Algebraic Topology/The fundamental group and covering spaces

Proposition (induced map by covering map is injective):

Let ${\displaystyle p:Z\to Y}$ be a covering space, and let ${\displaystyle z_{0}\in Z}$. Then the induced map

${\displaystyle p_{*}:\pi (Z,z_{0})\to \pi (Y,y_{0})}$

is injective.

Proof: Suppose that ${\displaystyle \rho ,\eta :[0,1]\to Z}$ are loops at ${\displaystyle z_{0}}$ such that ${\displaystyle p_{*}([\rho ])=p_{*}([\eta ])}$. Then there exists a homotopy ${\displaystyle H}$ from ${\displaystyle p\circ \rho }$ to ${\displaystyle p\circ \eta }$, which are both loops at ${\displaystyle x_{0}}$. This homotopy lifts uniquely to a homotopy ${\displaystyle {\tilde {H}}}$ such that ${\displaystyle {\tilde {H}}_{0}=\rho }$. Moreover ${\displaystyle {\tilde {H}}}$ fixes the basepoint, because the maps ${\displaystyle t\mapsto {\tilde {H}}_{t}(0)}$ and ${\displaystyle t\mapsto {\tilde {H}}_{t}(1)}$ are continuous and hence map connected sets to connected sets, and the preimage of a point under a covering map bears the discrete topology. By uniqueness of path lifting, ${\displaystyle {\tilde {H}}_{1}}$ will be equal to ${\displaystyle \eta }$, so that ${\displaystyle [\rho ]=[\eta ]}$. ${\displaystyle \Box }$

Proposition (fundamental group criterion for existence of lifting):

Let ${\displaystyle p:Z\to Y}$ be a covering and let ${\displaystyle f:X\to Y}$ be a continuous function, where ${\displaystyle X}$ is path-connected and strongly locally connected. Let ${\displaystyle y_{0}\in Y}$ and ${\displaystyle x_{0}\in X}$ so that ${\displaystyle f(x_{0})=y_{0}}$, and let ${\displaystyle z_{0}}$ be a lift of ${\displaystyle y_{0}}$ (ie. ${\displaystyle p(z_{0})=y_{0}}$). Then the following are equivalent:

• There exists a unique lift ${\displaystyle {\tilde {f}}:X\to Z}$ of ${\displaystyle f}$ so that ${\displaystyle {\tilde {f}}(x_{0})=z_{0}}$
• ${\displaystyle f_{*}(\pi (X,x_{0}))\subseteq p_{*}(\pi (Z,z_{0}))}$

Proof: Suppose that ${\displaystyle {\tilde {f}}}$ lifts ${\displaystyle f}$, so that ${\displaystyle {\tilde {f}}(x_{0})=z_{0}}$. Then ${\displaystyle f_{*}(\pi (X,x_{0}))=p_{*}{\tilde {f}}_{*}(\pi (X,x_{0}))}$ so that, since ${\displaystyle {\tilde {f}}_{*}}$ maps ${\displaystyle \pi (X,x_{0})}$ to ${\displaystyle \pi (Z,z_{0})}$, we indeed have ${\displaystyle f*(\pi _{X},x_{0})\subseteq p_{*}(\pi (Z,z_{0}))}$.

