# On the boundaries of highly connected, almost closed manifolds

@article{Burklund2019OnTB, title={On the boundaries of highly connected, almost closed manifolds}, author={Robert Burklund and Jeremy Hahn and Andrew Senger}, journal={arXiv: Algebraic Topology}, year={2019} }

Building on work of Stolz, we prove for integers $0 \le d \le 3$ and $k>232$ that the boundaries of $(k-1)$-connected, almost closed $(2k+d)$-manifolds also bound parallelizable manifolds. Away from finitely many dimensions, this settles longstanding questions of C.T.C. Wall, determines all Stein fillable homotopy spheres, and proves a conjecture of Galatius and Randal--Williams. Implications are drawn for both the classification of highly connected manifolds and, via work of Krannich and Kreck…

## 24 Citations

### A $v_1$-banded vanishing line for the mod 2 Moore spectrum

- Mathematics
- 2020

The mod 2 Moore spectrum $C(2)$ is the cofiber of the self-map $2: \mathbb{S} \to \mathbb{S}$. Building on work of Burklund, Hahn, and Senger, we prove that above a line of slope $\frac{1}{5}$, the…

### Inertia groups of $(n-1)$-connected $2n$-manifolds

- Mathematics
- 2022

. In this paper, we compute the inertia groups of ( n − 1) -connected, smooth, closed, oriented 2 n -manifolds where n ≥ 3 . As a consequence, we complete the diﬀeomorphism classiﬁcation of such…

### On the high-dimensional geography problem

- Mathematics
- 2020

In 1962, Wall showed that smooth, closed, oriented, $(n-1)$-connected $2n$-manifolds of dimension at least $6$ are classified up to connected sum with an exotic sphere by an algebraic refinement of…

### The Adams differentials on the classes $h_j^3$

- Mathematics
- 2023

In filtration 1 of the Adams spectral sequence, using secondary cohomology operations, Adams computed the differentials on the classes $h_j$, resolving the Hopf invariant one problem. In Adams…

### The $\mathbb C$-motivic Adams-Novikov spectral sequence for topological modular forms

- Mathematics
- 2023

We analyze the $\mathbb{C}$-motivic (and classical) Adams-Novikov spectral sequence for the $\mathbb{C}$-motivic modular forms spectrum $\mathit{mmf}$ (and for the classical topological modular forms…

### C2-equivariant topological modular forms

- Mathematics
- 2022

We compute the homotopy groups of the C2 fixed points of equivariant topological modular forms at the prime 2 using the descent spectral sequence. We then show that as a TMF-module, it is isomorphic…

### Some smooth circle and cyclic group actions on exotic spheres

- Mathematics
- 2022

. Classical work of Lee, Schultz, and Stolz relates the smooth transformation groups of exotic spheres to the stable homotopy groups of spheres. In this note, we apply recent progress on the latter…

### Multiplicative structures on Moore spectra

- Mathematics
- 2022

. In this article we show that S / 8 is an E 1 -algebra, S / 32 is an E 2 -algebra, S /p n +1 is an E n -algebra at odd primes and, more generally, for every h and n there exist generalized Moore…

### How Big are the Stable Homotopy Groups of Spheres?

- Mathematics, Computer Science
- 2022

Bootstrapping from the stable case, it is proved that the size of the p-local homotopy groups of spheres is bounded by exp(O(log(n)3)), providing the first subexponential bound on the unstable stems.

## References

SHOWING 1-10 OF 93 REFERENCES

### Synthetic spectra and the cellular motivic category

- Mathematics
- 2018

To any Adams-type homology theory we associate a notion of a synthetic spectrum, this is a spherical sheaf on the site of finite spectra with projective E-homology. We show that the ∞-category SynE…

### volume 1116 of Lecture Notes in Mathematics

- Springer-Verlag, Berlin,
- 1985

### THE REGULAR COMPLEX IN THE BP 〈1〉–ADAMS SPECTRAL SEQUENCE

- Mathematics
- 1998

We give a complete description of the quotient complex C obtained by dividing out the Fp Eilenberg-Mac Lane wedge summands in the first term of the BP 〈1〉–Adams spectral sequence for the sphere…

### On relations between Adams spectral sequences, with an application to the stable homotopy of a moore space

- Mathematics
- 1981

### Vanishing lines in generalized Adams spectral sequences are generic

- Mathematics
- 1999

We show that in a generalized Adams spectral sequence, the presence of a vanishing line of xed slope (at some term of the spectral sequence, with some intercept) is a generic property.

### arXiv e-prints

- page arXiv:1701.01528, Jan
- 2017

### volume 242 of Springer Proc

- Math. Stat., pages 269–311. Springer, Cham,
- 2018