Strong surjectivity of mappings of some 3-complexes into 3-manifolds
Volume 192 / 2006
Fundamenta Mathematicae 192 (2006), 195-214
MSC: Primary 55M20;
Secondary 55S35, 55N25.
DOI: 10.4064/fm192-3-1
Abstract
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 ${\rm MR}[f,a]\neq 0$.
