20.500.12556/RUL-174318 Veliki eksotični 4-prostor magistrsko delo A large exotic 4-space Konstruiramo primer velikega eksotičnega $\mathbb{R}^4$, ki je komplement neke topološko vložene sfere $S^2$ v $\mathbb{C} P^2$. V mnogoterosti $\mathbb{C} P^2 \# 9 \overline{\mathbb{C} P^2}$ lahko neki homološki $2$-razred predstavimo s topološko vloženo $2$-sfero $S$ z uporabo Cassonovih ročajev in Freedmanovega izreka \cite{freedman_paper}. Vsak Cassonov ročaj se gladko vloži v neki $2$-ročaj, zato obstaja neka gladka vložitev $\iota: U \rightarrow \mathbb{C} P^2$ neke odprte okolice $U$ sfere $S$ v $\mathbb{C} P^2$. Komplement $R = \mathbb{C} P^2 - \iota(S)$ je po Freedmanovem delu homeomorfen $\mathbb{R}^4$, vendar iz Donaldsonovega izreka \cite{donaldson} sledi, da ne obstaja gladka vložitev $R$ v $\mathbb{R}^4$, torej je $R$ veliki eksotični $\mathbb{R}^4$. We construct an example of a large exotic $\mathbb{R}^4$ which is the complement of a topologically embedded sphere $S^2$ in $\mathbb{C}P^2$. We can represent a certain homological $2$-class of the manifold $\mathbb{C}P^2 \# 9 \overline{\mathbb{C} P^2}$ by a topologically embedded $2$-sphere $S$ using Casson handles and Freedman's theorem \cite{freedman_paper}. Every Casson handle embeds smoothly in a $2$-handle, therefore there exists a smooth embedding $\iota: U \rightarrow \mathbb{C}P^2$ of an open neighborhood $U$ of the sphere $S$ in $\mathbb{C}P^2$. The complement $R = \mathbb{C} P^2 - \iota(S)$ is by the work of Freedman homeomorphic to $\mathbb{R}^4$. However, it follows by Donaldson's theorem \cite{donaldson} there does not exist a smooth embedding of $R$ in $\mathbb{R}^4$, so $R$ is a large exotic $\mathbb{R}^4$. 4-mnogoterost presečna forma Cassonov ročaj eksotična gladka struktura 4-manifold intersection form Casson handle exotic smooth structure true false false Slovenski jezik Angleški jezik Magistrsko delo/naloga 2025-10-01 08:15:34 2025-10-01 08:15:35 2025-11-20 03:49:24 0000-00-00 00:00:00 2025 0 0 0000-00-00 NiDoloceno NiDoloceno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 515.1 155097 250903043 19815.pdf 19815.pdf 1 FD995BFCF3B1F7D38534CC536EE8FD4E 330ba7948251ff618e2e7f25d4e6b04cdd926c93ab2e237c5a0e886f97c0081d d68f5968-9e8d-11f0-9328-0050569b8976 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=219107 Fakulteta za matematiko in fiziko 0 0 0