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\centerline{\Head Mathematics 205B, Winter 2019, Assignment 1}
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This will be due on {\bf Monday, February 4, 2018, at 9 :10 A.M.}
at the beginning class or by prior arrangement in my mailbox or at 
the front desk of Surge 202 at the same time.  If you wish to use 
some version of \TeX\ in writing up your answers, please feel free 
to do so.
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{\sl You must show the work behind or reasons for your answers. }
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{\bf 1. }\quad 
Let $X$ be a space which is Hausdorff, connected and locally 
arcwise connected, let $f:(X,x)\to (S^1,1)$ be a continuous 
mapping, and let $p:\bbR\to S^1$ be the map $p(t)=\exp(2\pi it)$.  
Prove that $f$ lifts to a map $F:X\to\bbR$ if and only if $f$ is 
basepoint preservingly homotopic to a constant mapping.
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{\bf 2. }\quad 
Let $X$ be a space which is Hausdorff, connected and locally 
arcwise connected, let $x\in X$, and assume further that 
$\pi_1(X,x)$ is finite of order $n$.  Prove that, up to equivalence, 
there are only finitely many connected covering spaces of $X$, 
and give an upper bound $B(n)$ for the number of equivalence 
classes of coverings as an explicit, elementary function of $n$.
\bye
{\sf In the lectures there is a classification of arcwise connected 
Hausdorff 
covering spaces over a given space.  The point of this exercise is 
to prove a generalization for Hausdorff covering spaces which are not 
necessarily arcwise connected. }
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$(a)$ 
Let $B$ be a space which is Hausdorff, locally arcwise connected 
and connected, and let $b_0\in B$.  Suppose further that $p:
(E,e_0)\to (B,b_0)$ is a covering space projection such that $E$ 
is Hausdorff (note that it is automatically locally arcwise connected).  
If $E_0$ is the arc component of $e_0$, prove that $P|E_0$ is also 
a covering space projection.
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$(b)$ 
Let $p:(E,e_0)\to (B,b_0)$ be as in $(a)$.  Prove that $E$ is a 
union of pairwise disjoint open closed subspaces $E_\alpha$ such  
that each $E_\alpha$ is arcwise connected and each restriction 
$p|E_\alpha$ is a covering space projection.
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$(c)$ 
Let $p:(E,e_0)\to (B,b_0)$ be as in $(a)$ once more, and now assume 
that $B$ is simply connected.  Prove that $E$ is homeomorphic to 
$B\times F$, where $F$ is the fiber $p^{-1}\left[\{b_0\}\right]$.  
[{\it Hint: } Why is each restriction $p|E_\alpha$ a homeomorphism?]
\bye
