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A collection of earlier and more recent papers related to the Arf-Kervaire invariant problem


Arf's original paper
Early work on the stable 2-stem
The Arf-Kervaire invariant in differential topology
The Arf-Kervaire invariant in homotopy theory
Papers relevant to our proof
Papers written and talks given since our announcement

Arf's original paper on quadratic forms in characteristic 2, 1941
Untersuchungen uber quadratische Formen in Korpern der Charakteristik 2. (Teil I.)
On the Arf invariant in historical perspective by Falko Lorenz and Peter Roquette, 2010. The historical context of Arf's work on quadratic forms in characteristic 2, including a biography, his correspondance with his thesis adviser Helmut Hasse and an appendix correcting in error in the 1941 paper.
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
A manifold which does not admit any differentiable structure by Michel A. Kervaire, 1960. MathSciNet Review. 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
WHITEHEAD PRODUCTS AND COHOMOLOGY OPERATIONS by Edgar H. Brown, Jr. and Franklin P. Peterson, 1964. The construction of quadratic cohomology operations.
NOTE ON AN INVARIANT OF KERVAIRE by Edgar H. Brown, Jr., 1965. The Kervaire invariant is defined for Spin manifolds.
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 homotopy 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.
Im(J)-theory and the Kervaire invariant by Karlheinz Knapp, 1993. Account of how θj and its odd primary analog are detected by Im(J)-theory. Includes detailed proofs of some assertions made in Section 1.5 of the green book.
The Kervaire invariant one element and the double transfer by Nori Minami, 1995. A proof that θj for j>4 does not factor through the double transfer RP∧RP→S0.
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.
Papers relevant to our proof
K-theory and reality by M. F. Atiyah, 1966. A pioneering paper on C2-equivariant homotopy theory. Atiyah's real K-theory is similar to the real cobordism we consider.
Equivariant stable homotopy theory by John Greenlees and J. Peter May, 1995. A very helpful introduction to the subject.
Real-oriented homotopy theory and an analogue of the Adams-Novikov spectral sequence by Po Hu and Igor Kriz, 2001. This paper studies MU as a C2-equivariant spectrum. Their Proposition 4.9 is our Reduction Theorem (in Section 7 of our preprint) for the group C2. They also prove a special case of our Slice Differentials Theorem (10.9).
Equivariant orthogonal spectra by Michael A. Mandell and J. Peter May, 2002. See also EQUIVARIANT ORTHOGONAL SPECTRA, 2010, a 39 page version of the above written for the MSRI workshop. Reference for the definition of equivariant spectra used in our paper.
An Atiyah-Hirzebruch spectral sequence for KR-theory by Daniel Dugger, 2003. A special case of the slice spectral sequence used in our preprint.
Papers written and talks given by others since our announcement
See the links in the Harvard-MIT Summer (2009) Seminar on the Kervaire Invariant. The talks by Haynes Miller on June 24, and Martin Frankland on July 1 (with additional comments by Miller) are written up in detail. They describe some homotopy theoretic background for the problem.
Lecture notes on a proof (with Jean Lannes) of Browder's theorem using characteristic numbers for manifolds with corners by Haynes Miller, June, 2009. A new proof of Browder's theorem.
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.
History of the Kervaire invariant problem by William Browder, January, 2010. A summary of the history of the problem up to 1969.
The Kervaire invariant in homotopy theory by Mark Mahowald and Paul Goerss, January, 2010. A description of Mahowald's work on the possible role of the Kervaire invariant elements in the EHP sequence.
The Kervaire invariant, slides for a talk given by John Jones in Edinburgh, June 14, 2010. We now know that the only dimensions in which there are framed manifolds with Kervaire invariant one are 2, 6, 14, 30, 62 and possibly 126.
"The hunt to find examples in these six special cases has begun!"
Differential Topology 46 years later (AMS Notices, July, 2011) by John Milnor. "In the 1965 Hedrick Lectures, I described the state of Differential Topology, a field which was then young but growing very rapidly. .... The question as to just when Φk = 0 was the last major unsolved problem in understanding the group of homotopy spheres. It has recently been solved in all but one case by Hill, Hopkins, and Ravenel."
Lectures on equivariant stable homotopy theory, 2011, by Stefan Schwede. Writeup of a course given in Barcelona in 2010. It includes a proof (in Remark 2.7) that the category of orthogonal equivariant G-spectra, defined by Mandell-May and used by us, is equivalent to the category of orthogonal spectra with G-action.
KERVAIRE INVARIANT ONE, Bourbaki Seminar 2010 by Haynes Miller, 30 pages. "Hill, Hopkins, and Ravenel (hereafter HHR) marshall three major developments in stable homotopy theory in their attack on the Kervaire invariant problem:"
  • The chromatic perspective based on work of Novikov and Quillen and pioneered by Landweber, Morava, Miller, Ravenel, Wilson, and many more recent workers;
  • The theory of structured ring spectra, implemented by May and many others; and
  • Equivariant stable homotopy theory, as developed by May and collaborators.
"The specific application of equivariant stable homotopy theory was inspired by analogy with a fourth development, the motivic theory initiated by Voevodsky and Morel, and uses as a starting point the theory of 'Real bordism' investigated by Landweber, Araki, Hu and Kriz. In their application of these ideas, HHR require significant extensions of the existing state of knowledge of this subject, and their paper provides an excellent account of the relevant parts of equivariant stable homotopy theory."
The Equivariant Slice Filtration: a Primer by Mike Hill We present an introduction to the equivariant slice filtration. After reviewing the definitions and basic properties, we determine the slice dimension of various families of naturally arising spectra. This leads to an analysis of pullbacks of slices defined on quotient groups, producing new collections of slices. Building on this, we determine the slice tower for the Eilenberg-Mac Lane spectrum associated to a Mackey functor for a cyclic p-group. We then relate the Postnikov tower to the slice tower for various spectra. Finally, we pose a few conjectures about the nature of slices and pullbacks.

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