Homotopy Spheres Research Articles

Classical Poincaré Conjecture via 4D Topology

The classical Poincaré conjecture that every homotopy 3-sphere is diffeomorphic to the 3-sphere is confirmed by Perelman in arXiv papers solving Thurston’s program on geometrizations of 3-manifolds. A new confirmation of this conjecture is given by a method of 4D topology. For this proof, the spun torus-knot of every knot in every homotopy 3-sphere is observed to be a ribbon torus-knot in the 4-sphere, where Smooth 4D Poincaré Conjecture and Ribbonness of a spherelink with (not necessarily meridian-based) free fundamental group are used. By examining a disk-chord system of a ribbon solid torus bounded by the spun torus-knot, it is proved that the knot belongs to a 3-ball in the homotopy 3-sphere. Then by Bing’s result, it is confirmed that the homotopy 3-sphere is diffeomorphic to the 3-sphere.

Families of relatively exact Lagrangians, free loop spaces and generalised homology

AbstractWe prove that (under appropriate orientation conditions, depending on R) a Hamiltonian isotopy $$\psi ^1$$ ψ 1 of a symplectic manifold $$(M, \omega )$$ ( M , ω ) fixing a relatively exact Lagrangian L setwise must act trivially on $$R_*(L)$$ R ∗ ( L ) , where $$R_*$$ R ∗ is some generalised homology theory. We use a strategy inspired by that of Hu et al. (Geom Topol 15:1617–1650, 2011), who proved an analogous result over $${\mathbb {Z}}/2$$ Z / 2 and over $${\mathbb {Z}}$$ Z under stronger orientation assumptions. However the differences in our approaches let us deduce that if L is a homotopy sphere, $$\psi ^1|_L$$ ψ 1 | L is homotopic to the identity. Our technical set-up differs from both theirs and that of Cohen et al. (in: Algebraic topology, Springer, Berlin, 2019) and Cohen (in: The Floer memorial volume, Birkhäuser, Basel). We also prove (under similar conditions) that $$\psi ^1|_L$$ ψ 1 | L acts trivially on $$R_*({\mathcal {L}}L)$$ R ∗ ( L L ) , where $${\mathcal {L}}L$$ L L is the free loop space of L. From this we deduce that when L is a surface or a $$K(\pi , 1)$$ K ( π , 1 ) , $$\psi ^1|_L$$ ψ 1 | L is homotopic to the identity. Using methods of Lalonde and McDuff (Topology 42:309–347, 2003), we also show that given a family of Lagrangians all of which are Hamiltonian isotopic to L over a sphere or a torus, the associated fibre bundle cohomologically splits over $${\mathbb {Z}}/2$$ Z / 2 .

Trace embeddings from zero surgery homeomorphisms

AbstractManolescu and Piccirillo (2023) recently initiated a program to construct an exotic or by using zero surgery homeomorphisms and Rasmussen's ‐invariant. They find five knots that if any were slice, one could construct an exotic and disprove the Smooth 4‐dimensional Poincaré conjecture. We rule out this exciting possibility and show that these knots are not slice. To do this, we use a zero surgery homeomorphism to relate slice properties of two knots stably after a connected sum with some 4‐manifold. Furthermore, we show that our techniques will extend to the entire infinite family of zero surgery homeomorphisms constructed by Manolescu and Piccirillo. However, our methods do not completely rule out the possibility of constructing an exotic or as Manolescu and Piccirillo proposed. We explain the limits of these methods hoping this will inform and invite new attempts to construct an exotic or . We also show that a family of homotopy spheres constructed by Manolescu and Piccirillo using annulus twists of a ribbon knot are all standard.

Inertia groups and smooth structures on quaternionic projective spaces

Abstract This paper deals with certain results on the number of smooth structures on quaternionic projective spaces, obtained through the computation of inertia groups and their analogues, which in turn are computed using techniques from stable homotopy theory. We show that the concordance inertia group is trivial in dimension 20, but there are many examples in high dimensions where the concordance inertia group is non-trivial. We extend these to computations of concordance classes of smooth structures. These have applications to 3-sphere actions on homotopy spheres and tangential homotopy structures.

