? ??????? , ??????? ? ???????? ?? ?????? ?????? , ???? ???? ??? ????? ?????? ??-?????, ?????, ??????? ???????? ?????? ?????? ??? ??? ??? ?? ????? ????? ? ????? ?????? . ?? ?? ?? ???? ????? ???? ???? ???? ?????????? ?????? ?????? ?? ?????, ????? ????????? ????? ???? ????? ????? ????. ??????, ???? ???? ???? ??????? ??? ????? ?? ????? . ??????? ?????? ??-??? ?? ????????? ?????? ???? .

?????? ?????? ?? ????? ?? ????? ???? ??? ????? ?????? ???????? ??????????? (?? ???????? ) ??? ??? ????? ???? ??????. ????? ???? ?????? ???? ?"???? ?????" ???? ??????? ????????? ?? ???????? ???????????, ?????? ???????? ?? ?????? , ??? ???????? ? ???? , ????? ????? ?????? ???????? ?? ????? ?????????? ?? ?????? ?????? ????, ?? ????? ?????? ???????? ?? ?????.

?? ???? ???? ??? ????? ??????? ??-?????, ?? ??? ???? ???? ???? (???? ?????????). ???? ?????? ????? ????? ????? ??????? ??-????? ????? ???? ?? ??? ?? ??? ??????????? . ??? ??????, ? ????? ?? ????? ?? ????? ????, ?? ???? ????? ? ????? ????? ???? ????? ????.

????? ??????? [ ????? ??? ???? | ????? ]

????????????? [ ????? ??? ???? | ????? ]

??????? ?????? ${\displaystyle f\,}$ ????? ???????????? ?? ??? ?????????? , ?? ?? ????? , ????????? ?????? ${\displaystyle f^{-1}\,}$ ?? ??? ??????????. ???? ?????? ??? ${\displaystyle f\,}$ ??? ?????????? ??? ?? ????? ?? ??? ????????????.

??? ????? [ ????? ??? ???? | ????? ]

??? ${\displaystyle X\,}$ ???? ???????? ???? .

??? (chart) ??? ????????????? ?????? ????? ????? ${\displaystyle U\subseteq X}$ ??? ????? ????? ?? ?????? ?????? ${\displaystyle \mathbb {C} }$.

???? ??? ???? ?? ???? ?? ??-?????? ????? ?? X, ?? ??????? ?? ??-??????? ???? ??? ????? ????, ????? ?? ????? ????? ????? ???? ??? ?????. ?? ????? X ??? ??? ????? ?? ???? ???? , ?? ????? "????" ????? ?? ??????? ?????? ??? ???? ??????: ???? ?? ????, ?? ??? ??? ????? ??? ??? ???? ???? ????, ??????? ?? ?? ?????? ???? ??? ??? ???? ???? ????.

???? ??? ??? ????, ??? ????? ?????? ?????? ????? (??????, ??? ?? ?????? ???? ?? ????, ?????? ?????? ?? ????? ). ???? ?????? ?????-???? (transition map) ??? ??? ??????? ????, ?????? ?? ??? ??? ?? ?????? ?? ???? ??????. ????? ??????, ???? ?? ${\displaystyle U,V\subseteq X}$ ?????? ??????, ??? ?????? ??? ???? ${\displaystyle f:U\rightarrow \mathbb {C} }$ ?- ${\displaystyle g:V\rightarrow \mathbb {C} }$. ????? ????? ??? ???????? ${\displaystyle f\circ g^{-1}:g(U\cap V)\to f(U\cap V)}$.

??? ???? ${\displaystyle f\,}$ ?- ${\displaystyle g\,}$ ?????? ?????? ?????????? , ?? ??? ????? ???? ??? ????????????.

???? ????? ???? ????????? , ?? ?? ????? ?? ?????? ??????????.

???? ???? [ ????? ??? ???? | ????? ]

?? X ??? ???? ???????? ????, ?-A ??? ???? ?????????, ?? ???? ${\displaystyle (X,A)\,}$ ???? ???? ????. ???? ??? ????? ?? ????? ????? ????? ?? ????? ???? ?- ${\displaystyle X\,}$.

