Links: Homotopy Groups of Spheres

Basic tools in homotopy of spaces

Theorems

Theorem: \(\pi_1 S^1 = {\mathbf{Z}}\)
- Proof: Covering space theory
Theorem: \(\pi_{1+k} S^1 = 0\) for all \(0 < k < \infty\)
- Proof: Use universal cover by \({\mathbf{R}}\)
- Theorem: \({\mathbf{R}}^n\) is contractible
- Theorem: \(R\) covers \(S^1\)
- Theorem: Covering spaces induce \(\pi_i X \cong \pi_i \tilde X, i \geq 2\)
Theorem: \(\pi_1 S^n = 0\) for \(n \geq 2\).
- \(S^n\) is simply connected.
Theorem: \(\pi_n S^n = {\mathbf{Z}}\)
- Proof: The degree map is an isomorphism. [G&M 4.1]
- Alternatively:
  - LES of Hopf fibration gives \(\pi_1 S^1 \cong \pi_2 S^2\)
  - Freudenthal suspension : \(\pi_k S^k \cong \pi_{k+1} S^{k+1}, k \geq 2\)
Theorem: \(\pi_k S^n = 0\) for all \(1 < k < n\)
- Proof: By cellular approximation: For \(k < n\),
  - Approximate \(S^k \xrightarrow{f} S^n\) by \(\tilde f\)
  - \(\tilde f\) maps the \(k{\hbox{-}}\)skeleton to a point,
  - Which forces \(\pi_k S^n = 0\)?
- Alternatively: Hurewicz
Theorem: \(\pi_k S^2 = \pi_k S^3\) for all \(k > 2\)
Theorem: \(\pi_k S^2 \neq 0\) for any \(2 < k < \infty\)
- Corollary: \(\pi_k S^3 \neq 0\) for any \(2 < k < \infty\)
Theorem: \(\pi_k S^2 = \pi_k S^3\)
- Proof: LES of Hopf fibration
Theorem: \(\pi_3 S^2 = {\mathbf{Z}}\)
- Proof: Method of killing homotopy
Theorem: \(\pi_4 S^2 = {\mathbf{Z}}_2\)
- Proof: Continued method of killing homotopy
Theorem: \(\pi_{n+1} S^n = {\mathbf{Z}}\) for \(n \geq 2\)?
- Proof: Freudenthal suspension in stable range?
Theorem: \(\pi_{n+2} S^n = {\mathbf{Z}}_2\) for \(n \geq 2\)?
- Proof: Freudenthal suspension in stable range?

Definitions

CW Complexes
Define homotopy
Define homotopy invariance
Classification of abelian groups
- Free and torsion
Define \(\pi_n(X)\)
- Show functoriality
- Show homotopy invariant
Unsorted/Whitehead theorem
- (homotopy and homology versions)
\(\pi_n\) for \(n\geq 2\) is abelian
Compute \(H_* S^n\)
Compute \(\pi_k S^1\)
Cellular approximation theorem
Show \(\pi_k S^n = 0\) for \(k<n\)
Show \(\pi_n\) only depends on n-skeleton
Hurewicz theorem
Show \(\pi_n S^n = Z\)
Show \(\pi_i S^n = 0\) for \(i < n\)
Define fibrations
Define Unsorted/list of fibrations
Define suspension and loop space
Show \(\Sigma \to \Omega\) adjunction
Show how to use \(\Sigma\) and \(\Omega\) to move between \(\pi_n\) using equalities
Hopf Fibration Talk
- Show \(\pi_k S^2 = \pi_k S^3\)
- Show \(\pi_3 S^2 = Z\)
- Killing homotopy groups
- Spectral Sequence of a filtration
Serre spectral sequence
- Compute algebra structure of \(CP^\infty\)
Compute \(\pi_4 S^2\)
Compute first stable \(\pi_k\)
Freudenthal Suspension
Eilenberg-MacLane space
- Representability:
- \(H^n (X; G) = [X, K(G, n)]\)
Summary of “easy” results:
- \(\pi_k S^1 = 0, i > 1\)
- \(\pi_n S^n = Z\)
- \(\pi_3 S^2 = Z\)
- \(\pi_k S^2 = \pi_k S^3\)
- \(\pi_i(S^n)\) is finite for \(i > n\)
  - Except for \(\pi_{4k-1}\)
Harder results
- \(\pi_n+1 S^n = Z\delta_2 + Z_2 \delta_{n \geq 3}\)
- \(\pi_n+2 S^n = Z_2\)
Exact sequences
Splitting and extension problem
Degree of a map to \(S^n\)
Lie algebra structure of \(\pi_*\)