# Useful fibrations

What are the most useful fibrations that one be familiar with in order to use spectral sequences effectively in algebraic topology? There's at least the four different Hopf fibrations and $S^1\to S^{2n+1}\to \mathbb{C}\textrm{P}^n$. Anything else that's useful?

I might add that one fibration I would like someone to explain is the homotopy fiber of a map. I have trouble wrapping my head around it.

• You should not memorize any fibrations. Jan 15 '12 at 5:20
• (I find this question orders of magnitude more unsettling that the «How to remember the trigonometric identities» of a little time ago!) Jan 15 '12 at 5:24
• @MarianoSuárez-Alvarez: I don't know anything about fibrations (aside from the definition) and am therefore quite curious: why shouldn't one memorize fibrations? Jan 15 '12 at 6:02
• I would also like to say that when doing computations some stuff should fire automatically in your brain and I do consider that to be a level of memorization...
– asdf
Jan 15 '12 at 13:30
• I think this is a very very good question, and am baffled by the fact that it hasn't attracted enough attention. Having few examples to play with is the main reason I dislike fibrations somewhat.
– Leo
Jun 15 '13 at 0:30

## 4 Answers

There's the path fibration $\Omega B \to PB \to B$, where for basepoint $* \in B$,

$PB = \{\gamma:[0,1]\to B \ |\ \gamma(0) = *\}$ is called the path space of $B$, and

$\Omega B = \{\gamma:[0,1]\to B \ |\ \gamma(0) = \gamma(1) = *\}$ is the loop space of $B$.

The map $p:PB \to B$ is the endpoint map $\gamma \mapsto \gamma(1)$.

I'm not quite sure what you mean by memorizing'' a fibration, but this is a useful one to understand.

• In fact, by working the Serre spectral sequence backwards, we can compute the homology of $\Omega S^n$. Jun 20 '13 at 13:22

I recommend Introduction to Homotopy Theory, Arkowitz, Springer Universitext 2011, $\S$3.4, p.93, and Lecture Notes in Algebraic Topology, Davis & Kirk, AMS GSM 35 2002, $\S$4.3, 6.14, 7.7.

Let $\mathbb{F}=\mathbb{R},\mathbb{C},\mathbb{H}$ and $d=1,2,4=\dim_\mathbb{R}\mathbb{F}$.

$\bullet$ Let $n\!\in\!\mathbb{N}$. Let $\mathbb{S}^n_\mathbb{F}=\{(x_0,\ldots,x_n)\!\in\!\mathbb{F}^{n+1};\, \sum_{i=0}^n\!|x_i|^2\!=\!1\}=\mathbb{S}^{d(n+1)-1}$ and $\mathbb{P}_\mathbb{F}^n=\frac{\mathbb{F}^{n+1}\setminus\{0\}}{x\sim \lambda x;\, \lambda\in\mathbb{F}\setminus\{0\}}$. Then $$\mathbb{S}^{0}_\mathbb{F} \longrightarrow \mathbb{S}^n_\mathbb{F} \overset{p}{\longrightarrow} \mathbb{P}^{n}_\mathbb{F}$$ is a fiber bundle, the Hopf fibration, where $p(x_0,\ldots,x_n)\!=\!(x_0\!:\ldots:\!x_n)$. There holds $\mathbb{P}^1_\mathbb{F}=\mathbb{S}^d$.

$\bullet$ Let $0\!\leq\!l\!\leq\!k\!\leq\!n$. Let $V_\mathbb{F}^{k,n}$ $=$ $\{A\!\in\!\mathbb{F}^{n\times k};\, A\overline{A}^t\!=\!I_n\}$ $=$ $\{(v_1,\ldots,v_k)\!\in\!\mathbb{F}^{n\times k};\, v_1,\ldots,v_k\text{ are pairwise orthonormal}\}$ $\subseteq\mathbb{F}^{nk}$ with the subspace topology, the Stiefel manifold of real dimension $dnk\!-\!k\!-\!d\binom{k}{2}$, since $\|v_i\|\!=\!1$ gives $k$ equations over $\mathbb{R}$, and $v_iv_j\!=\!0$ for $i\!<\!j$ gives $\binom{k}{2}$ equations over $\mathbb{F}$. Then $$V^{k-l,n-l}_\mathbb{F} \!\!\longrightarrow V^{k,n}_\mathbb{F} \overset{p}{\longrightarrow} V^{l,n}_\mathbb{F}$$ is a fiber bundle, where $p(v_1,\ldots,v_k)\!=\!(v_{k-l+1},\ldots,v_k)$. There holds $V_\mathbb{F}^{1,n}\!\!=\!\mathbb{S}_\mathbb{F}^{n-1}$ and $V_\mathbb{R}^{n-1,n}\!\!\approx\!\mathrm{SO}^n, V_\mathbb{C}^{n-1,n}\!\!\approx\!\mathrm{SU}^n$ and $V_\mathbb{R}^{n,n}\!\approx\!\mathrm{O}^n, V_\mathbb{H}^{n,n}\!\approx\!\mathrm{U}^n, V_\mathbb{H}^{n,n}\!\approx\!\mathrm{Sp}^n$. Thus as a particular case of $l\!=\!1,k\!=\!n$ and $l\!=\!1,k\!=\!n\!-\!1$, we obtain fiber bundles $$\begin{array}{ccc} \mathrm{O}^{n-1}\longrightarrow\mathrm{O}^n\longrightarrow\mathbb{S}^{n-1}, &\mathrm{U}^{n-1}\longrightarrow\mathrm{U}^n\longrightarrow\mathbb{S}^{2n-1}, &\mathrm{Sp}^{n-1}\longrightarrow\mathrm{Sp}^n\longrightarrow \mathbb{S}^{4n-1},\\ \mathrm{SO}^{n-1}\longrightarrow\mathrm{SO}^n\longrightarrow\mathbb{S}^{n-1}, &\mathrm{SU}^{n-1}\longrightarrow\mathrm{SU}^n\longrightarrow\mathbb{S}^{2n-1}. &\\ \end{array}$$ $$\mathrm{SO}^n\longrightarrow\mathrm{O}^n\overset{\det}{\longrightarrow}\mathbb{S}^0,\hspace{8mm}\mathrm{SU}^n\longrightarrow\mathrm{U}^n\overset{\det}{\longrightarrow}\mathbb{S}^1$$

