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Note that the condition of being a strong deformation retract is strictly stronger than being a deformation retract. For instance, let X be the subspace of R 2 {\displaystyle \mathbb {R} ^{2}} consisting of closed line segments connecting the origin and the point ( 1 / n , 1 ) {\displaystyle (1/n,1)} for n a positive integer, together with the ...
Given a map :, the mapping cylinder is a space , together with a cofibration ~: and a surjective homotopy equivalence (indeed, Y is a deformation retract of ), such that the composition equals f. Thus the space Y gets replaced with a homotopy equivalent space M f {\displaystyle M_{f}} , and the map f with a lifted map f ~ {\displaystyle {\tilde ...
Inclusion maps are seen in algebraic topology where if is a strong deformation retract of , the inclusion map yields an isomorphism between all homotopy groups (that is, it is a homotopy equivalence). Inclusion maps in geometry come in different kinds: for example embeddings of submanifolds.
It is called a strong deformation retract if, in addition, satisfies the requirement that | is the identity. For example, a homotopy h t : B → B , x ↦ ( 1 − t ) x {\displaystyle h_{t}:B\to B,\,x\mapsto (1-t)x} exhibits that the origin is a strong deformation retract of an open ball B centered at the origin.
Indeed, since U is a tubular neighborhood, there is a smooth strong deformation retract from U to V; i.e., there is a smooth homotopy : from the projection to the identity such that is the identity on V. Then we have the homotopy formula on U:
Since the union of any + faces of + is a strong deformation retract of +, any continuous function defined on these faces can be extended to +, which shows that () is a Kan complex. [ 5 ] Relation with geometric realization
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4. If A is a strong deformation retract of a topological space X, then the inclusion map from A to X induces an isomorphism between fundamental groups (so the fundamental group of X can be described using only loops in the subspace A).