Return to Arf-Kervaire page.
A collection of earlier papers related to the Arf-Kervaire invariant problem

Arf's original paper on quadratic forms in characteristic 2, 1941
Untersuchungen uber quadratische Formen in Korpern der Charakteristik 2. (Teil I.)
Early work on the stable 2-stem.
Thanks to Andrew Ranicki for finding these.
Sur les transformations des spheres en spheres by L. S. Pontrjagin, 1936. At the Oslo ICM Pontrjagin announced that π4(S2)=Z2 and πn+2(Sn)=0 for n>2. He could not attend, and his lecture was given by Lefschetz.
A classification of continuous transformations of a complex into a sphere, 1 and 2 by L. S. Pontrjagin, 1938. In the second of these two Doklady notes Pontrjagin gave the false proof. Challenge: find the mistake!
Uber die Klassen der Spharenabbildungen I. Groβe Dimensionen by Hans Freudenthal, 1938. Freudenthal proved π4(S2)=Z2. Note the use of "k-Stamm" (k-stem) in the first paragraph: it may have been the first!
Homotopy classification of the mappings of an (n+2)-dimensional sphere on an n-dimensional one (Russian) by L. S. Pontrjagin, 1950. Pontrjagin proved πn+2(Sn)=Z2. He uses the quadratic form, but does not have the Arf formula.
The (n+2)nd homotopy group of the n-sphere by George W. Whitehead, 1950. Another proof that πn+2(Sn)=Z2.
Smooth manifolds and their applications in homotopy theory by L. S. Pontrjagin, 1955. A detailed account of the isomorphism Ωnfr ≅ πnS, including π2S= Z2 and the Arf formula on page 101. So this was the first occurence of the Kervaire invariant, albeit only in dimension 2.
The Arf-Kervaire invariant in differential topology
Andrew Ranicki's Exotic spheres page with links to many more papers on the subject
MathSciNet review of A manifold which does not admit any differentiable structure by Michel A. Kervaire, 1960. The paper where the Arf-Kervaire invariant is first defined in higher dimensions.
Groups of homotopy spheres: I by Michel A. Kervaire and John W. Milnor, 1963. The classification of exotic spheres, in which the the Arf-Kervaire invariant plays a crucial role.
Problems in differential and algebraic topology. Seattle Conference, 1963 edited by R. Lashof. Problem 34 (page 581) is the first known explicit statement of the Arf-Kervaire invariant problem. Thanks again to Andrew Ranicki for finding this.
The Arf-Kervaire invariant in homotopy theory
THE KERVAIRE INVARIANT OF (8k + 2)-MANIFOLDS by Edgar H. Brown, Jr. and Franklin P. Peterson, 1966. They prove that the Kervaire invariant vanishes in half of all cases.
The Kervaire invariant of framed manifolds and its generalization by William Browder, 1969. The paper that related the problem to stable homotpy theory and the Adams spectral sequence.
Generalizations of the Kervaire invariant by Edgar H. Brown, Jr. 1972. A variant with values in Z/8.
A GEOMETRICAL OBSERVATION ON THE ARF INVARIANT OF A FRAMED MANIFOLD by Nigel Ray 1972. This paper shows that any framed manifold with Arf-Kervaire invaraint one is cobordant to one that can be reframed so that its invariant is zero.
Kervaire's invariant for framed manifolds (1978) and A note on the Kervaire invariant (1975) by John Jones and Elmer Rees. These give an alternate proof of Browder's theorem.
The Kervaire invariant of extended power manifolds by John Jones, 1978. He shows that an extended fourth power of S7 realizes θ4 and that higher extended powers of it will not realize higher θs.
The nonexistence of odd primary Arf invariant elements in stable homotopy, 1978, solved an analogous problem at primes larger than 3. This material can also be found in Section 6.4 of the green book. Solution to an analogous problem at primes larger than 3. The methods used there to detect the elements in question (Theorem 6.4.4) were similar in spirit to our proof of the Detection Theorem. The problem at the prime 3 remains open.
Sur l'invariant de Kervaire des varietes fermees stablement parallelisees by Jean E. Lannes, 1981. Another proof of Browder's theorem.
RELATIONS AMONGST TODA BRACKETS AND THE KERVAIRE INVARIANT IN DIMENSION 62 by Michael Barratt, John Jones and Mark Mahowald, 1984. The paper that established the existence of θ5.
The Kervaire invariant of immersions by Ralph Cohen, John Jones and Mark Mahowald, 1985. A generalization of Browder's Theorem to immersed manifolds having small structure groups in their normal bundles.
The Kervaire invariant one element and the double transfer by Nori Minami, 1993. A proof that θj for j>4 does not factor through the double transfer RP ∞∧RP ∞→S 0.
The Kervaire invariant and surgery theory by Edgar H. Brown, Jr., 2000. A survey of the Kervaire invariant and its generalizations with emphasis on its applications to surgery and to the existence of stably parallelizable manifolds with Kervaire invariant one.
The Arf-Kervaire invariant of framed manifolds by Victor P. Snaith, October 2009. A survey with some history and brief comments on our work and Akhmeti'ev's.