How Does Induced Mapping Function in Algebraic Topology?

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SUMMARY

Induced mapping in Algebraic Topology refers to the functorial nature of mappings between topological spaces. Given a continuous function f: X → Y and groups G/N in X and G'/N' in Y, the induced map i* is defined as i*([a]) = ([f([a])]), which translates homotopy classes from the fundamental group π₁(X) to π₁(Y). This concept is crucial for understanding how properties of spaces are preserved under continuous mappings.

PREREQUISITES
  • Understanding of basic concepts in Algebraic Topology
  • Familiarity with homotopy and homology groups
  • Knowledge of continuous functions between topological spaces
  • Basic grasp of functoriality in category theory
NEXT STEPS
  • Study the properties of functors in category theory
  • Learn about homotopy groups and their applications in Algebraic Topology
  • Explore examples of induced maps in various topological contexts
  • Investigate the relationship between homology and cohomology theories
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Mathematicians, students of Algebraic Topology, and anyone interested in the structural properties of topological spaces and their mappings.

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Given example for what is induced mapping ? In basic level
 
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srgmath2905 said:
Given example for what is induced mapping ? In basic level
Without context this is impossible to answer. The same term is used differently in different contexts.
 
Maybe only commonality is that it's functorial in nature *
*As well as in the wild ;).
Edit: In Algebraic Topology, given f: X-->Y, and groups G/N in X, G'/N' in Y( some (co) homology or Homotopy groups associated to each of X,Y, with N normal in G. N' normal in G), often designated as i* I believe they're of the form:
i*([a])=([f([a])]), from G/N to G'/N'. So a map between cosets.

So, e.g., the homotopy class [a] of ## \pi_1(X)## as above, is sent to the class [f(a)] in ##\pi_1(Y)##.
 
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