Conversely, suppose that ${\displaystyle f_{*}(\pi (X,x_{0}))\subseteq p_{*}(\pi (Z,z_{0}))}$. Then we define a lift ${\displaystyle {\tilde {f}}:X\to Y}$ of ${\displaystyle f}$ as follows: For each ${\displaystyle x\in X}$, choose a path ${\displaystyle \gamma :[0,1]\to X}$ so that ${\displaystyle \gamma (0)=x_{0}}$ and ${\displaystyle \gamma (1)=x}$ by path-connectedness of ${\displaystyle X}$. By path lifting, ${\displaystyle f\circ \gamma }$ lifts uniquely to a path ${\displaystyle \rho :[0,1]\to Z}$. Then set ${\displaystyle {\tilde {f}}(x):=\rho (1)}$. We have to show that this definition does not depend on the choice of ${\displaystyle \gamma }$. Indeed, let ${\displaystyle \gamma '}$ be another path like ${\displaystyle \gamma }$. Then ${\displaystyle \gamma *{\overline {\gamma '}}}$ is a loop at ${\displaystyle x_{0}}$ that induces an equivalence class ${\displaystyle [\gamma *{\overline {\gamma '}}]\in \pi (X,x_{0})}$. This in turn induces an equivalence class ${\displaystyle [(f\circ \gamma )*{\overline {(f\circ \gamma ')}}]\in \pi (Y,y_{0})}$; indeed, ${\displaystyle {\overline {(f\circ \gamma ')}}=(f\circ {\overline {\gamma '}})}$, since both are composition with the map ${\displaystyle t\mapsto 1-t}$. By the assumption, there exists a loop ${\displaystyle \eta :[0,1]\to Z}$ in ${\displaystyle z_{0}}$ such that ${\displaystyle p_{*}([\eta ])=[(f\circ \gamma )*{\overline {(f\circ \gamma ')}}]}$. Moreover, ${\displaystyle (f\circ \gamma )*{\overline {(f\circ \gamma ')}}}$ is a path in ${\displaystyle Y}$ that may be lifted to a path ${\displaystyle \eta '}$ in ${\displaystyle Z}$. Since we may lift homotopies, we may lift a homotopy between ${\displaystyle p_{*}([\eta ])}$ and ${\displaystyle (f\circ \gamma )*{\overline {(f\circ \gamma ')}}}$ to a homotopy between ${\displaystyle \eta }$ and ${\displaystyle \eta '}$, which, similarly to the proof of injectivity of ${\displaystyle p_{*}}$, leaves the endpoints fixed. Hence, ${\displaystyle \eta '}$ is a loop, and in particular ${\displaystyle \eta '}$ restricted to ${\displaystyle [1/2,1]}$ yields, when direction is reversed, a lift of ${\displaystyle \gamma '}$ that connects ${\displaystyle z_{0}}$ to ${\displaystyle \rho (1)}$. We conclude well-definedness.

It remains to prove continuity. Hence, let ${\displaystyle x_{1}\in X}$ be arbitrary; we shall prove continuity at ${\displaystyle x_{1}}$. Pick an evenly covered neighbourhood ${\displaystyle V}$ about ${\displaystyle f(x_{1})}$. Let ${\displaystyle W\subseteq Z}$ be the open set mapping homeomorphically to ${\displaystyle V}$ which contains ${\displaystyle {\tilde {f}}(x_{1})}$. Let ${\displaystyle A}$ be any open neighbourhood of ${\displaystyle {\tilde {f}}(x_{1})}$. Set ${\displaystyle {\tilde {A}}\subseteq Y}$ to be the image of ${\displaystyle A\cap W}$ (which is open) under ${\displaystyle p}$, so that ${\displaystyle {\tilde {A}}}$ is itself open. By continuity of ${\displaystyle f}$, there exists an open neihbourhood ${\displaystyle B}$ of ${\displaystyle x_{1}}$ that is mapped by ${\displaystyle f}$ into ${\displaystyle {\tilde {A}}}$. By strong local connectedness, choose ${\displaystyle C\subseteq B}$ to be connected. Recall that maps from connected domains lift uniquely; this shows that on ${\displaystyle C}$, we have ${\displaystyle {\tilde {f}}=(p|_{V})^{-1}\circ f}$. Hence, ${\displaystyle {\tilde {f}}}$ maps ${\displaystyle C}$ into ${\displaystyle A}$.

Finally, connectedness of ${\displaystyle X}$ implies that ${\displaystyle {\tilde {f}}}$ is unique. ${\displaystyle \Box }$