How many proofs that $\pi_n(S^n)=\mathbb{Z}$ are there? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T10:54:45Z http://mathoverflow.net/feeds/question/121130 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/121130/how-many-proofs-that-pi-nsn-mathbbz-are-there How many proofs that $\pi_n(S^n)=\mathbb{Z}$ are there? Mike Shulman 2013-02-08T00:27:51Z 2013-02-15T21:36:59Z <p>Offhand I can think of two ways in classical homotopy theory:</p> <ol> <li><p>Show that $\pi_k(S^n)=0$ for $k\lt n$ by deforming a map $S^k\to S^n$ to be non-surjective, then contracting it away from a point not in its image. Now use the Hurewicz theorem to show $\pi_n(S^n) = H_n(S^n) = \mathbb{Z}$, which is easy to calculate with cellular homology.</p></li> <li><p>Use the Freudenthal suspension theorem to induct up from $\pi_1(S^1)=\mathbb{Z}$, which you can prove using (say) the universal covering space $\mathbb{R}\to S^1$.</p></li> </ol> <p>What other ways are there to prove $\pi_n(S^n)=\mathbb{Z}$?</p> http://mathoverflow.net/questions/121130/how-many-proofs-that-pi-nsn-mathbbz-are-there/121131#121131 Answer by Chris Gerig for How many proofs that $\pi_n(S^n)=\mathbb{Z}$ are there? Chris Gerig 2013-02-08T00:38:02Z 2013-02-15T21:36:59Z <p>$\pi_n(S^n)=[S^n,S^n]=\lbrace$cobordism classes of framed 0-submanifolds$\rbrace$ by the Pontrjagin-Thom construction. These are collections of points (with sign) which add up to give the degree of the maps, so this set is precisely $\mathbb{Z}$.</p> http://mathoverflow.net/questions/121130/how-many-proofs-that-pi-nsn-mathbbz-are-there/121133#121133 Answer by Tom Goodwillie for How many proofs that $\pi_n(S^n)=\mathbb{Z}$ are there? Tom Goodwillie 2013-02-08T01:17:33Z 2013-02-08T01:24:08Z <p>A modified version of your 1: If you have the Hurewicz theorem for $\pi_n(S^n)$, then you also have it for $k &lt; n$, so you don't need the geometric argument for $\pi_k(S^n)=0$ when $k &lt; n$.</p> <p>Alternatively, after arguing geometrically that $\pi_k(S^n)=0$ when $k &lt; n$, you can use much the same idea to show that any map $S^n\to S^n$ is homotopic to one that takes everything to the basepoint except some little disks that are mapped in a standard way. From there you can go to seeing that $\pi_n(S^n)$ is generated by the class of the identity map. And then you don't need the full strength of Hurewicz, but just the fact that all the multiples of the identity have different effects on $H_n(S^n)$. (The first part of this is pretty close in spirit to the framed cobordism argument indicated by Chris Gerig.)</p> http://mathoverflow.net/questions/121130/how-many-proofs-that-pi-nsn-mathbbz-are-there/121158#121158 Answer by Mark Grant for How many proofs that $\pi_n(S^n)=\mathbb{Z}$ are there? Mark Grant 2013-02-08T07:25:21Z 2013-02-08T07:25:21Z <p>This can be proved using the <a href="http://mathworld.wolfram.com/BrouwerDegree.html" rel="nofollow">Brouwer degree</a>. Smooth maps $f,g\colon\thinspace S^n\to S^n$ are homotopic if and only if they have the same Brouwer degree, and any continuous map is homotopic to a smooth one.</p> <p>On second thoughts, this might be identical to Chris Gerig's answer. Milnor's lovely book "Topology from the Differentiable Viewpoint" contains the details.</p> http://mathoverflow.net/questions/121130/how-many-proofs-that-pi-nsn-mathbbz-are-there/121172#121172 Answer by Ricardo Andrade for How many proofs that $\pi_n(S^n)=\mathbb{Z}$ are there? Ricardo Andrade 2013-02-08T11:16:16Z 2013-02-10T06:14:36Z <p>A rather roundabout method for computing the fundamental group of $S^n$ comes from using Kan's loop group construction as briefly described in <a href="http://mathoverflow.net/questions/31004/computational-complexity-of-computing-homotopy-groups-of-spheres/96397#96397" rel="nofollow">this answer by John Klein</a>. The basic theory of the Kan loop group construction justifying this method can be found in Kan's article "A combinatorial definition of homotopy groups". The actual calculation of $\pi_n(\Delta^n/\partial\Delta^n)$ as the homology of the "normalized chain complex" (the complex $N_\ast$ in the linked answer) of the Kan loop group is quite easy and quick.