Past exam of the topology and groups course of the University of Oxford / 2023 / 3 / b

A finite tree is a nonempty connected graph with no graph cycles. We show by mathematical induction on its number $v$ of vertices that it has $v-1$ edges. For $v=1$ there are no edges, so the assertion holds.

If $v>1$, take a path in a graph of maximal length among those without repeated vertices. Each endpoint is a leaf of a tree: an additional incident edge would either extend the path in a graph or create a graph cycle. Delete one such leaf of a tree and its unique incident edge. The remaining graph stays connected, since a path in a graph between remaining vertices never needs that terminal edge, and it has no graph cycles. It is therefore a tree with $v-1$ vertices and, by mathematical induction, $v-2$ edges. Restoring the deleted edge gives $e=v-1$.

Thus its Euler characteristic is

Choose a spanning tree $T$ of the finite connected graph $X$. It contains every vertex of $X$ and has $v(X)-1$ edges by part (i). Collapsing $T$ to a point is a homotopy equivalence: $T$ is a contractible subcomplex, and collapsing a contractible cofibration in a CW complex preserves homotopy type.

Every edge outside $T$ then becomes a circle attached to the common point. Hence the quotient space $X/T$ is a bouquet of circles withpetals. The fundamental group of a bouquet of circles is the free group with one member of a free basis of a group for each petal. Therefore the rank of the fundamental group of a finite connected graph isMore concretely, choose a vertex as base point; follow the unique path in a graph in $T$ to one end of a non-tree edge, traverse it, and return through $T$. These based loops give the free basis of a group. Thus $r$ is the number of free group generators, equivalently the minimum size of a generating set of a group, rather than the size of every possible redundant generating set of a group. Minimality also follows by abelianization, since $F_r$ has abelianization ${\mathbb{Z}}^r$.

Assume $p$ has a common finite degree of a covering map $d$. Declare the lifted vertices to be $p^{-1}(V(X))$, so there are exactly $dv(X)$ of them. Each open edge is homeomorphic to $(0,1)$, a simply connected interval. The restriction of the covering map above it is a disjoint union of $d$ copies of that open edge.

These copies acquire endpoints as follows. Lift the parametrizing path $[0,1]\to X$ of each oriented edge, starting at each of the $d$ points over its initial vertex. The path lifting theorem gives a unique lift all the way to $1$. Its endpoint lies over the terminal vertex, and the interiors of these lifted paths are exactly the $d$ open edges already described. This also handles a graph loop, whose two endpoints may coincide downstairs or upstairs. Local covering sheets above a small star at a vertex show that the lifted incidence relations have precisely the original topology. Thus the graph structure lifted through a covering map is a finite topological graph withSubtracting gives the Euler characteristic under a finite covering formulaThe formula holds even when $\tilde{X}$ is disconnected. If a disconnected base is allowed to have degrees $d_C$ that vary with its connected components $C$, the precise replacement isThe printed formula with a single $d$ uses the constant-degree convention explained in part (a)(ii).