---- > [!definition] Definition. ([[isomorphism of covering spaces]]) > Let $p: E \to B$ and $p': E' \to B$ be [[covering space|covering maps]]. They are said to be **equivalent** or **isomorphic**, denoted $p \cong p'$, if there exists a [[homeomorphism]] $h:E \to E'$ such that $p=p' \circ h$. $h$ is called an **equivalence of covering maps/covering spaces**. >```tikz >\usepackage{tikz-cd} >\usepackage{amsmath} >\begin{document} >% https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAFEQBfU9TXfIRQAmclVqMWbdgHJuvEBmx4CRAIyk14+s1aIQAIW7iYUAObwioAGYAnCAFskZEDghJRE3WzQhqDOgAjGAYABX4VIRBbLDMACxx5G3snRA1Xd0RPHSl9NDkeZMdnajckdJy9EDjjLiA >\begin{tikzcd} E \arrow[rd, "p"'] \arrow[rr, "h"] & & E' \arrow[ld, "p'"] \\ > & B & >\end{tikzcd} >\end{document} >``` > When $E=E'$ and $p=p'$, $h$ is called a **deck transformation** or **covering automorphism** of $p$. Together with composition of maps, the collection of deck transformations forms a [[group]] denoted $\text{Deck}(p)$ or $\text{Aut}(E / B)$. ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```