I have encountered the Hopf fibration $S^1\hookrightarrow S^3\twoheadrightarrow S^2$ when studying smooth principal $G$-bundles. Whenever I google the Hopf fibration, I encounter a remark which boils down to:

"... which induces an exact sequence in homotopy $$\ldots\pi_2(S^1)\to\pi_2(S^3)\to\pi_2(S^2)\to\pi_1(S^1)\to\pi_1(S^3)\to\pi_1(S^1)\ldots$$ from which the higher homotopy groups of spheres could be computed ..."

Now I know exact sequences, some algebraic topology and how continuous maps induce maps on homology and how Mayer-Vietoris extends this (under certain conditions) to a long exact sequence, and how the Hurewicz theorem gets you from homology to homotopy, but the machinery behind the above statement seems to be some other result/theorem/piece of theory. A very crude guess is that $\pi_n$ is a functor which preserves exact sequences, and some kind of Mayer-Vietoris then gives the long exact sequence.

If someone has a nice reference for this material, that would be nice. To motivate you guys, here is a pretty picture.

enter image description here

  • $\begingroup$ I yhink that the Hurewicz theorem to get you from homotopy to homology $\endgroup$ – m.idaya Jun 26 '16 at 9:54
  • 1
    $\begingroup$ That picture a screenshot from this video, taken at 2:26. $\endgroup$ – Arthur Jun 26 '16 at 10:01
  • $\begingroup$ Which is awesome. $\endgroup$ – B. Pasternak Jun 26 '16 at 10:07

Let $p :E \to B$ be a Serre fibration with path-connected base $B$. For $b_0 \in B$, set $F = p^{-1}(b_0)$ and denote the inclusion $F \hookrightarrow E$ by $i$. For any $x_0 \in F$, we have the long exact sequence in homotopy

$$\dots \to \pi_{n+1}(B, b_0) \to \pi_n(F, x_0) \xrightarrow{i_*} \pi_n(E, x_0) \xrightarrow{p_*} \pi_n(B, b_0) \to \pi_{n-1}(F, x_0) \to \dots \to \pi_0(F, x_0) \xrightarrow{i_*} \pi_0(E, x_0) \to 0.$$

Note, as $p$ is a Serre fibration, the homotopy type of $F$ is independent of the choice of $b_0 \in B$.

A proof of this fact can be found in Hatcher's Algebraic Topology, Theorem $4.41$. It combines the long exact sequence in homotopy associated to the pointed pair $(E, F, x_0)$

$$\dots \to \pi_{n+1}(E, F, x_0) \to \pi_n(F, x_0) \xrightarrow{i_*} \pi_n(E, x_0) \to \pi_n(E, F, x_0) \to \pi_{n-1}(F, x_0) \to \dots \to \pi_0(F, x_0) \xrightarrow{i_*} \pi_0(E, x_0)$$

with a proof of the fact that $\pi_n(E, F, x_0) \cong \pi_n(B, b_0)$. Finally, the path-lifting property, together with the fact that $B$ is path-connected, shows that the map $\pi_0(F, x_0) \to \pi_0(E, x_0)$ is surjective.

Another reference is May's A Concise Course in Algebraic Topology, page $66$, which takes a more functorial approach; in particular, it uses the loop space functor.

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  • $\begingroup$ Also Switzer's book, Algebraic Topology: Homotopy and Homology, his chapter 4 or 5 develops the long exact of relative homotopy groups and enough machinery to get the appropriate isomorphisms, and gives the long exact sequence of a fibration. He gives this as one of the examples of stable homotopy and develops fruedenthal's theorem from just homotopy. $\endgroup$ – Benjamin Gadoua Jul 6 '16 at 1:24

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