Homotopy of maps
A homotopy is a continuous transformation from one continuous map into another.
Let
The homotopy relation
Proof
Clearly
is homotopic to itself via π β π³ π π ( π , π ) , so β ( π₯ , π‘ ) = π ( π₯ ) is reflexive. If β is a homotopy from β : π Γ [ 0 , 1 ] β π to π then π is a homotopy from β β² : ( π₯ , π‘ ) β¦ β ( π₯ , 1 β π‘ ) to π , so π is symmetric. If β is a homotopy from β to π and π is a homotopy from β β² to π , then π β β² β β = { β ( 2 π‘ ) 0 β€ π‘ β€ 1 2 β β² ( 2 π‘ β 1 ) 1 2 β€ π‘ β€ 1 is a homotopy from
to π , so π is transitive. Therefore β is an equivalence relation. To show β is a congruence relation, let β with π 1 , π 2 : π β π and β 1 : π 1 β π 2 with π 1 , π 2 : π β π . Then β 2 : π 1 β π 2 , and similarly π 2 β 1 : π 2 π 1 β π 2 π 2 . Thus β 2 ( π ( β ) , β ) : π 1 π 1 β π 2 π 1 , as required. π 1 π 1 β π 2 π 2
Homotopy class
The congruence classes of homotopic maps are called homotopy classes of maps,
and form the morphisms in the NaΓ―ve homotopy category
Other kinds of topological homotopy
Further terminology
- A map
π