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User:MWinter4/van Kampen obstruction

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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

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

  • Any two embeddings of can be transformed into each other by so-called finger moves. A finger move moves an edge "across" a 2-cell.
  • Let be the set of pairs , where are disjoint 2-cells.
  • The intersection vector of a mapping 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 and 2-cell , applying a finger move that pulls across changes the intersection vector in a way that only depends on and , but not their embeddings. More precisely, .

Suppose we are given a mapping . If there also exists an embedding , then there exists a sequence of finger moves transforming into . This means that can be written as a linear combination of .

Formulation using deleted products

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Formulation using homology

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Generalizations

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References

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