# h-cobordism and s-cobordism Theorems: Transfer over Semialgebraic and Nash categories, Uniform bound and Effectiveeness

## TL;DR

This paper establishes semialgebraic and Nash versions of the h-cobordism and s-cobordism theorems, providing uniform complexity bounds and extending their validity over any real closed field, with implications for effectiveness in real algebraic geometry.

## Contribution

It proves semialgebraic and Nash h- and s-cobordism theorems with uniform complexity bounds, extending classical results to broader categories and fields.

## Key findings

- Uniform bounds on complexity of cobordism homeomorphisms/diffeomorphisms.
- Non-recursive nature of the uniform bound in semialgebraic h-cobordism.
- Validity of theorems over any real closed field.

## Abstract

The h-cobordism theorem is a noted theorem in differential and PL topology. A generalization of the h-cobordism theorem for possibly non simply connected manifolds is the so called s-cobordism theorem. In this paper, we prove semialgebraic and Nash versions of these theorems. That is, starting with semialgebraic or Nash cobordism data, we get a semialgebraic homeomorphism (respectively a Nash diffeomorphism). The main tools used are semialgebraic triangulation and Nash approximation. One aspect of the algebraic nature of semialgebraic or Nash objects is that one can measure their complexities. We show h and s-cobordism theorems with a uniform bound on the complexity of the semialgebraic homeomorphism (or Nash diffeomorphism) obtained in terms of the complexity of the cobordism data. The uniform bound of semialgebraic h-cobordism cannot be recursive, which gives another example of non effectiveness in real algebraic geometry see [ABB]. Finally we deduce the validity of the semialgebraic and Nash versions of these theorems over any real closed field.