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A pair of maps whose composites are homotopic to the corresponding identity maps.
wp-homotopy-equivalencenlab-homotopy-equivalencehatcher-algebraic-topologyEach relation below ends at this concept.
No direct relations are authored in this direction.
Each relation below starts at this concept.
This is an authored directed relation from the source endpoint to the target endpoint.
Homotopy-equivalent spaces have isomorphic singular cohomology groups.
Record a genuine theorem implication that is not part of the target definition; these edges may point toward a weaker structure.
hatcher-algebraic-topologyThis is an authored directed relation from the source endpoint to the target endpoint.
Homotopy-equivalent spaces have isomorphic singular homology groups.
Record a genuine theorem implication that is not part of the target definition; these edges may point toward a weaker structure.
hatcher-algebraic-topologyThis is an authored directed relation from the source endpoint to the target endpoint.
A homotopy equivalence consists of maps whose composites are homotopic to the relevant identity maps; retaining a witnessing deformation gives homotopy data.
Pass canonically from a stronger object to structure it determines, or forget part of the data while retaining a valid weaker structure. The carrier may change under a canonical induced construction.
hatcher-algebraic-topologyThis is an authored directed relation from the source endpoint to the target endpoint.
A homotopy equivalence is the relation-generating data used to classify spaces by homotopy type; the construction retains invariants such as homotopy and homology groups.
A mathematical concept supplies part of the formal language, state space, representation, or analytic machinery used by a scientific or mathematical-physics concept. The source is the mathematical predecessor; this does not claim that physical content follows from mathematics alone.
hatcher-algebraic-topology