# On the existence of a $v^{32}_2$-self map on $M(1,4)$ at the prime 2

@article{Behrens2007OnTE, title={On the existence of a \$v^\{32\}\_2\$-self map on \$M(1,4)\$ at the prime 2}, author={Mark Joseph Behrens and Michael A. Hill and Michael J. Hopkins and Mark E. Mahowald}, journal={Homology, Homotopy and Applications}, year={2007}, volume={10}, pages={45-84} }

Let M(1) be the mod 2 Moore spectrum. J.F. Adams proved that M(1) admits a minimal v_1-self map v_1^4: Sigma^8 M(1) -> M(1). Let M(1,4) be the cofiber of this self-map. The purpose of this paper is to prove that M(1,4) admits a minimal v_2-self map of the form v_2^32: Sigma^192 M(1,4) -> M(1,4). The existence of this map implies the existence of many 192-periodic families of elements in the stable homotopy groups of spheres.

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## References

SHOWING 1-10 OF 25 REFERENCES

### v1- AND V2-PERIODICITY IN STABLE HOMOTOPY THEORY

- Mathematics
- 1981

this paper we construct some self-maps related to the elements v1 E 7r2(BP) and v2 E 76(BP) and use them to obtain families in the 2-primary stable homotopy of spheres. In particular, we obtain…

### On the Cohomology of the Steenrod algebra

- Mathematics
- 1993

L etA be the dual of the mod p Steenrod algebra. A = Fp(1;:::). Let A(n) be the subalgebra generated by 1;::: ;n. We show that there exists a family of nite p-groups G(n;r) whose group algebra…

### Computation of the homotopy of the spectrum tmf

- Mathematics
- 2003

This paper contains a complete computation of the homotopy ring of the spectrum of topological modular forms constructed by Hopkins and Miller. The computation is done away from 6, and at the…

### Nilpotence and Periodicity in Stable Homotopy Theory.

- Mathematics
- 1992

"Nilpotence and Periodicity in Stable Homotopy Theory" describes some major advances made in algebraic topology in recent years, centering on the nilpotence and periodicity theorems, which were…

### Brown-Comenetz duality and the Adams spectral sequence

- Mathematics
- 1999

<abstract abstract-type="TeX"><p>We show that the class of <i>p</i>-complete connective spectra with finitely presented cohomology over the Steenrod algebra admits a duality theory related to…

### The rigid analytic period mapping, Lubin-Tate space, and stable homotopy theory

- Mathematics
- 1994

The geometry of the Lubin-Tate space of deformations of a formal group is studied via an \'etale, rigid analytic map from the deformation space to projective space. This leads to a simple description…

### Some root invariants at the prime 2

- Mathematics
- 2005

The first part of this paper consists of lecture notes which summarize the machinery of filtered root invariants. A conceptual notion of "homotopy Greek letter element" is also introduced, and…

### Complex Cobordism and Stable Homotopy Groups of Spheres

- Mathematics
- 1986

An introduction to the homotopy groups of spheres Setting up the Adams spectral sequence The classical Adams spectral sequence $BP$-theory and the Adams-Novikov spectral sequence The chromatic…

### H Ring Spectra and Their Applications

- Mathematics, Chemistry
- 1986

Extended powers and H? ring spectra.- Miscellaneous applications in stable homotopy theory.- Homology operations for H? and Hn ring spectra.- The homotopy theory of H? ring spectra.- The homotopy…

### On A-groups

- MathematicsMathematical Proceedings of the Cambridge Philosophical Society
- 1949

A sequence of papers by P. Hall ((1)–(4)) on some fundamental properties of soluble groups was followed in 1940 by his account of a new construction theory for such groups (5). We shall have frequent…