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There is an additivity theorem for algebraic K-theory. My question is is there an additivity theorem for algebraic L-theory?

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Let me just ignore the appearing decorations. I think the argument should work with all decorations. The $L$-groups of an additive category $\mathcal{A}$ with involution are defined as the cobordism classes or certain chain complexes over $\mathcal{A}$. The group structure is induced by direct sum of chain complexes.

If we have a additive functor between two additive categories $\mathcal{A}\rightarrow \mathcal{A}'$, it induces a group homomorphism $L_n(\mathcal{A})\rightarrow L_n(\mathcal{A}')$ by applying it degreewise. Now if we have two such functors $F,F'$ and an element $[C_*]\in L_n(\mathcal{A})$ then $$L_n(F\oplus F')([C_*]):=[(F(C_*)\oplus F'(C_*)]=[(F(C_*)]+[F'(C_*)]=(L_n(F)+L_n(F'))(C_*).$$ Thus additivity follows quite immediately from the definition. Definitions of the L-groups can be found for example in Ranicki, The algebraic theory of surgery, I Fuoundations, Prop. 3.2

I think $K$-theory is much harder, since the $K$-groups can only be defined as the homotopy groups of some space/spectrum. For $L$-theory those groups can be defined purely algebraically. However the $L$-theory spectra are still needed - for example to define homology with coefficients in L-theory.

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  • $\begingroup$ Thanks, Henrik! I agree with you, there should be an additivity theorem for all decorations. Probably some algebraic argument following the above argument you gave. $\endgroup$
    – Kun Wang
    Commented Oct 21, 2013 at 21:37
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    $\begingroup$ The additivity theorem in K-theory says that for an exact category $\mathscr{C}$ the functor $s\times t \colon SES(\mathscr{C}) \to \mathscr{C} \times \mathscr{C}$ induces a homotopy equivalence on K-theory spaces. Here $SES(\mathscr{C})$ is the category of short exact sequences in $\mathscr{C}$ and $s$ and $t$ project on the first and last elements of a sequence, respectively. Is there something like this in L-theory? $\endgroup$
    – K.J. Moi
    Commented Oct 22, 2013 at 13:33
  • $\begingroup$ Jacob Lurie defined L-theory for stable $\infty$ categories with some additional structure. Maybe you can find something in his lecture notes. $\endgroup$
    – HenrikRüping
    Commented Oct 23, 2013 at 0:11

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