Math 2.0

K-theory Definition

Assembly map

Builds on K-theory spectrumGroup ringClassifying spaceSmash productWhitehead group

Definition

Let GG be a group and let RR be a ring. Let BGBG be the classifying space of GG and let R[G]R[G] be the group ring. Write K(−)K(-) for the K-theory spectrum and Ω0∞\Omega^\infty_0 for the base point component of the infinite loop space. The inclusion G→GL1(ℤ[G])G \to GL_1(\Z[G]) gives a map BG→BGL(ℤ[G])+≃Ω0∞K(ℤ[G])BG \to BGL(\Z[G])^+ \simeq \Omega^\infty_0 K(\Z[G]). Its adjoint is a map of spectra Σ∞BG+→K(ℤ[G])\Sigma^\infty BG_+ \to K(\Z[G]). The assembly map is the composite map of spectra BG+∧K(R)→K(ℤ[G])∧K(R)→K(ℤ[G]⊗R)=K(R[G]),BG_+ \wedge K(R) \to K(\Z[G]) \wedge K(R) \to K(\Z[G] \otimes R) = K(R[G]), where the second map is the product in K-theory and ∧\wedge is the smash product. On homotopy groups it gives homomorphisms Hn(BG;K(R))→Kn(R[G])H_n(BG; K(R)) \to K_n(R[G]).

Remarks

Example: for GG trivial, BG+=S0BG_+ = S^0 and the assembly map is the identity of K(R)K(R). Example: for R=ℤR = \Z and n=1n = 1, the image of the assembly map is generated by the classes of ±g\pm g for g∈Gg \in G, and its cokernel is the Whitehead group Wh(G)Wh(G). Non-example of an isomorphism: for G=ℤ/23G = \Z/23 and R=ℤR = \Z, the image on K0K_0 is ℤ\Z, but K0(ℤ[ℤ/23])K_0(\Z[\Z/23]) also contains the class group of ℤ[e2πi/23]\Z[e^{2\pi i/23}], which has order 33. Davis and Lück defined assembly maps HnG(E;𝐊R)→Kn(R[G])H^G_n(E; \mathbf{K}_R) \to K_n(R[G]) for every GG-CW complex EE, induced by the map E→ptE \to \mathrm{pt}. The map above is the case E=EGE = EG; the Farrell–Jones conjecture uses EE built from virtually cyclic subgroups. The same construction works for L-theory of group rings with involution.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

Show the path as a graph
Classifying space, concept, no evidence levelClassifying spaceconceptGroup ring, concept, no evidence levelGroup ringconceptK-theory spectrum, concept, no evidence levelK-theory spectrumconceptSmash product, concept, no evidence levelSmash productconceptWhitehead group, concept, no evidence levelWhitehead groupconceptAssembly map, definition, no evidence levelAssembly mapdefinition
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. Classifying space concept
  2. Group ring concept
  3. K-theory spectrum concept
  4. Smash product concept
  5. Whitehead group concept
  6. Assembly map definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Assembly_map
About this page
TagSPAS5GPRVM
URIm2://SPAS5GPRVM
Kinddefinition
WritingW0 stub
MSC19D50, 19B28, 55N20
DefinesAssembly map
UsesK-theory spectrum, Group ring, Classifying space, Smash product, Whitehead group
Changed3 November 2026

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