</p> <p>However, one needs to first justify --- as done in Kan's article --- that the Kan loop group of $X$ gives the homotopy of $X$ in the manner described by John Klein's answer. This requires showing that the Kan loop group is weakly equivalent to the loop space of (a fibrant replacement of) $X$. The only proofs I have seen of this fact use the Whitehead theorem that a simply connected space (or simplicial set) which has trivial reduced homology is weakly contractible.</p> http://mathoverflow.net/questions/121130/how-many-proofs-that-pi-nsn-mathbbz-are-there/121175#121175 Answer by Ronnie Brown for How many proofs that $\pi_n(S^n)=\mathbb{Z}$ are there? Ronnie Brown 2013-02-08T11:32:19Z 2013-02-08T22:25:33Z <p>In the paper </p> <p>R. Brown and P.J. Higgins Colimit theorems for relative homotopy groups'', <em>J. Pure Appl. Algebra</em> 22 (1981) 11-41.</p> <p>we generalised to all dimensions the formulation and proof due to Crowell (1959) of the Seifert-van Kampen Theorem for the fundamental group. Our generalised proof <em>did not involve singular homology</em>, nor simplicial approximation. Consequences were: </p> <ol> <li><p>The Brouwer Degree Theorem (determination for $n \geqslant 1$ of $\pi_r(S^n)$ for $r \leqslant n$; and also the determination of $\pi_n(S^n \vee W)$ where $W$ is a wedge of circles (without using covering spaces); </p></li> <li><p>the Relative Hurewicz Theorem (as a determination of $\pi_n(X \cup CA,x)$ if $(X,A)$ is $(n-1)$-connected); </p></li> <li><p>J.H.C. Whitehead's determination of $\pi_2(X \cup e^2_\lambda,X,x)$ as the free crossed $\pi_1(X,x)$-module on the $2$-cells $e^2_\lambda$ (Whitehead's proof involved transversality and knot theory, and there are maybe three other proofs); </p></li> <li><p>a generalisation of 3. to determine $\pi_2(X \cup_f CA, X,x)$ as a crossed module induced by $f_*: \pi_1(A,a) \to \pi_1(X,x)$ (for $A,X$ connected --at present no other proof is available). </p></li> </ol> <p>The full story is also given in the <a href="http://pages.bangor.ac.uk/~mas010/nonab-a-t.html" rel="nofollow">EMS Tract Vol 15</a>. </p> <p>The proofs very much involve the use of filtered spaces, the algebra of the multiple compositions of cubes, and the relation with the classical invariants of relative homotopy groups. </p> http://mathoverflow.net/questions/121130/how-many-proofs-that-pi-nsn-mathbbz-are-there/121257#121257 Answer by Jeremy Miller for How many proofs that $\pi_n(S^n)=\mathbb{Z}$ are there? Jeremy Miller 2013-02-09T00:17:22Z 2013-02-09T00:17:22Z <p>I think this can be made into a proof without being circular. In "Configuration spaces and iterated loop spaces," Segal proves that the group completion of the monoid of configurations of distinct unordered points in $\mathbb{R}^n$ is $\Omega^n S^n$. $\pi_0$ of this monoid is the natural numbers so its Grothendieck group is the integers. This gives $\pi_k(S^k)=\mathbb{Z}$. Since the abelianization of the braid groups is the integers, we have that $\pi_3(S^2)=\mathbb{Z}$. Since the abelianization of the symmetric groups is $\mathbb{Z}/2$, we have that $\pi_{k+1}(S^k)=\mathbb{Z}/2$ for k>2.</p> http://mathoverflow.net/questions/121130/how-many-proofs-that-pi-nsn-mathbbz-are-there/121943#121943 Answer by Ricardo Andrade for How many proofs that $\pi_n(S^n)=\mathbb{Z}$ are there? Ricardo Andrade 2013-02-15T21:19:56Z 2013-02-15T21:31:37Z <p>A stable version of Jeremy Miller's answer uses instead the Barratt-Priddy-Quillen theorem about $\Omega^\infty\Sigma^\infty S^0$ (for example, as stated in Graeme Segal's "Categories and cohomology theories"). We immediately conclude that $\pi_0(\Omega^\infty\Sigma^\infty S^0)\cong {\mathbb Z}$, as it is the Grothendieck group of $\pi_0(\coprod_{n\in{\mathbb N}} B\Sigma_n)={\mathbb N}$. By the way, we can also compute $\pi_1\cong{\mathbb Z}/2$ as the abelianization of $\Sigma_n$ for large $n$, and $\pi_2$ as the second homology group of the alternating group $A_n$ for large $n$. Applying the Freudenthal suspension theorem to $\pi_0(\Omega^\infty\Sigma^\infty S^0)\cong {\mathbb Z}$, we obtain an isomorphism between $\pi_n(S^n)$ and $\mathbb Z$ for $n\geq 2$.</p> <p>This argument is weaker and more indirect than Jeremy's answer, since it only obtains stable information, and it thus requires using the Freudenthal suspension theorem.</p>