# The triviality of the 61-stem in the stable homotopy groups of spheres

@article{Wang2016TheTO, title={The triviality of the 61-stem in the stable homotopy groups of spheres}, author={Guozhen Wang and Zhouli Xu}, journal={arXiv: Algebraic Topology}, year={2016} }

We prove that the 2-primary $\pi_{61}$ is zero. As a consequence, the Kervaire invariant element $\theta_5$ is contained in the strictly defined 4-fold Toda bracket $\langle 2, \theta_4, \theta_4, 2\rangle$.
Our result has a geometric corollary: the 61-sphere has a unique smooth structure and it is the last odd dimensional case - the only ones are $S^1, S^3, S^5$ and $S^{61}$.
Our proof is a computation of homotopy groups of spheres. A major part of this paper is to prove an Adams…

## 24 Citations

### Some extensions in the Adams spectral sequence and the 51–stem

- MathematicsAlgebraic & Geometric Topology
- 2018

We show a few nontrivial extensions in the classical Adams spectral sequence. In particular, we compute that the 2-primary part of $\pi_{51}$ is $\mathbb{Z}/8\oplus\mathbb{Z}/8\oplus\mathbb{Z}/2$.…

### Intersection forms of spin 4-manifolds and the pin(2)-equivariant Mahowald invariant

- MathematicsCommunications of the American Mathematical Society
- 2022

In studying the “11/8-Conjecture” on the Geography Problem in 4-dimensional topology, Furuta proposed a question on the existence of
Pin
(
2
)
\operatorname {Pin}(2)
-equivariant…

### The special fiber of the motivic deformation of the stable homotopy category is algebraic

- MathematicsActa Mathematica
- 2021

For each prime $p$, we define a $t$-structure on the category $\widehat{S^{0,0}}/\tau\text{-}\mathbf{Mod}_{harm}^b$ of harmonic $\mathbb{C}$-motivic left module spectra over $\widehat{S^{0,0}}/\tau$,…

### The motivic lambda algebra and motivic Hopf invariant one problem

- Mathematics
- 2021

. We investigate forms of the Hopf invariant one problem in motivic homotopy theory over arbitrary base ﬁelds of characteristic not equal to 2. Maps of Hopf invariant one classically arise from…

### The higher structure of unstable homotopy groups

- Mathematics
- 2020

We construct certain unstable higher-order homotopy operations indexed by the simplex categories of $\Delta^{n}$ for ${n\geq 2}$ and prove that all elements in the homotopy groups of a wedge of…

### Towards a Browder theorem for spherical classes in $\Omega^lS^{n+l}$

- Mathematics
- 2017

According to Browder if $4n+2\neq 2^{t+1}-2$ then the Kervaire invariant of the cobordism class of a $(4n+2)$-dimensional manifold $M^{4n+2}$ vanishes and $M^{2^{t+1}-2}$ is of Kervaire invariant one…

### The 2-primary Hurewicz image of tmf.

- Mathematics
- 2020

We determine the image of the 2-primary tmf-Hurewicz homomorphism, where tmf is the spectrum of topological modular forms. We do this by lifting elements of tmf_* to the homotopy groups of the…

### Classical and Motivic Adams Charts

- History
- 2014

This document contains large-format Adams charts that compute 2-complete stable homotopy groups, both in the classical context and in the motivic context over C. The charts are essentially complete…

### Property FW, differentiable structures and smoothability of singular actions

- MathematicsJournal of Topology
- 2020

We provide a smoothening criterion for group actions on manifolds by singular diffeomorphisms. We prove that if a countable group Γ has the fixed point property FW for walls (for example, if it has…

### A note on the divisibility of the Whitehead square

- Mathematics
- 2021

. We show that if we suppose n ≥ 4 and π S 2 n − 1 has no 2-torsion, then the Whitehead squares of the identity maps on S 2 n +1 and S 4 n +3 are divisible by 2. Applying the result of G. Wang and Z.…

## References

SHOWING 1-10 OF 58 REFERENCES

### The strong Kervaire invariant problem in dimension 62

- Mathematics
- 2014

Using a Toda bracket computation $\langle \theta_4, 2, \sigma^2\rangle$ due to Daniel C. Isaksen [11], we investigate the $45$-stem more thoroughly. We prove that $\theta_4^2=0$ using a $4$-fold Toda…

### The decomposition of stable homotopy.

- MathematicsProceedings of the National Academy of Sciences of the United States of America
- 1967

It is shown that, under the operation of higher Toda bracket (which will be defined in ? 2), certain classes in the stable homotopy ring of spheres w*(S) generate all of w-(S), called the Hopf classes, which are those detected by primary cohomology operations.

### The order of the image of the $J$-homomorphism

- Mathematics
- 1970

The set üS can be identified with the set of all base point preserving maps of $ into itself. SO(n)> acting on S as R with a point a t infinity, is also a set of base point preserving maps of S onto…

### The algebraic Atiyah-Hirzebruch spectral sequence of real projective spectra

- Mathematics
- 2016

In this note, we use Curtis's algorithm and the Lambda algebra to compute the algebraic Atiyah-Hirzebruch spectral sequence of the suspension spectrum of $\mathbb{R}P^\infty$ with the aid of a…

### The Adams spectral sequence of the real projective spaces.

- Mathematics
- 1988

and converges to the stable homotopy i^ί(P) = ί(P) where A denotes the mod 2 Steenrod algebra and H*(P) is the reduced mod 2 cohomology of P. We simply write Ext*'(P) for Ext*'(#*(P),Z/2) and…

### RELATIONS AMONGST TODA BRACKETS AND THE KERVAIRE INVARIANT IN DIMENSION 62

- Mathematics
- 1984

In this paper we use relations amongst Toda brackets and a lot of detailed information about the homotopy groups of spheres to show that there exists a 62-dimensional framed manifold with Kervaire…

### Secondary compositions and the Adams spectral sequence

- Mathematics
- 1970

In each term of the Adams spectral sequence [1, 2] Massey products [10] can be formed and in the E2-term it has been found convenient to describe specific elements by means of these products. In…

### Generalized Poincare's Conjecture in Dimensions Greater Than Four

- Mathematics
- 1961

Poincare has posed the problem as to whether every simply connected closed 3-manifold (triangulated) is homeomorphic to the 3-sphere, see [18] for example. This problem, still open, is usually called…

### THE HOMOTOPY GROUPS OF tmf AND OF ITS LOCALIZATIONS

- Mathematics
- 2009

(1) π∗(S) −→ π∗(tmf ) −→ MF∗ that we now describe. Both maps are surprisingly close to being isomorphisms (even though π∗(S) and MF∗ have nothing to do with each other). The first map (1) is the…