# 7.01 Covering spaces

Below the video you will find accompanying notes and some pre-class questions.

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# Notes

## Intuition for covering spaces

*(0.30)*Take \(\mathbf{C}\setminus\{0\}\) and consider the map \(p\colon\mathbf{C}\setminus\{0\}\to\mathbf{C}\setminus\{0\}\) given by \(p(z)=z^2\). This map is surjective and 2-to-1. Nonetheless, we do have something like an inverse locally: the square root function (we actually have two local inverses differing by a sign \(\pm\sqrt{}\)).

*(2.26)* More precisely, if we excise a half-line
\(B=\{z\in\mathbf{C}\setminus\{0\}\ :\ Im(z)=0,\ Re(z)<0\}\) then on
\(\mathbf{C}\setminus B\) we have two functions
\[q_+,q_-\colon\mathbf{C}\setminus B\to\mathbf{C}\setminus\{0\}\]
such that \(p(q_\pm(z))=z\). In this example, \(q_-=-q_+\),
e.g.\(q_+(1)=1\), \(q_-(1)=-1\).

*(4.00)*We could equally have taken a branch cut \(\{z\in\mathbf{C}\setminus\{0\}\ :\ Im(z)=0,\ Re(z)>0\}\) and we would have obtained two different local inverses for \(p\). The point of taking a branch cut is that \(\mathbf{C}\setminus\{0\}\) is not simply-connected but \(\mathbf{C}\setminus B\) is simply-connected, which is what lets us define an inverse on the complement of the branch-cut.

*(5.11)*Take \(f\colon\mathbf{C}\mapsto\mathbf{C}\setminus\{0\}\), \(f(z)=e^{iz}\). This map is surjective and \(\infty\)-to-1 (for example \(f^{-1}(1)=\{2\pi n\ :\ n\in\mathbf{Z}\}\)). On the complement of a branch cut \(B\) we have infinitely many well-defined local inverses \(q_n=2\pi n-i\log)\colon\mathbf{C}\setminus B\to\mathbf{C}\) satisfying \(q_n(1)=2\pi n\) and \(p(q_n(z))=z\) (that is \(\exp(i(2\pi n-i\log z))=z\)). [The video is missing a factor of \(-i\).]

*(9.00)*Roughly speaking, a covering map is a map \(p\colon Y\to X\) which is \(N\)-to-1 for some \(N\) and which has \(N\) locally-defined inverses \(q_n\) defined on simply-connected subsets of the target (satisfying \(p\circ q_n=id\)).

*(10.40)*Notice that in the second example, the domain of the covering map is \(\mathbf{C}\), which is simply-connnected, and the local inverses \(q_n\) are indexed by the integers \(\mathbf{Z}\). This is, roughly speaking, how you deduce that the target of the covering map has \(\pi_1\cong\mathbf{Z}\).

## Definition

*(11.20)*Let \(p\colon Y\to X\) be a continuous map.

- A subset \(U\subset X\) is called an
*elementary neighbourhood*if there is a discrete set \(F\) and a homeomorphism \(h\colon p^{-1}(U)\to U\times F\) such that \(pr_1\circ h=p|_{p^{-1}(U)}\) (where \(pr_1\colon U\times F\to U\) is the projection to the first factor).*(13.30)*You should think of an elementary neighbourhood as playing the role of the complement of a branch cut in our earlier examples: \(U\) is the subset on which you have locally-defined inverses to \(p\).*(13.57)*The locally-defined inverses are parametrised by the discrete set \(F\); in our examples we had \(F=\{-1,1\}\) and \(F=\mathbf{Z}\). For each point \(m\in F\) we get a map \(q_m\colon U\to U\times F\to p^{-1}(U)\) defined by \(q_m(x)=h^{-1}(x,m)\) which satisfies \(p(q_m(x))=pr_1(h(h^{-1}(x,m)))=x\).

*(16.07)*We say \(p\) is a*covering map*if \(X\) is covered by elementary neighbourhoods (i.e. every point \(x\in X\) is contained in some elementary neighbourhood).- Note that the word
*cover*can now mean two things: covering map and open cover (like in the definition of compactness). Hopefully it will be clear what is meant from the context.

- Note that the word
*(17.40)*We say that a subset \(V\subset Y\) is an*elementary sheet*if it is path-connected and \(p(V)\) is an elementary neighbourhood.

*(18.36)*Here is a picture of the exponential map \(p(z)=e^{iz}\) restricted to the real axis (so its image is the unit circle in \(\mathbf{C}\)):

*(21.30)*Here is a picture of the squaring map \(p_2(z)=z^2\) (restricted to the unit circle). In the circle, the green arc is an elementary sheet; its preimage is a union of two arcs, each of which is an elementary sheet.

## More examples

*(24.05)*Consider the figure 8, with its two loops drawn in red and blue. The following picture defines a 2-to-1 covering space of the figure 8:

*(26.22)*Another covering space of the figure 8 is given by this picture:

# Pre-class questions

- Are there any other 2-to-1 covers of the figure 8? Remember, in each
case, the cross-point has two preimages which look like
cross-points, each edge has two preimages which connect the
cross-points.

# Navigation

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