Groups of Homotopy Spheres, I

@inproceedings{Kervaire2015GroupsOH,
  title={Groups of Homotopy Spheres, I},
  author={Michel Kervaire and John W. Milnor},
  year={2015}
}
DEFINITION. Two closed n-manifolds M, and M2 are h-cobordant1 if the disjoint sum M, + (- M2) is the boundary of some manifold W, where both M1 and (-M2) are deformation retracts of W. It is clear that this is an equivalence relation. The connected sum of two connected n-manifolds is obtained by removing a small n-cell from each, and then pasting together the resulting boundaries. Details will be given in ? 2. 
Simply-connected manifolds with large homotopy stable classes
For every k ≥ 2 and n ≥ 2 we construct n pairwise homotopically inequivalent simply-connected, closed 4k-dimensional manifolds, all of which are stably diffeomorphic to one another. Each of these
Cobordism categories and moduli spaces of odd dimensional manifolds
We prove that the stable moduli space of $(n-1)$-connected, $n$-parallelizable, $(2n+1)$-dimensional manifolds is homology equivalent to an infinite loopspace for $n \geq 4, n \neq 7$. The main novel
Realizing congruence subgroups inside the diffeomorphism group of a product of homotopy spheres
Let M be a smooth manifold which is homeomorphic to the n-fold product of S^k, where k is odd. There is an induced homomorphism from the group of diffeomorphisms of M to the automorphism group of H k
Some finiteness results for groups of automorphisms of manifolds
We prove that in dimensions not equal to 4, 5, or 7, the homology and homotopy groups of the classifying space of the topological group of diffeomorphisms of a disk fixing the boundary are finitely
On smooth manifolds with the homotopy type of a homology sphere
Abstract In this paper we study M ( X ) , the set of diffeomorphism classes of smooth manifolds with the simple homotopy type of X, via a map Ψ from M ( X ) into the quotient of K ( X ) = [ X , B S O
The topology of Stein fillable manifolds in high dimensions II
We give a bordism-theoretic characterisation of those closed almost contact (2q+1)-manifolds (with q > 2) which admit a Stein fillable contact structure. Our method is to apply Eliashberg's
The classification of 2‐connected 7‐manifolds
We present a classification theorem for closed smooth spin 2-connected 7-manifolds M. This builds on the almost-smooth classification from the first author's thesis. The main additional ingredient is
Stable existence of incompressible 3-manifolds in 4-manifolds
Abstract Given a separating embedded connected 3-manifold in a closed 4-manifold, the Seifert–van Kampen theorem implies that the fundamental group of the 4-manifold is an amalgamated product along
Surgery and Singularities in Codimension Two By Yukio MATSUMOT 0
1. Statement of results. Throughout this paper, W /2 denotes a compact connected 1-connected PL m+ 2-manifold which is a Poincar6 complex of formal dimension m. A closed PL submanifold L of W with
Fe b 20 16 LINKING FORMS AND STABILIZATION OF DIFFEOMORPHISM GROUPS OF MANIFOLDS OF DIMENSION 4 n + 1
Let n ≥ 2. We prove a homological stability theorem for the diffeomorphism groups of (4n + 1)-dimensional manifolds, with respect to forming the connected sum with certain (2n − 1)connected,
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 53 REFERENCES
Killing the middle homotopy groups of odd dimensional manifolds
The main object of this paper is to prove the theorem: If W is an mparallelisable (2m+1)-manifold, whose boundary has no homology in dimensions m, m+1; then W is x-equivalent to an m-connected
THE STABLE HOMOTOPY OF THE CLASSICAL GROUPS.
  • R. Bott
  • Mathematics, Medicine
    Proceedings of the National Academy of Sciences of the United States of America
  • 1957
TLDR
The index of s, denoted by X(s), is the properly counted sum of the conjugate points of P in the interior of s which occurs as the index of some geodesic from P to Q in the class h.
BERNOULLI NUMBERS, HOMOTOPY GROUPS, A N D A THEOREM OF ROHLIN
A homomorphism J: 7Tk_1(SOw) -> nm+k_1(S ) from the homotopy groups of rotation groups to the homotopy groups of spheres has been defined by H. Hopf and G. W. Whitehead. This homomorphism plays an
Generalized Poincare's Conjecture in Dimensions Greater Than Four
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
Obstructions to the smoothing of piecewise-differentiable homeomorphisms
Since the publication in 1956 of John Milnor's fundamental paper [l ] in which he constructs differentiable structures on S7 nondiffeomorphic to the standard one, several further results concerning
The generalised Poincaré conjecture
The lemma is useful in a variety of contexts. For the application that we need here, choose A q to be the g-skeleton of M and B to be a point; then a regular neighbourhood of C is an w-ball
On the Structure of Manifolds
In this paper, we prove a number of theorems which give some insight into the structure of differentiable manifolds. The methods, results and some notation of [13], hereafter referred to as GPC, and
RELATIVE CHARACTERISTIC CLASSES.
In the proof, we shall make use of a not quite classical form of Whitney duality, involving Stiefel-Whitney characteristic classes which have to be considered as relative cohomology classes. Since
An Interpretation of G. Whitehead's Generalization of H. Hopf's Invariant
In the present paper', the generalized H. Hopf's invariant H: 7rdwn++(Sn+l) w7rd +n+i(Sqn+ ), due to G. Whitehead [10], is given a new definition which has some similarity with the original H. Hopf's
Topologie de certains espaces de plongements
© Bulletin de la S. M. F., 1961, tous droits réservés. L’accès aux archives de la revue « Bulletin de la S. M. F. » (http://smf. emath.fr/Publications/Bulletin/Presentation.html) implique l’accord
...
1
2
3
4
5
...