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For example, a map from the unit circle to any space is null-homotopic precisely when it can be continuously extended to a map from the unit disk to that agrees with on the boundary. It follows from these definitions that a space X {\displaystyle X} is contractible if and only if the identity map from X {\displaystyle X} to itself—which is ...
Two maps , are called homotopic relative to A if they are homotopic by a basepoint-preserving homotopy : [,] such that, for each p in and t in [,], the element (,) is in A. Note that ordinary homotopy groups are recovered for the special case in which A = { x 0 } {\displaystyle A=\{x_{0}\}} is the singleton containing the base point.
For example, given a space , for each integer , let be the set of all maps from the n-simplex to . Then the sequence S n X {\displaystyle S_{n}X} of sets is a simplicial set. [ 22 ] Each simplicial set K = { K n } n ≥ 0 {\displaystyle K=\{K_{n}\}_{n\geq 0}} has a naturally associated chain complex and the homology of that chain complex is the ...
Two chain homotopic maps f and g induce the same maps on homology because (f − g) sends cycles to boundaries, which are zero in homology. In particular a homotopy equivalence is a quasi-isomorphism. (The converse is false in general.)
For example, the homotopy pushout encountered above always maps to the ordinary pushout. This map is not typically a weak equivalence, for example the join is not weakly equivalent to the pushout of X 0 ← X 0 × X 1 → X 1 {\displaystyle X_{0}\leftarrow X_{0}\times X_{1}\rightarrow X_{1}} , which is a point.
Low-dimensional examples: A connected map (0-connected map) is one that is onto path components (0th homotopy group); this corresponds to the homotopy fiber being non-empty. A simply connected map (1-connected map) is one that is an isomorphism on path components (0th homotopy group) and onto the fundamental group (1st homotopy group).
Any two knots in 3-space are equivalent by regular homotopy, though not by isotopy. This curve has total curvature 6π, and turning number 3.. The Whitney–Graustein theorem classifies the regular homotopy classes of a circle into the plane; two immersions are regularly homotopic if and only if they have the same turning number – equivalently, total curvature; equivalently, if and only if ...
If (,) has the homotopy extension property, then the simple inclusion map : is a cofibration.. In fact, if : is a cofibration, then is homeomorphic to its image under .This implies that any cofibration can be treated as an inclusion map, and therefore it can be treated as having the homotopy extension property.