Geometric formulas for Smale invariants of codimension two immersions

Cobordism of fold maps, stably framed manifolds and immersions

We give complete geometric invariants of cobordisms of fold maps with oriented singular set and cobordisms of even codimensional fold maps. These invariants are given in terms of cobordisms of stably framed manifolds and cobordisms of immersions with prescribed normal bundles defined by the author in his earlier works.

投影浸入的配边群上的计算

Computations on cobordism groups of projected immersions

本文通过 Szűcs 和 Terpai 的分类空间构造推广 Salomonsen 正合序列，研究具有指定奇点的超平面投影浸入的配边群，并在小余维和大余维情形下给出计算结果。

链环合流蕴含链环同伦

Link concordance implies link homotopy

本文证明了余维数至少为二时链环合流蕴含链环同伦，通过扩展分层Morse理论与Cerf理论并引入手指移动消除障碍，证明了高维链环的同伦平凡性。

The Hopf invariant of a Haefliger knot

We show that Haefliger's differentiable (6,3)-knot bounds, in 6-space, a 4-manifold (a Seifert surface) of arbitrarily prescribed signature. This implies, according to our previous paper, that the Seifert surface has been prolonged in a prescribed direction near its boundary. This aspect enables us to understand a resemblance between Ekholm-Szucs' formula for the Smale invariant and Boechat-Haefliger's formula for Haefliger knots. As a consequence, we show that an immersion of the 3-sphere in 5-space can be regularly homotoped to the projection of an embedding in 6-space if and only if its Sma…

The geometric Hopf invariant and double points

The geometric Hopf invariant of a stable map F is a stable Z_2-equivariant map h(F) such that the stable Z_2-equivariant homotopy class of h(F) is the primary obstruction to F being homotopic to an unstable map. In this paper we express the geometric Hopf invariant of the Umkehr map F of an immersion f:M^m \to N^n in terms of the double point set of f. We interpret the Smale-Hirsch-Haefliger regular homotopy classification of immersions f in the metastable dimension range 3m<2n-1 (when a generic f has no triple points) in terms of the geometric Hopf invariant.

Geometric approach towards stable homotopy groups of spheres. The Steenrod-Hopf invariant I

In this paper a geometric approach toward stable homotopy groups of spheres, based on the Pontrjagin-Thom construction is proposed. From this approach a new proof of Hopf Invariant One Theorem by J.F.Adams for all dimensions except $15,31,63,127$ is obtained. It is proved that for $n>127$ in the stable homotopy group of spheres $Π_n$ there is no elements with Hopf invariant one. The new proof is based on geometric topology methods. The Pontrjagin-Thom Theorem (in the form proposed by R.Wells) about the representation of stable homotopy groups of the real projective infinite-dimensional space (…

How to Fold a Manifold

Techniques for constructing codimension 2 embeddings and immersions of the 2 and 3-fold branched covers of the 3 and 4-dimensional spheres are presented. These covers are in braided form, and it is in this sense that they are folded. More precisely the composition of the embedding (or immersion) and the canonical projection induces the branched covering map. In the case of the 3-sphere, the branch locus is a knotted or linked curve in space, the 2-fold branched cover always embeds, and the 3-fold branch cover might be immersed. In the case of the 4-sphere, the branch locus is a knotted or link…

Cobordism invariants of fold maps

This is a survey paper of author's results on cobordism groups and semigroups of fold maps and simple fold maps. The results include: establishing a relation between fold maps and immersions through geometrical invariants of cobordism classes of fold maps and simple fold maps in terms of immersions with prescribed normal bundles, detecting stable homotopy groups of spheres as direct summands of the cobordism semigroups of fold maps, Pontryagin-Thom type construction for -1 codimensional fold maps and estimations about the cobordism classes of manifolds which have fold maps into stably parallel…

Embeddings, Normal Invariants and Functor Calculus

This paper investigates the space of codimension zero embeddings of a Poincare duality space in a disk. One of our main results exhibits a tower that interpolates from the space of Poincare immersions to a certain space of "unlinked" Poincare embeddings. The layers of this tower are described in terms of the coefficient spectra of the identity appearing in Goodwillie's homotopy functor calculus. We also answer a question posed to us by Sylvain Cappell. The appendix proposes a conjectural relationship between our tower and the manifold calculus tower for the smooth embedding space.

利用 g-圈（g=2,3）图的构型空间积分不变量检测到的长嵌入空间的若干非平凡循环

Some non-trivial cycles of the space of long embeddings detected by configuration space integral invariants using g-loop (g =2, 3) graphs

本文用改进的 Bott-Cattaneo-Rossi 图复形构型空间积分及2、3-圈带毛图弦图构造并检测长嵌入空间的非平凡循环，进而证明相关有理同伦群的非有限生成性。