Homotopy theory MOC

Homotopy of paths

Let 𝛼,𝛽 :𝕀 →𝑋 be paths with common endpoints, i.e. 𝛼(0) =𝛽(0) and 𝛼(1) =𝛽(1). Then a homotopy of paths 𝐻 :𝛼 ≃𝛽 is a homotopy of maps with the additional constraint that the endpoints are the same for all 𝑡, homotopy i.e. 𝐺 :[0,1] ×[0,1] →𝑋 with

𝐻(𝑢,0)=𝛼(𝑢)𝐻(𝑢,1)=𝛽(𝑢)𝐻(0,𝑡)=𝛼(0)=𝛽(0)𝐻(1,𝑡)=𝛼(1)=𝛽(1)

This is equivalent to homotopy relative {0,1}. Homotopy classes of paths are the morphisms of the Fundamental groupoid.

Properties

• Path traversal lemma

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