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The Barratt-Quillen-Priddy theorem says in one interpretation that there is a weak equivalence of spectra $K(FinSet) \simeq \mathbb{S}^0$. In other words K-theory groups of finite sets are the stable homotopy groups of spheres $\pi_*^s$.

If $A$ is a commutative ring, $K_0(A)$ has a simple definition as the free abelian group of projective finitely generated $A$-modules modulo exact sequences. On this group we use the exterior powers $\Lambda^k$ to get so-called Lambda-operations $\lambda^k$. These have nice properties and one can use them to alternatively construct Adams operations $\Psi^i$. This construction can be extended to all $K_n(A)$, giving $K_*(A)$ the structure of a Lambda-ring. This can found in sections II.4 and IV.5 of Weibel's book.

There is a strong analogy between finite sets and vector spaces. This tells you that an analogue of the exterior power $\Lambda^k$ is given by construction that sends a finite set $X$ to its set of $k$-element subsets ${X \choose k}$. This gives the standard Lambda-ring structure on $\mathbb{Z} = \pi_0^s$, i.e. the one on Wikipedia.

It seems that Weibel's construction of the Lambda-operations on higher K-theory groups works in this context as well. Is this correct? If so, we get $\lambda^i$ and $\Psi^i$ on the stable homotopy groups of spheres. What is known about these? Have they been used for anything?

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I think that Stefan Schwede has studied this question, but I am not sure if he has published anything. – Neil Strickland Nov 4 '12 at 19:24
up vote 14 down vote accepted

You can refine this. Let's take $k=2$ to give the idea. To a based set $X$ you can associate $(X\wedge X)/X$, a based set with free action of $\Sigma_2$. This leads to an operation going from stable homotopy of $S^0$ to stable homotopy of $B\Sigma_2$, such that when followed by transfer it gives the difference between the identity and squaring.

This leads to a proof of the Kahn-Priddy Theorem, a proof due to Kahn and Priddy I believe. (EDIT: No, I guess it was Segal.)

Waldhausen adapted the same idea to prove an important result about his $A(X)$, the "vanishing of the mystery homology theory". (That's why I know about it.) This involved extending the construction of these operations from the algebraic $K$-theory of sets to the algebraic $K$-theory of spaces.

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The operation which sends a finite set $S$ to its set of $k$-element subsets, $\binom{S}{k}$, gives rise to the $k$-th stable Hopf invariant. There is additional structure in this: the set $\binom{S}{k}$ has a canonical $k$-fold covering so the operation is better viewed as a map $$ QS^0 \to Q(B\Sigma_k)_+ $$ rather than as a map $$ Q S^0 \to QS^0 , $$ where for a based space $X$, the space $QX$ is $\Omega^\infty\Sigma^\infty X$ is the representing space for the stable homotopy of $X$, i.e., $\pi_j(QX) = \pi_j^{\text{st}}(X)$. So the operation induces a homomorphism $$ \pi_j^{\text{st}}(S^0) \to \pi_j^{\text{st}}((B\Sigma_k)_+) . $$

These operations satisfy certain axioms (Cartan Formula, compatibility with transfers, etc.). A good place to read about these operations is:

Segal, Graeme: Operations in stable homotopy theory. New developments in topology (Proc. Sympos. Algebraic Topology, Oxford, 1972), pp. 105–110. London Math Soc. Lecture Note Ser., No. 11, Cambridge Univ. Press, London, 1974.

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Dear Sander

Maybe this paper by Pierre Guillot can help you:

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Let me add that, for some strange reason, when I uploaded this to the arxiv there was a bug with the latex compilation, and all the lambda's appear as L's. (On one other occasion I had the O in O_n, the orthogonal group, come out as the empty set. I think some commands such as \O and \l should not be redefined...). If you send me an email I can get you a copy with proper lambdas ! – Pierre Nov 5 '12 at 9:32

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