??????? [ ????? ??? ???? | ????? ]

• ?????? ?????? ${\displaystyle \mathbb {C} }$ ??? ?????? ?????? ????? ????? ????. ???????? ????? ${\displaystyle z\mapsto z}$ ?????? ??? (???? ?? ????) ?? ${\displaystyle \mathbb {C} }$. ???????? ????? ?????? ${\displaystyle g(z)={\overline {z}}}$ ?? ??? ??? (???? ?? ????) ?? ${\displaystyle \mathbb {C} }$, ??? ????? ?? ???? ?? ???? ????. ???? ?? ?? ${\displaystyle f\,}$ ?- ${\displaystyle g\,}$ ???? ???? ?????? ??????????, ???? ??? ??? ????? ???? ?????.
• ????? ???????, ?? ????? ????? ${\displaystyle U\subseteq \mathbb {C} }$ ??? ???? ????, ???? ???????? ????? ????? ??? ????? ???? ????. ????? ???? ????, ?? ??-????? ????? ?? ???? ???? ??? ???? ????.
• ?????? ?? ???? ??? ?????? ?????? ????? ?? ???? ???? ??????? : ????? ${\displaystyle X=\mathbb {C} \cup \{\infty \}}$. ??? ${\displaystyle f:X-\{\infty \}\rightarrow \mathbb {C} }$ ?????? ?? ??? ${\displaystyle z\mapsto z}$, ???? ${\displaystyle g:X-\{0\}\rightarrow \mathbb {C} }$ ?????? ?? ??? ${\displaystyle z\mapsto 1/z}$, ???? ????? ${\displaystyle 1/\infty =0}$. ?? ${\displaystyle f\,}$ ?- ${\displaystyle g\,}$ ?? ???? ?? ${\displaystyle X\,}$, ?- ${\displaystyle \{f,g\}\,}$ ??? ???? ?????????, ????? ?- ${\displaystyle X\,}$ ???? ?? ???? ???? ???????. ?????? ????????? ???? ?????? ?- ${\displaystyle X\,}$ ??????????? ? ????? .
• ?? ???? ?????? ??? ??? ??????? ???????? ??? ???? ???? ???????.

?????? ??????? ?????? [ ????? ??? ???? | ????? ]

• ??????? ${\displaystyle \,f:X\rightarrow Y}$ ??? ??? ????? ???? ????? ?????????? ?? ??? ??? ${\displaystyle \ g}$ ?? ${\displaystyle \ X}$ ???? ??? ${\displaystyle \ h}$ ?? ${\displaystyle \ Y}$ ???????? ${\displaystyle h\circ f\circ g^{-1}}$ (???? ??????? ?????? ) ??? ??????????.
• ??? ????? ???? ${\displaystyle \ X}$ ?- ${\displaystyle \ Y}$ ?????? ??????????? ?? ????? ??????? ?????????? ${\displaystyle \,f:X\rightarrow Y}$, ?? ? ${\displaystyle \,f}$ ??? ?? ?? ????? , ?? , ????????? ${\displaystyle \,f^{-1}:Y\rightarrow X}$ ?? ??? ?????????? (????? ??, ??? ????? ??????, ???? ?- ${\displaystyle \,f}$ ???????????? ). ?????, ??? ????? ???? ??????????? ?? ????.

???? ????????????? ????? ????? ?? ????? ???? [ ????? ??? ???? | ????? ]

?? ?? ???? ?????????????? , ?? ???? ???? ???? ??? ????????? ???? ?????? ??????:

• ${\displaystyle \mathbb {C} }$,
• ?????? ?? ???? ${\displaystyle \ S^{2}}$, ??
• ????? ?????? ????? ${\displaystyle \{z\in \mathbb {C} :|z|<1\}}$ (???? ????????? ? ??? ?????? ?????? ).

?????, ???? ?????????????? ???? ?? ??? ???? ???? ????? ?????? ?????? ????? ???????? ????? ?? ???????? ????? ????? ???? (????? ??????), 1 (????? ????) ?? ${\displaystyle \ -1}$ (????? ??????). ???? ??? ???????? ????? 1 ???? ???? ???? ???????? . ???? ???? ?? ???????? 0 ???? ?????? (???? ${\displaystyle \mathbb {C} }$, ???????). ???? ?? ???????? 1 ???? ?????? .

?????? ??????? ?? ?? ???? ???? ?????? ?????????? ? ???? ????? 2 ?- ${\displaystyle \ \mathbb {C} }$. ??? ??, ? ????? ?????????? ?? ?? ???? ??? ????????? ?- ${\displaystyle \ \mathbb {C} }$. ???? ???? ?? ???? ??? ?? ??? ???? ???? ${\displaystyle \mathbb {C} /\Gamma }$, ???? ${\displaystyle \ \Gamma }$ ??? ????. ?????? ????????? ???? ???? ??????????? ??????.

?????? ??????? ?? ?? ???? ???? ???????? ?????????? ? ????? ???? . ???? ??? ???? ???? ?? ??? ???? ???? ${\displaystyle \ H/\Gamma }$ ???? ${\displaystyle \ H}$ ??? ??? ?????? ?????? ?- ${\displaystyle \ \Gamma }$ ??? ????? ???? (??? ?? ???????? ????????? ).

???? ????? ????? ???? ?????? (?? ??? ???????????) ??? ?????? ?? ????.

