--- date: 2021-10-27 19:35 modification date: Sunday 10th April 2022 14:28:52 title: "2021-04-28 Homotopy groups of spheres, talk outline" aliases: [2021-04-28_Homotopy_Groups_of_Spheres_Talk] --- Last modified date: <%+ tp.file.last_modified_date() %> --- - Tags: - #homotopy/of-spheres - Refs: - #todo/add-references - Links: - [homotopy theory](homotopy%20theory.md) - [Homotopy Groups of Spheres](Homotopy%20Groups%20of%20Spheres.md) - [Unsorted/Algebraic Topology (Subject MOC)](Unsorted/Algebraic%20Topology%20(Subject%20MOC).md) --- # Homotopy groups of spheres, talk outline ## Big Points - Homotopy as a means of classification somewhere between homeomorphism and [cobordism](cobordism.md) - Comparison to homology - Higher homotopy groups of spheres exist - Homotopy groups of spheres govern gluing of [CW](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](special%20values%20of%20L%20functions.md) - 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 $\CC$) - 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](Unsorted/Freudenthal%20suspension%20theorem.md) theorem, investigation into "stable" phenomena. - 1938: Pontrayagin shows link between homotopy groups of spheres and [framed cobordism](framed%20cobordism) 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]([obstruction%20theory)] - 1949: Whitehead introduces [homotopy types](homotopy%20types), [CW](CW) complexes, equivalences = [weak equivalences](weak%20equivalences). Importance of homotopy classes of maps between spheres becomes apparent - 1951: Serre uses [spectral sequences](spectral%20sequences) to show that *all* groups $\pi_k S^n$ are torsion except $k=n$, and $k\equiv 3\mod 4, n\equiv 0 \mod 2$. - In first case, $\ZZ$, in second case, $\ZZ \oplus T$ for some torsion group. - Tight bounds on where $p\dash$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](http://philogb.github.io/page/hopf/#) A bundle Visualization: the same way $S^2\setminus\pt \to \RR^2$ via stereographic projection, we take $S^3\setminus\pt \to \RR^3$. 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 $X \mapsvia{f} Y$ satisfies $f(X^{(n)}) \subseteq Y^{(n)}$, then $f$ is said to be a *cellular map*. Any map $X \mapsvia{f} Y$ 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 $n\dash$dimensional, $Y$ may be chosen to be $n\dash$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 $n\dash$ 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: $\pi_1 S^1 = \ZZ$ - *Proof*: Covering space theory - Theorem: $\pi_{1+k} S^1 = 0$ for all $0 < k < \infty$ - *Proof*: Use universal cover by $\RR$ - 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: $\pi_n S^n = \ZZ$ - 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 $S^k \mapsvia{f} S^n$ by $\tilde f$ - $\tilde f$ maps the $k\dash$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: $\pi_3 S^2 = \ZZ$ - *Proof*: Method of killing homotopy - Theorem: $\pi_4 S^2 = \ZZ_2$ - *Proof*: Continued method of killing homotopy - Theorem: $\pi_{n+1} S^n = \ZZ$ for $n \geq 2$? - *Proof*: Freudenthal suspension in stable range? - Theorem: $\pi_{n+2} S^n = \ZZ_2$ for $n \geq 2$? - *Proof*: Freudenthal suspension in stable range?