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 $$v(\phi)\in \Bbb Z_2^{\mathcal C}$$ of a mapping $$\phi:X\to\Bbb R^4$$ 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 $$\Delta_{e,c}$$.