Useful fibrations

Question

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.

Answer 1

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.

Answer 2

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 $S^3$.

Answer 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.