Intermediate curvatures and highly connected manifolds

We show that after forming a connected sum with a homotopy sphere, all (2j-1)-connected 2j-parallelisable manifolds in dimension 4j+1, j > 0, can be equipped with Riemannian metrics of 2-positive Ricci curvature. The condition of 2-positive Ricci curvature is defined to mean that the sum of the two smallest eigenvalues of the Ricci tensor is positive at every point. This result is a counterpart to a previous result of the authors concerning the existence of positive Ricci curvature on highly connected manifolds in dimensions 4j-1 for j > 1, and in dimensions 4j+1 for j > 0 with torsion-free cohomology.

Caustics of Lagrangian homotopy spheres with stably trivial Gauss map

For each positive integer $n$, we give a geometric description of the stably trivial elements of the group $\pi_n U_n/O_n$. In particular, we show that all such elements admit representatives whose tangencies with respect to a fixed Lagrangian plane consist only of folds. By the h-principle for the simplification of caustics, this has the following consequence: if a Lagrangian distribution is stably trivial from the viewpoint of a Lagrangian homotopy sphere, then by an ambient Hamiltonian isotopy one may deform the Lagrangian homotopy sphere so that its tangencies with respect to the Lagrangian distribution are only of fold type. Thus the stable triviality of the Lagrangian distribution, which is a necessary condition for the simplification of caustics to be possible, is also sufficient. We give applications of this result to the arborealization program and to the study of nearby Lagrangian homotopy spheres.

Stable homotopy groups of spheres

We discuss the current state of knowledge of stable homotopy groups of spheres. We describe a computational method using motivic homotopy theory, viewed as a deformation of classical homotopy theory. This yields a streamlined computation of the first 61 stable homotopy groups and gives information about the stable homotopy groups in dimensions 62 through 90. As an application, we determine the groups of homotopy spheres that classify smooth structures on spheres through dimension 90, except for dimension 4. The method relies more heavily on machine computations than previous methods and is therefore less prone to error. The main mathematical tool is the Adams spectral sequence.

Moduli Spaces of Metrics of Positive Scalar Curvature on Topological Spherical Space Forms

AbstractLet $M$ be a topological spherical space form, i.e., a smooth manifold whose universal cover is a homotopy sphere. We determine the number of path components of the space and moduli space of Riemannian metrics with positive scalar curvature on $M$ if the dimension of $M$ is at least 5 and $M$ is not simply-connected.

Face numbers of Engström representations of matroids

A classic problem in the theory of matroids is to find a subspace arrangement, such as a hyperplane or pseudosphere arrangement, whose intersection poset is isomorphic to a prescribed geometric lattice. Engström gave an explicit construction for an infinite family of such arrangements, indexed by the set of finite regular CW complexes. In this paper, we compute the face numbers of these topological representations in terms of the face numbers of the indexing complexes and give upper bounds on the total number of faces in these objects. Moreover, for a fixed rank, we show that the total number of faces in the Engström representation corresponding to a codimension one homotopy sphere arrangement is bounded above by a polynomial in the number of elements of the matroid, whose degree is one less than the matroid’s rank.

On characteristic classes of exotic manifold bundles

Given a closed simply connected manifold M of dimension 2nge 6, we compare the ring of characteristic classes of smooth oriented bundles with fibre M to the analogous ring resulting from replacing M by the connected sum Msharp Sigma with an exotic sphere Sigma . We show that, after inverting the order of Sigma in the group of homotopy spheres, the two rings in question are isomorphic in a range of degrees. Furthermore, we construct infinite families of examples witnessing that inverting the order of Sigma is necessary.

