arXiv research

New methods show surfaces in HNN extensions have complexity at least their boundary complexity.

Kervaire's sphere-link is equivalent to a ribbon sphere-link, simplifying complex 2-complexes.

Conjecture $\mathbb{Z}$ is a knot theoretical equivalent form of the Kervaire Conjecture. We say that a knot have property $\mathbb{Z}$ if it satisfies Conjecture $\mathbb{Z}$ for that specific knot. In this work, we show that alternating Montesinos knots with three tangles have property $\mathbb{Z}$. We also show that…

The (4k+2)-dimensional Kervaire manifold is a closed, piecewise linear (PL) manifold with Kervaire invariant 1 and the same homology as the product of two (2k+1)-dimensional spheres. We show that a finite group of odd order acts freely on a Kervaire manifold if and only if it acts freely on the corresponding product of…

Proves existence of manifolds with Kervaire invariant one in specific dimensions.

We present Michel Kervaire work on knots in higher dimensions.

We show that Kervaire invariant one elements in the homotopy groups of spheres exist only in dimensions at most 126. By Browder's Theorem, this means that smooth framed manifolds of Kervaire invariant one exist only in dimensions 2, 6, 14, 30, 62, and possibly 126. With the exception of dimension 126 this resolves a lo…

New method classifies spin 4-manifolds using Kervaire-Milnor invariant.

It is proved that there exists an integer $L$ such that a framed manifold of dimension $2^l-2$, $l\le L$ has the trivial Kervaire Invariant.

In this note we study the relative Kervaire semi-characteristic and prove its invariance under cut-and-past operation. Our approach is analytic and follow very closely the method introduced by W. Zhang

Analyzes semi-characteristics on specific manifolds, proving a vanishing theorem.

We prove that no $14$-connected (resp. $30$-connected) stably parallelizable manifold $N^{30}$ (resp. $N^{62}$) of dimension $30$ (resp. $62$) with the Arf-Kervaire invariant 1 can be smoothly embedded into $\mathbb{R}^{36}$ (resp. $\mathbb{R}^{83}$).

The paper finds manifold structures on complex spaces.

Completes classification of high-dimensional manifolds up to exotic sums.

In contrast to the homogeneous case, we show that there are compact cohomogeneity one manifolds, that do not support invariant metrics of non-negative sectional curvature. In fact we exhibit infinite families of such manifolds including the exotic Kervaire spheres. Such examples exist for any codimension of the singula…

Proves surface embedding theorem for 4-manifolds with good fundamental group.

It was proved by Chern, Hirzebruch and Serre that the signature of a fibre bundle is multiplicative if the fundamental group of the base acts trivially on the cohomology ring of the fibre with real coefficients, in which case the signature of the total space equals the product of the signatures of base and fibre. Hambl…

The paper finds infinite families of exotic spheres with free actions.

Explains a 2D color exchange invariant correspondence to 3D linking numbers.

In this paper, we exploit a subtle indeterminacy in the definition of the spherical Kervaire-Milnor invariant which was discovered by R. Stong to construct non-spin 4-manifolds with even intersection form and prescribed signature.

The paper computes inertia groups of certain high-dimensional manifolds.

Here we generalize the Gromoll-Meyer construction of an exotic 7-sphere by producing geometric models of exotic 8, 10 and Kervaire spheres as quotients of sphere bundles over spheres by free isometric actions. We give a geometric application at the end.

One of the important theorems in homotopy theory is the Hilton splitting. In this paper we will construct all the Hilton homomorphisms by geometrical means and prove a family of sharper symmetry relations of linking coefficients which desuspend and generalize the relations of Kervaire, Haefliger and Steer.

We provide a topological procedure to obtain geometric realizations of both classical and `exotic' $G$-manifolds, such as spheres, bundles over spheres and Kervaire manifolds. As an application, we apply the process known as Cheeger deformations to produce new metrics of both positive Ricci and almost non-negative curv…

In superstring theory spin structures are present on both the 2-dimensional worldsheet and 10-dimensional spacetime. We present a new proposal for the B-field in superstring theory and demonstrate its interaction with worldsheet spin structures. Our formulation generalizes to orientifolds, where various twistings appea…

We show that the Hopf elements, the Kervaire classes, and the $\barκ$-family in the stable homotopy groups of spheres are detected by the Hurewicz map from the sphere spectrum to the $C_2$-fixed points of the Real Brown-Peterson spectrum. A subset of these families is detected by the $C_2$-fixed points of Real Johnson-…

Classical Delaunay surfaces are highly symmetric constant mean curvature (CMC) submanifolds of space forms. We prove the existence of Delaunay-type hypersurfaces in a large class of compact manifolds, using the geometry of cohomogeneity one group actions and variational bifurcation techniques. Our construction speciali…

