Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Some remark on real algebraic maps which are topologically special generic maps and generalize the canonical projections of the unit spheres

Naoki Kitazawa

TL;DR

This work addresses constructing explicit real algebraic maps that realize topologically special generic maps, generalizing the canonical projections of unit spheres $S^m$ into Euclidean spaces. The author develops a reconstruction framework: starting from a product-organized special generic map $f:M\to N$ and a smooth immersion into $N$, one obtains an $m$-dimensional real algebraic manifold $M_0$ and a real algebraic map $f_0:M_0\to \mathbb{R}^n$ satisfying $f_0=\phi\circ f$ for a homeomorphism $\phi$, with $S(f)$ corresponding to $S(f_0)$. The construction proceeds in three steps: isotoping an embedding to create a real-algebraic boundary (Step 1), building a polynomial-based model to define $f_0$ (Step 2), and verifying that the resulting map is topologically product-organized special generic (Step 3); the approach yields explicit, embedded-image maps bridging real algebraic geometry and differential topology. The results extend previous work on explicit real algebraic realizations of special generic maps and illuminate how algebraic data can realize and classify singular map structures. Overall, the paper provides a concrete method for producing real algebraic models of special generic maps and highlights potential connections to Reeb graphs and Hilbert-type questions in real algebraic geometry.

Abstract

Morse functions with exactly two singular points on homotopy spheres and canonical projections of spheres are generalized as special generic maps. A special generic map is, roughly, a smooth map represented as the composition of a smooth surjection onto a manifold whose preimages are diffeomorphic to a unit sphere in the interior of the manifold and single point sets on the boundary with a smooth immersion of codimension $0$. This paper constructs real algebraic maps topologically special generic maps whose images are smoothly embedded manifolds. We are also interested in construction of explicit and meaningful smooth maps in differential topology and recently ones in real algebraic geometry. This has been an important and difficult problem. In such stories, we have previously constructed real algebraic maps topologically regarded as special generic maps. This paper is a kind of additional short remark on such maps.

Some remark on real algebraic maps which are topologically special generic maps and generalize the canonical projections of the unit spheres

TL;DR

This work addresses constructing explicit real algebraic maps that realize topologically special generic maps, generalizing the canonical projections of unit spheres into Euclidean spaces. The author develops a reconstruction framework: starting from a product-organized special generic map and a smooth immersion into , one obtains an -dimensional real algebraic manifold and a real algebraic map satisfying for a homeomorphism , with corresponding to . The construction proceeds in three steps: isotoping an embedding to create a real-algebraic boundary (Step 1), building a polynomial-based model to define (Step 2), and verifying that the resulting map is topologically product-organized special generic (Step 3); the approach yields explicit, embedded-image maps bridging real algebraic geometry and differential topology. The results extend previous work on explicit real algebraic realizations of special generic maps and illuminate how algebraic data can realize and classify singular map structures. Overall, the paper provides a concrete method for producing real algebraic models of special generic maps and highlights potential connections to Reeb graphs and Hilbert-type questions in real algebraic geometry.

Abstract

Morse functions with exactly two singular points on homotopy spheres and canonical projections of spheres are generalized as special generic maps. A special generic map is, roughly, a smooth map represented as the composition of a smooth surjection onto a manifold whose preimages are diffeomorphic to a unit sphere in the interior of the manifold and single point sets on the boundary with a smooth immersion of codimension . This paper constructs real algebraic maps topologically special generic maps whose images are smoothly embedded manifolds. We are also interested in construction of explicit and meaningful smooth maps in differential topology and recently ones in real algebraic geometry. This has been an important and difficult problem. In such stories, we have previously constructed real algebraic maps topologically regarded as special generic maps. This paper is a kind of additional short remark on such maps.
Paper Structure (6 sections, 3 theorems, 5 equations)

This paper contains 6 sections, 3 theorems, 5 equations.

Table of Contents

  1. Introduction.
  2. Smooth manifolds and maps.
  3. Our main result and the structure of our paper.
  4. On Main Theorems.
  5. Additional comments on our new result.
  6. Acknowledgement.

Key Result

Proposition 1

Let $m>n \geq 1$ be integers. For a special generic map $f:M \rightarrow N$ from an $m$-dimensional closed manifold $M$ into an $n$-dimensional smooth manifold $N$ with no boundary, we have a decomposition $f=\bar{f} \circ q_f$ as follows. Note that the case $N:={\mathbb{R}}^n$ is mainly discussed i

Theorems & Definitions (5)

  • Proposition 1: saeki2
  • Proposition 2: saeki2
  • Proposition 3
  • proof : A proof of Main Theorems \ref{['mthm:1']} and \ref{['mthm:2']}
  • proof : A proof of Main Theorem \ref{['mthm:3']}