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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 ${\displaystyle X}$. The van Kampen obstruction is based on a series of observations

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

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

Formulation using deleted products

Formulation using homology

Generalizations

References

External links

• www.example.com