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The homomorphism on fundamental groups induced by a pointed continuous map
Definition
Let be continuous and let . Composition sends a loop at to the loop at . Using the loop classes and fundamental group of Based loops and the fundamental group, the proposed induced homomorphism is
The next theorem proves that this value is independent of the representative, that it is a group homomorphism in the sense of Monoid homomorphism and group homomorphism, and that induced maps respect identities, composition and homotopies that fix the basepoint.
Depends on
Used by
- Semilocally simply connected spaces with explicit basepoint convention Definition
- Changing the point over a fixed basepoint conjugates the induced covering subgroup Lemma
- Homotopic-loop factorizations have the same value in the group pushout Lemma
- Loops over a two-set path-connected open cover factor through the covering sets Lemma
- A based morphism between connected coverings exists exactly when the induced subgroups are included Proposition
- The fundamental group is a functor π₁:Top_*toGrp Proposition
- A covering map induces an injective homomorphism on fundamental groups Theorem
- Induced fundamental-group maps are well defined, functorial and invariant under based homotopy Theorem
- Lifting criterion for maps from path-connected locally path-connected spaces Theorem
- Seifert–van Kampen identifies the fundamental group with a group pushout Theorem
- π₁(X× Y,(x₀,y₀))≅π₁(X,x₀)×π₁(Y,y₀) Theorem
Dependency tree · two levels
18 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- A. Hatcher, Algebraic Topology, Chapter 1, Induced Homomorphisms (standard reference, not scraped)