Topology (specifically homotopy) question

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SUMMARY

This discussion focuses on two topology problems related to homotopy. The first problem establishes that any continuous map \( e: X \to \mathbb{R}^n \) is homotopic to a constant map, leveraging the contractibility of \( \mathbb{R}^n \). The second problem demonstrates that if \( f: X \to S^n \) is not surjective, it is also homotopic to a constant map, utilizing the result from the first problem. Both solutions emphasize the concept of homotopy and the properties of contractible spaces.

PREREQUISITES
  • Understanding of homotopy theory
  • Familiarity with continuous maps in topology
  • Knowledge of contractible spaces, specifically \( \mathbb{R}^n \)
  • Basic concepts of algebraic topology, including the properties of spheres \( S^n \)
NEXT STEPS
  • Study the concept of homotopy equivalence in algebraic topology
  • Learn about contractible spaces and their implications in topology
  • Explore the properties of continuous functions and their mappings
  • Investigate the role of surjectivity in topological mappings
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Mathematicians, particularly those specializing in topology, students preparing for topology exams, and anyone interested in the properties of homotopy and continuous mappings.

timboj2008
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Could anybody help me with this topology question?

i) Prove that every map e: X-> R^n is homotopic to a constant map.

ii) If f: X->S^n is a map that is not onto (surjective), show that f is homtopic to a constant map.

It's part of a past exam paper but it does not come with solutions. Any help on the solution would be greatly appreciated.

Thanks.
 
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i) Try to find a homotopy H:I x X-->R^n between e and the map that sends everything in X to 0.

ii) Hint: Use (i)
 
Basically you need to realize that R^n is contractible. This is pretty easy to visualize: e.g. take R^2 or R^3 and send every point to the origin along the straight line between them.
 
these are the same question.
 

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