Trisections of 4-manifolds

The study of n-dimensional manifolds has seen great advances in the last half century. In dimensions greater than four, surgery theory has reduced classification to homotopy theory except when the fundamental group is nontrivial, where serious algebraic issues remain. In dimension 3, the proof by Perelman of Thurston’s Geometrization Conjecture (1) allows an algorithmic classification of 3-manifolds. The work of Freedman (2) classifies topological 4-manifolds if the fundamental group is not too large. Also, gauge theory in the hands of Donaldson (3) has provided invariants leading to proofs that some topological 4-manifolds have no smooth structure, that many compact 4-manifolds have countably many smooth structures, and that many noncompact 4-manifolds, in particular 4D Euclidean space R 4 , have uncountably many. However, the gauge theory invariants run into trouble with small 4-manifolds, such as those with the same homology groups as the 4D sphere, S 4 . In particular, the smooth 4D Poincare Conjecture, the last remaining case of that hallowed conjecture, is still open. (In higher dimensions, the smooth Poincare Conjecture is sometimes true in the following sense. In dimensions 3, 5, 6, 12, and 61, a homotopy sphere is diffeomorphic to the standard one, and in all other known cases, there are increasingly many exotic smooth structures on the topological sphere; however, it is possible that there may be more high-dimensional cases with no exotic spheres.) The gauge theory invariants are very good at distinguishing smooth 4-manifolds that are homotopy equivalent but do not help at showing that they are diffeomorphic. What is missing is the equivalent of the higher-dimensional s-cobordism theorem, a key to the successes in higher dimensions. The s-cobordism theorem states that, if M 0 m and M 1 m are the two boundary components of an m + 1 -dimensional manifold W and if … [↵][1]1Email: kirby{at}math.berkeley.edu. [1]: #xref-corresp-1-1

On the topology of CR-warped product submanifolds

We obtain a necessary condition for homology group to be zero on CR-warped product submanifold in Euclidean spaces in terms of second fundamental form of the submanifold and warping function. By using this condition, we show that such CR-warped product submanifold is a homotopy sphere.

Standard special generic maps of homotopy spheres into Euclidean spaces

A so-called special generic map is by definition a map between smooth manifolds all of whose singularities are definite fold points. It is in general an open problem posed by Saeki in 1993 to determine the set of integers p for which a given homotopy sphere admits a special generic map into Rp.By means of the technique of Stein factorization we introduce and study certain special generic maps of homotopy spheres into Euclidean spaces called standard. Modifying a construction due to Weiss, we show that standard special generic maps give naturally rise to a filtration of the group of homotopy spheres by subgroups that is strongly related to the Gromoll filtration. Finally, we apply our result to some concrete homotopy spheres, which in particular answers Saeki's problem for the Milnor 7-sphere.

Positive Ricci curvature on highly connected manifolds

For $k \geq 2$, let $M^{4k-1}$ be a closed $(2k-2)$-connected manifold. If $k \equiv 1 \mod 4$ assume further that $M$ is $(2k-1)$-parallelisable. Then there is a homotopy sphere $\Sigma^{4k-1}$ such that $M \sharp \Sigma$ admits a Ricci positive metric. This follows from a new description of these manifolds as the boundaries of explicit plumbings.

Representation rings for fusion systems and dimension functions

We define the representation ring of a saturated fusion system \(\mathcal F\) as the Grothendieck ring of the semiring of \(\mathcal F\)-stable representations, and study the dimension functions of \(\mathcal F\)-stable representations using the transfer map induced by the characteristic idempotent of \(\mathcal F\). We find a list of conditions for an \(\mathcal F\)-stable super class function to be realized as the dimension function of an \(\mathcal F\)-stable virtual representation. We also give an application of our results to constructions of finite group actions on homotopy spheres.

Quotient manifolds of flows

This paper investigates which smooth manifolds arise as quotients (orbit spaces) of flows of vector fields. Such quotient maps were already known to be surjective on fundamental groups, but this paper shows that every epimorphism of countably presented groups is induced by the quotient map of some flow, and that higher homology can also be controlled. Manifolds of fixed dimension arising as quotients of flows on Euclidean space realize all even (and some odd) intersection pairings, and all homotopy spheres of dimension at least two arise in this manner. Most Euclidean spaces of dimensions five and higher have families of topologically equivalent but smoothly inequivalent flows with quotient homeomorphic to a manifold with flexibly chosen homology. For [Formula: see text], there is a topological flow on (ℝ2r+1 − 8 points) × ℝm that is unsmoothable, although smoothable near each orbit, with quotient an unsmoothable topological manifold.

