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Dold–Thom Theorem
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How does the Dold–Thom Theorem relate to singular homology?
The theorem relates homotopy groups of the infinite symmetric product SP^ fty(X) to the singular homology groups of the underlying space X with integer coefficients.
What role do pointed spaces play in the theorem?
The pointed space notion is crucial because the base point allows the construction of the symmetric product and ensures coherence between the base points of homotopy groups.
Give an example of an application of the Dold–Thom Theorem.
The theorem can be applied to compute the homotopy groups of spheres by considering the infinite symmetric product of a bouquet of circles, which simplifies certain homotopy calculations.
Define the Dold–Thom Theorem.
The Dold–Thom Theorem states that the homotopy groups of an infinite symmetric product of a pointed space are isomorphic to the homology groups of that space.
What is the infinite symmetric product SP^ fty(X)?
SP^ fty(X) is the space obtained by taking all finite subsets of points in X, including multiplicities, and making identifications using the symmetric group action.
What does the Dold–Thom Theorem imply for the homotopy group ?
According to the Dold–Thom Theorem, is isomorphic to the singular homology group for all integers k.
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