User:MWinter4/van Kampen obstruction

In topology the van Kampen obstruction is a computationally checkable obstruction to the embeddability of a 2-dimensional CW complex into 4-dimensional Euclidean space.

Fundamental idea

Given a 2-dimensional CW complex $X$. The van Kampen obstruction is based on a series of observations

• Any two embeddings of $X$ can be transformed into each other by so-called finger moves. A finger move moves an edge "across" a 2-cell.
• Let ${\mathcal {C}}$ be the set of pairs $(c_{1},c_{2})$, where $c_{1},c_{2}\subseteq X$ are disjoint 2-cells.
• The intersection vector $v(\phi )\in {\mathbb {Z}}_{2}^{\mathcal {C}}$ of a mapping $\phi :X\to {\mathbb {R}}^{4}$ records whether the two 2-cells in a pair intersect (and we can assume that all such intersections are transversal). The intersection vector of an embedding is zero.
• For any edge $e\subseteq X$ and 2-cell $c\subseteq X$, applying a finger move that pulls $e$ across $c$ changes the intersection vector in a way that only depends on $e$ and $c$, but not their embeddings. More precisely, $v(\phi ')=v(\phi )+\Delta _{e,c}$.

Suppose we are given a mapping $\phi$. If there also exists an embedding $\psi$, then there exists a sequence of finger moves transforming $\phi$ into $\psi$. This means that $v(\phi )$ can be written as a linear combination of $\Delta _{e,c}$.

External links

• www.example.com