???????? ?? ????? ???? [ ????? ??? ???? | ????? ]

???????? ??????????? [ ????? ??? ???? | ????? ]

?????? ???? ???? ${\displaystyle X\,}$, ??????? ????????? ${\displaystyle f:X\rightarrow \mathbb {C} }$ (?????? ${\displaystyle X}$ ????? ${\displaystyle \mathbb {C} }$) ????? ??????? ?????????? ?? ${\displaystyle X\,}$ (???? ????? ?? ?????). ???? ?????? ? ???? ?????? ??? ?? ???? ???????? ??????????? ?? ????? ????, ?????? ?? ??????? ?????????? ?? ???? ???? ??????? ??? ?????. ????? ???, ?? ?? ???? ???? ?? ??????? ?? ???????? ??????????? ?? ??????. ?? ${\displaystyle X\,}$ ???? ???? ????? ??????? ?- ${\displaystyle x,y\in X}$ ??? ?????? ?????, ?? ????? ??????? ?????????? ${\displaystyle f\,}$ ?? ${\displaystyle X\,}$ ?? ?- ${\displaystyle \,f(x)\neq f(y)}$. ????? ??????? ??????? ???? ?? ????? ????? ????? . ???? ?????? ??????? ????? ???? ??? ????? ????? ????? ${\displaystyle n\,}$ ????? ?????? ??????????? ???-????? ?? ${\displaystyle \mathbb {C} ^{2n+1}}$, ?????? ?? ???? ???? ?? ??????? ???? ?????? ??????????? ???-????? ?? ${\displaystyle \mathbb {C} ^{3}}$.

???????? ?????????? [ ????? ??? ???? | ????? ]

??? ????? ? ????? ?? ???????? ?????????? ?? ???? ???? ???????

?????? ????? ????? ${\displaystyle S\subseteq X}$ ???????? ?????????? ${\displaystyle f:X-S\rightarrow \mathbb {C} }$, ?? ${\displaystyle f\,}$ ????? ??????? ?????????. ???? ??????? ?????????? ??? ?????????. ?? ???? ???? ??????? ??????? ??? ????????? ?? ??? ?????????? ??? ????? ??? ???? ????? ???? ?? ?????? (???? ??????????? ?? ????? ????? ??? ?????). ? ???? ???? ??? ???? ?? ?? ?? ???? ???? ??????? ?????? ???????? ?????????? ????? ??????. ?? ???? ???? ????? ???????, ???? ????? ?? ??????? ????????? ???? ?? ??? ???????? ???????????.

??? ????????? ??????????? [ ????? ??? ???? | ????? ]

???? ????????? ??????????? ?? ???? ???? ????? ??? , ?????? ?- ${\displaystyle {\mathcal {M}}(X)}$. ???? ?? ??? ??????? ????? ??? ?????? ?????????, ???? ${\displaystyle {\mathcal {M}}(X)}$ ??? ??? ????? ?? ${\displaystyle \mathbb {C} }$. ?? ${\displaystyle X\,}$ ???? ???? ???????, ???? ?????? ?- ${\displaystyle {\mathcal {M}}(X)}$ ??? ????? ?? ${\displaystyle \mathbb {C} }$ ????? ??????????? 1. ??? ????????? ?????????? ?? ???? ???? ??????? ???? ?? ????? ?? ??? ???????????, ????? ?? ${\displaystyle X\,}$ ?- ${\displaystyle Y\,}$ ?? ??? ????? ???? ????????? ?? ${\displaystyle X\cong Y}$ (X ????????? ? Y) ?? ??? ?? ${\displaystyle {\mathcal {M}}(X)\cong {\mathcal {M}}(Y)}$. ??????? ?? ???? ????????? ???? ?????? ????????? ?? ????? ???? ????????? (???? ?????????? ?? ???????? ???????????) ????? ???????? ?? ?????? ??? ????? ??????????? 1 ?? ??? ??????? ???????? .

????? ?? ???? [ ????? ??? ???? | ????? ]

???? ?????? ??? ???? ??????????? ??????? ??????????? ?????? ??? ??????? ???? "?????". ?????, ?? ???? ???? ??????? ???? ?????? ????????? ??? ????? ?? ?????. ???? ?"?????" ???? ? ???? ?? ????? ?????? ???? g . ????? ?? ???? ???? ??????? ??? ???? ????. ???? ??? ?? X ???? ???? ???????. ???? ? ${\displaystyle \,{\mathcal {F}}}$ ?? ????? ????????? ??????????? ?? X . ???? ?????? ?? ????? ?? ????? ???????? ${\displaystyle {\mathcal {H}}^{1}(X,{\mathcal {F}})}$ ( ???????????? ?'? ?? ????? ??) ???? ????? ?? ?????, ????? ${\displaystyle \dim {\mathcal {H}}^{1}(X,{\mathcal {F}})=g}$. ???? ???? ??? ?????? (????? ????????? ?????? ?? ????) ?? ??? ?? ????? ??? ??? 0. ??? ?????? ?? ??? ?? ????? ??? ??? 1. ???? ???? ?? ???? 1 ????? ???? ?????? . ?? ???? ???? ?? ???? ???? ?1 ??? ????????.

