The 3-primary Arf-Kervaire invariant problem
نویسندگان
چکیده
1 The main point of this talk The main point of this talk The 3-primary Arf-Kervaire invariant problem is still open. We have a program for solving it similar to what we did for p = 2. We are missing a crucial ingredient. Defining the problem The Arf-Kervaire invariant problem for a prime p is to determine the fate of the elements θ j = h 2 j for p = 2 b j−1 for p > 2 ∈ Ext 2,2p j (p−1) A (Z/p, Z/p) (1) 1 where A denotes the mod p Steenrod algebra. This Ext group is the E 2-term for the classical Adams spectral sequence converging to the p-component of the stable homotopy groups of spheres. In these bidegrees the groups are known to be isomorphic to Z/p in each case, generated by these elements. Browder's Theorem of 1969 states that for p = 2, h 2 j is a permanent cycle in the Adams spectral sequence if and only if there is a framed manifold with nontrivial Kervaire invariant manifold in dimension 2 j+1 − 2. Such manifolds are known to exist for 1 ≤ j ≤ 5. We recently showed that for p = 2, θ j does not exist for j ≥ 7. The case j = 6 remains open.
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