K-connectedness and the homotopy-lifting property

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SUMMARY

The discussion centers on the relationship between k-connectedness and the homotopy-lifting property in algebraic topology. Specifically, k-connectedness refers to the condition where the first through k-th fundamental groups are trivial. The conversation also touches on the extension of global sections from the j-th skeleton to the (j+1)-st skeleton, with a focus on pairs (A, X) where A is a subspace of X. The inquiry seeks clarity on whether the context involves a fixed triple or the general condition of surjective maps being fibrations.

PREREQUISITES
  • Understanding of k-connectedness in algebraic topology
  • Familiarity with fundamental groups and their properties
  • Knowledge of the homotopy-lifting property
  • Concept of skeletons in CW complexes
NEXT STEPS
  • Study the implications of k-connectedness on the structure of topological spaces
  • Explore the homotopy-lifting property in detail, particularly in relation to fibrations
  • Investigate the role of CW complexes and their skeletons in homotopy theory
  • Examine specific examples of pairs (A, X) and their homotopical properties
USEFUL FOR

Mathematicians, particularly those specializing in algebraic topology, as well as graduate students seeking to deepen their understanding of k-connectedness and homotopy theory.

Bacle
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Hi:
Is there a relation between k-connectedness ( meaning 1st, 2nd,..,k-th fundamental
groups are trivial.) and having the homotopy-lifting property.? . This may be
vaguely-related to being able to extend global sections from the j-th skeleton, to
the (j+1)-st skeleton (j<=k, obviously), but I am not sure.

How about k-connectedness for a pair (A,X) ( A a subspace of X).

Thanks.
 
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Could you be more specific? Are we fixing a triple, or are we asking about whether any surjective map is a fibration?
 

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