Homotopy groups of spheres, talk outline

Big Points

Homotopy as a means of classification somewhere between homeomorphism and cobordism
Comparison to homology
Higher homotopy groups of spheres exist
Homotopy groups of spheres govern gluing of CW complexes
CW complexes fully capture that homotopy category of spaces
There are concrete topological constructions of many important algebraic operations at the level of spaces (quotients, tensor products)
“Measuring stick” for current tools, similar to special values of L functions
Serre’s computation

History

1860s-1890s: (Roughly) defined by Jordan for complex integration, “combinatorial topology”
- Original motivation: when does a path integral depend on a specific path? (E.g. a contour integral in \({\mathbf{C}}\))
1895: Poincare, Analysis situs (“the analysis of position”) in analogy to Euler Geometria situs in 1865 on the Kongisberg bridge problem Attempts to study spaces arising from gluing polygons, polyhedra, etc (surfaces!), first use of “algebraic invariant theory” for spaces by introducing \(\pi_1\) and homology.
1920s: Rigorous proof of classification of surfaces (Klein, Möbius, Clifford, Dehn, Heegard), captured entirely by \(\pi_1\) (equivalently, by genus and orientability).
1925-1928: Noether, Mayer, Vietoris develop general algebraic theory of homology, now “algebraic topology”
1931: Hopf discovers a nontrivial (not homotopic to identity) map \(S^3 \to S^2\)
- Compare to homology: \(H^k S^n = 0\) for \(k\geq n\) is an easy theorem!
1932/1935: Cech (resp Hurewicz) introduce higher homotopy groups, gives a map relating homotopy to homology, shows they are abelian groups for \(n\geq 2\).
- Withdrew his paper because of this theorem!
1937: Freudenthal suspension theorem, investigation into “stable” phenomena.
1938: Pontrayagin shows link between homotopy groups of spheres and framed cobordism classes of submanifolds of \(S^n\).
- Direction of computation: using cobordism classes (geometric) to compute \(\pi_k S^n\). Modern day: run the computation backwards to compute cobordism groups.
1940: Eilenberg, [obstruction theory]
1949: Whitehead introduces homotopy types, CW complexes, equivalences = weak equivalences. Importance of homotopy classes of maps between spheres becomes apparent
1951: Serre uses spectral sequences to show that all groups \(\pi_k S^n\) are torsion except \(k=n\), and \(k\equiv 3\operatorname{mod}4, n\equiv 0 \operatorname{mod}2\).
- In first case, \({\mathbf{Z}}\), in second case, \({\mathbf{Z}}\oplus T\) for some torsion group.
- Tight bounds on where \(p{\hbox{-}}\)torsion can occur.
1953: Whitehead shows the homotopy groups of spheres split into stable and unstable ranges.
Today: We know \(\pi_{n+k}S^n\) for
- \(k \leq 64\) when \(n\geq k+2\) (stable range)
- \(k \leq 19\) when \(n < k+2\) (unstable range)
- We only have a complete list for \(S^0\) and \(S^1\), and know no patterns beyond this!

Actual Outline

Definitions of spheres and balls
Definition of homotopy of maps
- Motivations from complex analysis
Functoriality
Examples of spaces that are homotopy equivalent and aren’t.
Example where homotopy distinguishes homologically equivalent spaces

Images

Hopf Fibration Visualizer

A bundle

Visualization: the same way \(S^2\setminus{\operatorname{pt}}\to {\mathbf{R}}^2\) via stereographic projection, we take \(S^3\setminus{\operatorname{pt}}\to {\mathbf{R}}^3\). Realizes \(S^3\) as a family of circles parameterized by a 2-sphere, fiber above each point is a circle.

Theorems and Definitions

A map \(f: X \to Y\) is called a weak homotopy equivalence if the induced maps \(f^*_i: \pi_i(X, x_0) \to \pi_i(Y, f(x_0))\) are isomorphisms for every \(i \geq 0\). If a map \(X \xrightarrow{f} Y\) satisfies \(f(X^{(n)}) \subseteq Y^{(n)}\), then \(f\) is said to be a cellular map. Any map \(X \xrightarrow{f} Y\) between CW complexes is homotopic to a cellular map. For every topological space \(X\), there exists a CW complex \(Y\) and a weak homotopy equivalence \(f: X \to Y\). Moreover, if \(X\) is \(n{\hbox{-}}\)dimensional, \(Y\) may be chosen to be \(n{\hbox{-}}\)connected and is obtained from \(X\) by attaching cells of dimension greater than \(n\). Abbreviated statement: if \(X, Y\) are CW complexes, then any map \(f: X \to Y\) is a weak homotopy equivalence if and only if it is a homotopy equivalence. (Note: \(f\) must induce maps on all homotopy groups simultaneously.) If \(X\) is an \(n{\hbox{-}}\) connected CW complex, then there are maps \(\pi_i X \to \pi_{i+1} \Sigma X\) which is an isomorphism for \(i\leq 2n\) and a surjection for \(i=2n+1\).

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: 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: \(\pi_n S^n = {\mathbf{Z}}\)
- 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?