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• Tags
  • #homotopy
• Refs:
  • Lecture Notes: <https://webspace.science.uu.nl/~caval101/homepage/Geometry_and_Topology_2015_files/Groth - lecture notes on homotopy theory.pdf>
    • Includes Postnikov and Whitehead towers
  • https://www.math.purdue.edu/~zhang24/towers.pdf
  • Obstruction theory notes: https://stanford.edu/~sfh/282B.pdf#page=35
• Links:
  • #todo/create-links

Obstruction Theory

Motivations

The rough idea of obstruction theory is simple. Suppose we want to construct some kind of function on a CW complex \(X\). We do this by induction: if the function is defined on the k-skeleton \(X_k\), we try to extend it over the \((k + 1){\hbox{-}}\)skeleton \(X^{k+1}\). The obstruction to extending over a \((k + 1){\hbox{-}}\)cell is an element of \(\pi_k\) of something. These obstructions fit together to give a cellular cochain on \(X\) with coefficients in this \(π_k\). In fact this cochain is a cocycle, so it defines an “obstruction class” in \(H_{k+1}(X; π_k(something))\). If this cohomology class is zero, i.e. if there is a cellular \(k{\hbox{-}}\)cochain \(η\) with \(0 = δη\), then \(η\) prescribes a way to modify our map over the \(k{\hbox{-}}\)skeleton so that it can be extended over the \((k + 1){\hbox{-}}\)skeleton

Postnikov tower

Pasted image 20210505014637.png

Pasted image 20210505014732.png

k-invariant:

Pasted image 20210505014833.png

Construction

Whitehead tower

Unsorted