Free and properly discontinuous actions of groups $$G\rtimes {\mathbb {Z}}^m$$ G ⋊ Z m and $$G_1*_{G_0}G_2$$ G 1 ∗ G 0 G 2

We estimate the number of homotopy types of orbit spaces for all free and properly discontinuous cellular actions of groups \(G\rtimes {\mathbb {Z}}^m\) and \(G_1*_{G_0}G_2\). In particular, homotopy types of orbits of \((2n-1)\)-spheres \(\Sigma (2n-1)\) for such actions are analysed, provided the groups \(G_0, G_1, G_2\) and G are finite and periodic. This family of groups \(G\rtimes {\mathbb {Z}}^m\) and \(G_1*_{G_0}G_2\) contains properly the family of virtually cyclic groups. The possible actions of those groups on the top cohomology of the homotopy sphere are determined as well.

Isoparametric foliations, diffeomorphism groups and exotic smooth structures

In this paper, we are concerned with interactions between isoparametric theory and differential topology. Two foliations are called equivalent if there exists a diffeomorphism between the foliated manifolds mapping leaves to leaves. Using differential topology, we obtain several results towards the classification problem of isoparametric foliations up to equivalence. In particular, we show that each homotopy n-sphere has the “same” isoparametric foliations as the standard sphere Sn has except for n=4, reducing the classification problem on homotopy spheres to that on the standard sphere. Moreover, we prove the uniqueness up to equivalence of isoparametric foliations with two points as the focal submanifolds on each sphere Sn except for n=5. Besides, we show that the uniqueness holds on S5 if and only if π0(Diff(S4))≃Z2, i.e., pseudo-isotopy implies isotopy for diffeomorphisms on S4. At last, some ideas behind the proofs enable us to discover new exotic smooth structures on certain manifolds.

Richness of Smith equivalent modules for finite gap Oliver groups

Let $G$ be a finite group not of prime power order. Two real $G$-modules $U$ and $V$ are $\mathcal{P}(G)$-connectively Smith equivalent if there exists a homotopy sphere with smooth $G$-action such that the fixed point set by $P$ is connected for all Sylow subgroups $P$ of $G$, it has just two fixed points, and $U$ and $V$ are isomorphic to the tangential representations as real $G$-modules respectively. We study the $\mathcal{P}(G)$-connective Smith set for a finite Oliver group $G$ of the real representation ring consisting of all differences of $\mathcal{P}(G)$-connectively Smith equivalent $G$-modules, and determine this set for certain nonsolvable groups $G$.

INERTIA GROUPS AND SMOOTH STRUCTURES OF (n-1)-CONNECTED 2n-MANIFOLDS

Let $M^{2n}$ denote a closed ($n-1$)-connected smoothable topological $2n$-manifold. We show that the group $\mathcal{C}(M^{2n})$ of concordance classes of smoothings of $M^{2n}$ is isomorphic to the group of smooth homotopy spheres $\bar{\Theta}_{2n}$ for $n=4$ or $5$, the concordance inertia group $I_{c}(M^{2n})=0$ for $n=3$, $4$, $5$ or $11$ and the homotopy inertia group $I_{h}(M^{2n})=0$ for $n=4$. On the way, following Wall's approach [16] we present a new proof of the main result in [9], namely, for $n=4$, $8$ and $H^{n}(M^{2n};\mathbb{Z})\cong \mathbb{Z}$, the inertia group $I(M^{2n})\cong \mathbb{Z}_{2}$. We also show that, up to orientation-preserving diffeomorphism, $M^{8}$ has at most two distinct smooth structures; $M^{10}$ has exactly six distinct smooth structures and then show that if $M^{14}$ is a $\pi$-manifold, $M^{14}$ has exactly two distinct smooth structures.