Bordism Groups of Immersions and Classes Represented by Self-Intersections

Fold maps and immersions from the viewpoint of cobordism

We obtain complete geometric invariants of cobordism classes of oriented simple fold maps of (n+1)-dimensional manifolds into an n-dimensional manifold N in terms of immersions with prescribed normal bundles. We compute that this cobordism group of simple fold maps is isomorphic to the direct sum of the (n-1)th stable homotopy group of spheres and the (n-1)th stable homotopy group of the infinite dimensional projective space. By using geometric invariants defined in the author's earlier works, we also describe the natural map of the simple fold cobordism group to the fold cobordism group by na…

带线条页面上的涂鸦与斑点：曲面上穿越流的凸拟包络

Doodles and blobs on a lined page: convex quasi-envelops of traversing flows on surfaces

本文研究圆柱或带状区域中闭曲线和曲面的2-中等浸入配边，计算正则嵌入配边并构造不变量，对定向涂鸦给出近乎完整的阿贝尔群精确序列描述。

Bordism-finiteness and semi-simple group actions

We give bordism-finiteness results for manifolds with semi-simple group action. Consider the class of oriented manifolds which admit a circle action with isolated fixed points such that the action extends to an $S^3$-action with fixed point. We exhibit various subclasses, characterized by an upper bound for the Euler characteristic and properties of the first Pontrjagin class, which contain only finitely many oriented bordism types in any given dimension. Also we show finiteness results for homotopy complex projective spaces and complete intersections with $S^3$-action as above.

Self-intersections of Immersions and Steenrod Operations

We present a formula describing the action of a generalised Steenrod operation of $\Z_2$-type on the cohomology class represented by a proper self-transverse immersion $f\co M\imm X$, in terms of the equivariant double points of $f$ and the characteristic classes of its normal bundle. This generalises a classical result of R.\ Thom: If $α\in H^k(X;\Z_2)$ is the ordinary cohomology class represented by $f\co M\imm X$, then $\mathrm{Sq}^i(α)=f_* w_i(ν_f)$.

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

We develop a geometric approach to stable homotopy groups of spheres in the spirit of the work of Pontrjagin and Rokhlin. A new proof of the Hopf Invariant One Theorem by J.F.Adams is obtained in all dimensions except 15 and 31. To prove that the stable Hopf invariant H: Π_n \to Z/2 vanishes for n>31, we apply methods of geometric topology. The Pontrjagin-Thom construction along with Hirsch's compression lemma identify every α\in Π_n with the framed bordism class of a framed immersion of a closed n-manifold into R^{n+k}, for any given k>0. Its self-intersection M projects to an immersion f: M …

Burnside Condition on Some Intersection Subgroups

In this paper, using the notions graphs, core graphs, immersions and covering maps of graphs, introduced by Stallings in 1983, we prove the Burnside condition for the intersection of subgroups of free groups with Burnside condition.

Singular Seifert surfaces and Smale invariants for a family of 3-sphere immersions

A self-transverse immersion of the 2-sphere into 4-space with algebraic number of self intersection points equal to -n induces an immersion of the circle bundle over the 2-sphere of Euler class 2n into 4-space. Precomposing the circle bundle immersions with their universal covering maps, we get for n>0 immersions g_n of the 3-sphere into 4-space. In this note, we compute the Smale invariants of g_n. The computation is carried out by (partially) resolving the singularities of the natural singular map of the punctured complex projective plane which extends g_n. As an application, we determine th…

Geometric approach towards stable homotopy groups of spheres. The Kervaire invariant II

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 $f:M^{\frac{3n+q}{4}} \looparrowright \R^n$, $0 < q <<n$ (in the $\frac{3n}{4}+ε$-range) admits a geometrical $\Z/2 \oplus \Z/2$--control if the characteristic class of the skew-framing of this immersion admits a retraction of the order $q$, i.e. there exists a mapping $κ_0: M^{\frac{3n+q}{4}} \to \…

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 (…

Linking forms and stabilization of diffeomorphism groups of manifolds of dimension 4n+1

Let $n \geq 2$. We prove a homological stability theorem for the diffeomorphism groups of $(4n+1)$-dimensional manifolds, with respect to forming the connected sum with $(2n-1)$-connected, $(4n+1)$-dimensional manifolds that are stably parallelizable. Our techniques involve the study of the action of the diffeomorphism group of a manifold $M$, on the linking form associated to the homology groups of $M$. In particular, we construct a geometric model for the linking form using the intersections of embedded and immersed $\mathbb{Z}/k$-manifolds. In addition to our main homological stability theo…