The notion of the geometrical $\Z/2 \oplus \Z/2$--control of self-intersection of a skew-framed immersion and the notion of the $\Z/2 \oplus \Z/4$-structure (the cyclic structure) on the self-intersection manifold of a $\D_4$-framed immersion are introduced. It is shown that a skew-framed immersion $f:M^{\frac{3n+q}{4}…

We show that, for a closed orientable n-manifold, with n not congruent to 3 modulo 4, the existence of a CR-regular embedding into complex (n-1)-space ensures the existence of a totally real embedding into complex n-space. This implies that a closed orientable (4k+1)-manifold with non-vanishing Kervaire semi-characteri…

Using a Toda bracket computation $\langle θ_4, 2, σ^2\rangle$ due to Daniel C. Isaksen [11], we investigate the $45$-stem more thoroughly. We prove that $θ_4^2=0$ using a $4$-fold Toda bracket. By [2], this implies that $θ_5$ exists and there exists a $θ_5$ such that $2θ_5=0$. Based on $θ_4^2=0$, we simplify significan…

This paper extends widely the work in \cite{GT13}. Existence and non-existence results of isoparametric functions on exotic spheres and Eells-Kuiper projective planes are established. In particular, every homotopy $n$-sphere ($n>4$) carries an isoparametric function (with certain metric) with 2 points as the focal set,…

The subtle interplay between local and global charges for topological semimetals exactly parallels that for singular vector fields. Part of this story is the relationship between cohomological semimetal invariants, Euler structures, and ambiguities in the torsion of manifolds. Dually, a topological semimetal can be rep…

We uncover and highlight relations between the M-branes in M-theory and various topological invariants: the Hopf invariant over $\mathbb{Q}$, $\mathbb{Z}$ and $\mathbb{Z}_2$, the Kervaire invariant, the $f$-invariant, and the $ν$-invariant. This requires either a framing or a corner structure. The canonical framing pro…

In this paper we continue to study (`strong') Nielsen coincidence numbers (which were introduced recently for pairs of maps between manifolds of arbitrary dimensions) and the corresponding minimum numbers of coincidence points and pathcomponents. We explore compatibilities with fibrations and, more specifically, with c…

Program connects birational invariants with G-equivariant ones using Gromov-Witten theory.

New 4-manifold examples show necessary conditions for 4D Light Bulb Theorem.

The purpose of this paper is two-fold: On the one side we would like to close a gap on the classification of vector bundles over $5$-manifolds. Therefore it will be necessary to study quaternionic line bundles over $5$-manifolds which are in $1-1$ correspondence to elements in the first cohomotopy group $π^4(M)=[M,S^4]…

We call a Morse function $f$ on a closed manifold $k$-constrained if neither $f$ nor $-f$ has critical points of indefinite Morse index $< k$. In this paper we study bordism groups of $k$-constrained Morse functions, and thus interpolate between the case $k = 1$ of bordism groups of Morse functions (computed by Ikegami…

Based on the model of the space $Pol_3(n)$ of polygons in $R^3$ with limited number of vertex, which was proposed by Jean-Claude Hausmann and Allen Knutson, and developed by several authors: Jason Cantarella, Alexander Y. Grosberg, Robert Kusner, and Clayton Shonkwiler, we prove that there exists an isometric isotopy o…

The paper introduces a new invariant for cobordism classes of manifolds and extends cobordism groups.

This thesis is concerned with the residues modulo 4 and 8 of the signature of a 4k-dimensional oriented geometric Poincare complex. The Z_8-valued Brown-Kervaire invariant of Z_4-valued quadratic forms is used to prove that if the signature is divisible by 4, the divisibility by 8 is detected by the Arf invariant of a …

We prove that the 2-primary $π_{61}$ is zero. As a consequence, the Kervaire invariant element $θ_5$ is contained in the strictly defined 4-fold Toda bracket $\langle 2, θ_4, θ_4, 2\rangle$. Our result has a geometric corollary: the 61-sphere has a unique smooth structure and it is the last odd dimensional case - the o…

I. Hambleton, A. Korzeniewski and A. Ranicki proved that the signature of a fibre bundle of closed, connected, compatibly oriented PL manifolds is always multiplicative modulo 4. In this paper, we consider the Hirzebruch $χ_y$-genera for odd integers $y$ for a smooth fiber bundle such that the base, fibre, and total sp…

This article establishes the algebraic covering theory of quandles. For every connected quandle we explicitly construct a universal covering, which in turn leads us to define the algebraic fundamental group as the automorphism group of the universal covering. We then establish the Galois correspondence between connecte…

Relative Thom polynomials for maps around boundaries established.

This is the beginning of an obstruction theory for deciding whether a map f:S^2 --> X^4 is homotopic to a topologically flat embedding, in the presence of fundamental group and in the absence of dual spheres. The first obstruction is Wall's self-intersection number mu(f) which tells the whole story in higher dimensions…

Polyhedra volume conjecture supports Stoker conjecture weakly.

Survey on two non-Kähler geometry conjectures.