In mathematics, especially in algebraic topology, an induced homomorphism is a homomorphism derived in a canonical way from another map. For example, a continuous map from a topological space X to a topological space Y induces a group homomorphism from the fundamental group of X to the fundamental group of Y. More generally, in category theory, any functor by definition provides an induced morphism in the target category for each morphism in the source category. For example, fundamental groups, higher homotopy groups, singular homology, and De Rham cohomology are algebraic structures that are functorial, meaning that their definition provides a functor from (e.g.) the category of topological spaces to (e.g.) the category of groups or rings. This means that each space is associated with an algebraic structure, while each continuous map between spaces is associated with a structure-preserving map between structures, called an induced homomorphism. A homomorphism induced from a map h {\displaystyle h} is often denoted h ∗ {\displaystyle h_{*}} . Induced homomorphisms often inherit properties of the maps they come from; for example, two maps that are inverse to each other up to homotopy induce homomorphisms that are inverse to each other. A common use of induced homomorphisms is the following: by showing that a homomorphism with certain properties cannot exist, one concludes that there cannot exist a continuous map with properties that would induce it. Thanks to this, relations between spaces and continuous maps, often very intricate, can be inferred from relations between the homomorphisms they induce. The latter may be simpler to analyze, since they involve algebraic structures which can be often easily described, compared, and calculated in.
In fundamental groups
Let X {\displaystyle X} and Y {\displaystyle Y} be topological spaces with points x 0 {\displaystyle x_{0}} in X {\displaystyle X} and y 0 {\displaystyle y_{0}} in Y {\displaystyle Y} . Let h : X → Y {\displaystyle h:X\to Y} be a continuous map such that h ( x 0 ) = y 0 {\displaystyle h(x_{0})=y_{0}} . Then we can define a map h ∗ {\displaystyle h_{*}} from the fundamental group π 1 ( X , x 0 ) {\displaystyle \pi _{1}(X,x_{0})} to the fundamental group π 1 ( Y , y 0 ) {\displaystyle \pi _{1}(Y,y_{0})} as follows: any element of π 1 ( X , x 0 ) {\displaystyle \pi _{1}(X,x_{0})} , represented by a loop f {\displaystyle f} in X {\displaystyle X} based at x 0 {\displaystyle x_{0}} , is mapped to the loop in π 1 ( Y , y 0 ) {\displaystyle \pi _{1}(Y,y_{0})} obtained by composing with h {\displaystyle h} :
h ∗ ( [ f ] ) := [ h ∘ f ] {\displaystyle h_{*}([f]):=[h\circ f]}
Here [ f ] {\displaystyle [f]} denotes the equivalence class of f {\displaystyle f} under homotopy, as in the definition of the fundamental group. It is easily checked from the definitions that h ∗ {\displaystyle h_{*}} is a well-defined function π1(X, x0) → π1(Y, y0): loops in the same equivalence class, i.e. homotopic loops in X, are mapped to homotopic loops in Y, because a homotopy can be composed with h as well. It also follows from the definition of the group operation in fundamental groups (namely by concatenation of loops) that h ∗ {\displaystyle h_{*}} is a group homomorphism:
h ∗ ( [ f + g ] ) = h ∗ ( [ f ] ) + h ∗ ( [ g ] ) {\displaystyle h_{*}([f+g])=h_{*}([f])+h_{*}([g])}
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