?????? ?? ???? ???? [ ????? ??? ???? | ????? ]

??? ${\displaystyle X\,}$ ???? ????. ??????? ${\displaystyle D:X\rightarrow \mathbb {Z} }$ (???? ${\displaystyle \mathbb {Z} }$ - ??????? ?????? ) ????? ???? ?? ?????? ${\displaystyle \{p:D(p)\neq 0\}}$ (????? ????? ?? D) ??? ????? ?????. ????, ???? ?? ???? ???? ??????? ???? ?????? ?? ??? ???? ???? ${\displaystyle \textstyle \sum _{k=1}^{n}a_{i}p_{i}}$ ???? ${\displaystyle a_{i}\in \mathbb {Z} }$ ??? ???? ??? ? ${\displaystyle p_{i}\in X}$. ??????, ?? ${\displaystyle X\,}$ ??? ?????? ?? ???? ?? ????? ${\displaystyle 3\cdot 0+2\cdot \infty }$ ??? ????? ?? ${\displaystyle X\,}$ ?????? ?????? 0 ?? ???? 3, ?????? ??????? ?? ???? 2, ???? ????? ???? ?? ???? 0. ???? ???? ??? ?????? (?? ??? ????? ????????? ????????), ?????? ???? ??????? ?? ???? ???? ????? ????? ????? . ????? ??????? ?? ???? ???? ??????? ??? ????? ? ????? ?????? ??????? ?????? ?? ??? ??????? ?? ${\displaystyle X\,}$. ???? ???? ??????? ?? ??? ?? ?????.

???? ?? ??????? ????????? [ ????? ??? ???? | ????? ]

??? ${\displaystyle \ f}$ ??????? ????????? ????? ???????? ?????? ${\displaystyle \ 0}$ ?? ???? ???? ${\displaystyle \ X}$. ???? ?????? ???? ????? ${\displaystyle p\in X}$ ???? ???? ??? ???? - ${\displaystyle \ n}$, ???? ${\displaystyle g:U\to \mathbb {C} }$ (???? ${\displaystyle \ U}$ ????? ?? ${\displaystyle \ p}$), ?? ? ${\displaystyle \ g(p)=0}$ ??? ??????? ${\displaystyle \ f\circ g^{-1}(z)=z^{n}}$. ?? ${\displaystyle \ f}$ ?????????? ?- ${\displaystyle \ p}$ ?? ${\displaystyle \ n\geq 0}$, ?- ${\displaystyle \ n>0}$ ?? ??? ?? ${\displaystyle \ f(z)=0}$. ?? ${\displaystyle \ n<0}$ ???? ?? ${\displaystyle \ f}$ ?? ???? ?- ${\displaystyle \ p}$ ???? ${\displaystyle \ -n}$.

??? ??????? ????????? ${\displaystyle \ f}$ ???? ?????? ???? ${\displaystyle \ D}$ ????? ???: ?? ${\displaystyle \ f}$ ?????????? ? ${\displaystyle \ p}$ ????? ${\displaystyle \ D(p)=n}$, ???? ${\displaystyle \ n}$ ??? ???? ?? ${\displaystyle \ f}$ ?????? ${\displaystyle \ p}$. ????, ?? ? ${\displaystyle \ f}$ ?? ???? ? ${\displaystyle \ p}$ ???? ${\displaystyle \ n}$, ????? ${\displaystyle \ D(p)=-n}$. ?? ????? ?? ??????? ????????? ???? ???? ? ${\displaystyle \ (f)}$. ??????? ???? ???????? ????????? ??? ?????????? ?? ??? ?? ????? ??? ??? ???????.

???? ${\displaystyle \ D}$ ?????? ??????? ????????? ${\displaystyle \ f}$ ???? ????? ??? ???? ???? ????. ?????? ? - ${\displaystyle (f\cdot g)=(f)+(g)}$ ?- ${\displaystyle \textstyle \left({\frac {1}{f}}\right)=-(f)}$ ??? ????? ??????? ??????? ????? ??-????? ?? ????? ???????.

??? ?? [ ????? ??? ???? | ????? ]

• ????? ????
• ??????? ?????????
• ???? ???? ???
• ???? ?????? ?????????
• ???? CAT(0) ‎ - ????? ??????????? ?? ?????? ???? ????????? ??????

?????? ????? [ ????? ??? ???? | ????? ]

??????? ???????? [ ????? ??? ???? | ????? ]

• ???? ???? ???? PlanetMath
• ???? ???? , ???? MathWorld (???????)
• ????? ????? , ?? ??? ? ?????? ???????