Strong surjectivity of mappings of some 3-complexes into 3-manifolds

Claudemir Aniz¹

¹ Departamento de Matemática Universidade Federal de Mato Grosso do Sul - UFMS Caixa Postal 549 79070-900/Unidade V/Campo Grande, MS, Brasil

Fundamenta Mathematicae, Tome 192 (2006) no. 3, pp. 195-214

Résumé

Let $K$ be a $CW$-complex of dimension 3 such that $H^3(K;\mathbb Z)=0$, and $M$ a closed manifold of dimension~3 with a base point $a\in M$. We study the problem of existence of a map $f:K \to M$ which is strongly surjective, i.e. such that ${\rm MR} [f,a]\neq 0$. In particular if $M=S^1\times S^2$ we show that there is no $f:K \to S^1\times S^2$ which is strongly surjective. On the other hand, for $M$ the non-orientable $S^1$-bundle over $S^2$ there exists a complex $K$ and $f:K \to M$ such that ${\rm MR}[f,a]\neq 0$.

Détail

DOI : 10.4064/fm192-3-1

Keywords: cw complex dimension mathbb closed manifold dimension base point study problem existence map which strongly surjective neq particular times there times which strongly surjective other non orientable bundle there exists complex neq