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Fundamental Class
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How does the fundamental class relate to the orientation of a manifold?
The fundamental class is closely linked to the orientation of a manifold; a manifold with a fundamental class is orientable, and the choice of fundamental class corresponds to a choice of orientation.
In what homology group does the fundamental class of an -manifold live?
For an n-dimensional manifold, the fundamental class lives in the group, where is the dimension of the manifold.
How is the fundamental class used in intersection theory?
In intersection theory, the fundamental class allows for the computation of intersection numbers by evaluating the intersection pairing on the manifold's homology classes.
What is a fundamental class in the context of algebraic topology?
The fundamental class is a homology class that represents a topological space in the highest non-vanishing homology group, typically associated with oriented, closed manifolds.
What role does the fundamental class play in Poincaré duality?
In Poincaré duality, the fundamental class is used to establish an isomorphism between homology and cohomology, turning cycles into cocycles and vice versa.
Can a non-orientable manifold have a fundamental class?
Non-orientable manifolds do not have a well-defined fundamental class in their top singular homology group.
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