤ[π]\Z[\pi]-module. Wall's finiteness obstruction of XX is σ(X)=∑i(−1)i[Pi]∈K~0(ℤ[π]),\sigma(X) = \sum_i (-1)^i [P_i] \in \tilde K_0(\Z[\pi]), the class in the reduced K0 of the alternating sum. It does not depend on the choice of P∗P_*.

Remarks

Example: if XX is itself a finite CW complex, one can take P∗=C∗(X~)P_* = C_*(\tilde X), which consists of finitely generated free modules, so σ(X)=0\sigma(X) = 0. Non-example: an infinite wedge of circles is not finitely dominated, since its first homology is not finitely generated, so it has no finiteness obstruction. Wall's theorem says that σ(X)=0\sigma(X) = 0 if and only if XX is homotopy equivalent to a finite CW complex. Some authors define the unreduced class in K0(ℤ[π])K_0(\Z[\pi]); its image in K~0(ℤ[π])\tilde K_0(\Z[\pi]) is the obstruction above.

Reading path

To understand this page from the ground up, read these 10 pages in order. The foundations come first.

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CW complex, concept, no evidence levelCW complexconceptChain complex, concept, no evidence levelChain complexconceptChain homotopy, concept, no evidence levelChain homotopyconceptFinitely generated projective module, concept, no evidence levelFinitely generated pr…conceptFundamental group, concept, no evidence levelFundamental groupconceptGroup ring, concept, no evidence levelGroup ringconceptHomotopic maps, concept, no evidence levelHomotopic mapsconceptReduced K0, concept, no evidence levelReduced K0conceptUniversal cover, concept, no evidence levelUniversal coverconceptWall's finiteness obstruction, definition, no evidence levelWall's finiteness obs…definition
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. CW complex concept
  2. Chain complex concept
  3. Chain homotopy concept
  4. Finitely generated projective module concept
  5. Fundamental group concept
  6. Group ring concept
  7. Homotopic maps concept
  8. Reduced K0 concept
  9. Universal cover concept
  10. Wall's finiteness obstruction definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Wall%27s_finiteness_obstruction
About this page
Tag7KNNFSJDZQ
URIm2://7KNNFSJDZQ
Kinddefinition
WritingW0 stub
MSC19J05, 57Q10
DefinesWall's finiteness obstruction
UsesCW complex, Fundamental group, Universal cover, Group ring, Chain complex, Chain homotopy, Finitely generated projective module, Reduced K0, Homotopic maps
Changed3 November 2026

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