$\bullet$ Let $0\!\leq\!l\!\leq\!k\!\leq\!n$. Let $G_\mathbb{F}^{k,n}= \{k\text{-dimensional vector subspaces of }\mathbb{F}^n\}= \{V\!\leq\!\mathbb{F}^n;\, \dim_\mathbb{F}\!V\!=\!k\}$ with quotient topology obtained from the map $p\!:V_\mathbb{F}^{k,n}\!\rightarrow\!G_\mathbb{F}^{k,n}$ that sends $p(v_1,\ldots,v_k)\!=\!\langle v_1,\ldots,v_k\rangle$, the Grassmann manifold of real dimension ?. Then $$V^{k,k}_\mathbb{F} \longrightarrow V^{k,n}_\mathbb{F} \overset{p}{\longrightarrow} G^{k,n}_\mathbb{F}$$ is a fiber bundle, with $p$ given above. There holds $G^{1,n}_\mathbb{F}\!\!=\!\mathbb{P}_\mathbb{F}^{n-1}$, so Hopf fibrations are a special case.

$\bullet$ Let $n\!\in\!\mathbb{N}$ and $A\!\overset{\iota}{\longrightarrow}\!B\!\overset{p}{\longrightarrow}\!C$ be morphisms of groups (which are abelian if $n\!\geq\!2$) and $K(-,n)$ be the $n$-th Eilenberg–MacLane space functor. Then $0\!\longrightarrow\!A\!\overset{\iota}{\longrightarrow}\!B\!\overset{p}{\longrightarrow}\!C\!\longrightarrow\!0$ is an exact sequence of groups iff $$K(A,n)\: \overset{\iota_\ast}{\longrightarrow} \:K(B,n)\: \overset{p_\ast}{\longrightarrow} \:K(C,n)\text{ is a fibration.}$$

• Question: What does $K$ in the notation for the Eilenberg-MacLane space stand for? Is the notation $K_n(G)$ ever used in the literature? What does $V$ in the notation for the Stiefel manifold stand for?
– Leo
Jun 20 '13 at 12:00

The fibrations $O(n-1)\to O(n) \to S^{n-1}$, $U(n-1)\to U(n) \to S^{2n-1}$, and $Sp(n-1)\to Sp(n) \to S^{4n-1}$ from Bott periodicity are fairly important. Also mapping tori are fiber bundles. For example, the complement of a fibered knot in $S^3$.

You mentioned this in your post, but there is also the fibration associated to any map of based spaces $f\colon (X,x_0)\to (Y,y_0)$ using the homotopy fiber: $\operatorname{hofib}(f)\to P_f\to Y$. Here, $P_f=\{(x,\gamma)\in X\times Y^I | \gamma(0)=f(x)\}$, and $\operatorname{hofib}(f)$ is the (strict) pullback of the maps $P_f\to Y$ with $(x,\gamma)\mapsto \gamma(1)$ and $*\to Y$ is the inclusion of the base point. Alternatively, it's the homotopy pullback of $X\to Y\leftarrow*$. The strict pullback of this diagram is $f^{-1}(y_0)$, which is why we are justified in calling this the homotopy fiber. Explicitly, $$\operatorname{hofib}(f)=\{(x,\gamma)\in X\times Y^I | \gamma(0)=f(x),\gamma(1)=y_0\}$$ Note that because $P_f$ is homotopy equivalent to $X$ (shrink the paths $\gamma$ to constant paths), this fibration is generally written $\operatorname{hofib}(f)\to X\to Y$. The main reason we like this fibration is because it gives us a long exact sequence of homotopy groups for any map of spaces.

This generalizes in the theory of cubical diagrams. Namely, given a map of cubical diagrams $Z\colon X\to Y$, we get a fibration with total fibers: $\operatorname{tfib}(Z)\to\operatorname{tfib}(X)\to\operatorname{tfib}(Y)$. The case in the above paragraph is the case where $Z$ is a map of $0$-cubes. A $0$-cube is a space, and the total fiber of a map of spaces is the homotopy fiber.