In topology, an Akbulut cork is a structure that is frequently used to show that in 4-dimensions, the smooth h-cobordism theorem fails. It was named after Turkish mathematician Selman Akbulut. A compact contractible Stein 4-manifold with involution on its boundary is called an Akbulut cork, if extends to a self-homeomorphism but cannot extend to a self-diffeomorphism inside (hence a cork is an exotic copy of itself relative to its boundary). A cork is called a cork of a smooth 4-manifold , if removing from and re-gluing it via changes the smooth structure of (this operation is called "cork twisting"). Any exotic copy of a closed simply connected 4-manifold differs from by a single cork twist.
| Attributes | Values |
|---|
| rdfs:label
| - Akbulut-Korken (de)
- Akbulut cork (en)
|
| rdfs:comment
| - In topology, an Akbulut cork is a structure that is frequently used to show that in 4-dimensions, the smooth h-cobordism theorem fails. It was named after Turkish mathematician Selman Akbulut. A compact contractible Stein 4-manifold with involution on its boundary is called an Akbulut cork, if extends to a self-homeomorphism but cannot extend to a self-diffeomorphism inside (hence a cork is an exotic copy of itself relative to its boundary). A cork is called a cork of a smooth 4-manifold , if removing from and re-gluing it via changes the smooth structure of (this operation is called "cork twisting"). Any exotic copy of a closed simply connected 4-manifold differs from by a single cork twist. (en)
- In der Mathematik kommen Akbulut-Korken in der Theorie 4-dimensionaler Mannigfaltigkeiten vor. Insbesondere werden sie bei der Konstruktion exotischer 4-dimensionaler Räume (‘s) verwendet. Während für topologische 4-Mannigfaltigkeiten der gilt, ist dies für differenzierbare 4-Mannigfaltigkeiten nicht der Fall. Stattdessen gilt aber der folgende Satz von , Michael Freedman, Wu-Chung Hsiang, Robert Stong: Die 4-Mannigfaltigkeit ist homöomorph, aber nicht diffeomorph zur Vollkugel und wird als ein Akbulut-Korken bezeichnet. Er ist nach Selman Akbulut benannt. (de)
|
| dcterms:subject
| |
| Wikipage page ID
| |
| Wikipage revision ID
| |
| Link from a Wikipage to another Wikipage
| |
| sameAs
| |
| dbp:wikiPageUsesTemplate
| |
| has abstract
| - In der Mathematik kommen Akbulut-Korken in der Theorie 4-dimensionaler Mannigfaltigkeiten vor. Insbesondere werden sie bei der Konstruktion exotischer 4-dimensionaler Räume (‘s) verwendet. Während für topologische 4-Mannigfaltigkeiten der gilt, ist dies für differenzierbare 4-Mannigfaltigkeiten nicht der Fall. Stattdessen gilt aber der folgende Satz von , Michael Freedman, Wu-Chung Hsiang, Robert Stong: In jedem 5-dimensionalen h-Kobordismus zwischen 4-dimensionalen Mannigfaltigkeiten und gibt es kompakte, zusammenziehbare 4-dimensionale Untermannigfaltigkeiten mit Rand und einen in als Untermannigfaltigkeit mit Rand enthaltenen (kompakten und zusammenziehbaren) h-Kobordismus zwischen und , so dass außerhalb von ein trivialer Kobordismus ist, es also einen Diffeomorphismusgibt. kann so gewählt werden, dass es diffeomorph zur Vollkugel ist, dass einfach zusammenhängend ist und dass es einen Diffeomorphismus gibt, dessen Einschränkung auf den Rand eine Involution ist. Zwei h-kobordante 4-Mannigfaltigkeiten müssen also nicht diffeomorph sein, man kann aber aus gewinnen durch Ausschneiden einer kompakten, zusammenziehbaren Untermannigfaltigkeit und Wiedereinkleben mittels einer Involution von . Die 4-Mannigfaltigkeit ist homöomorph, aber nicht diffeomorph zur Vollkugel und wird als ein Akbulut-Korken bezeichnet. Er ist nach Selman Akbulut benannt. Jeder Akbulut-Korken kann in einen exotischen eingebettet werden. Genauer kann man im obigen Satz einen enthaltenden und außerhalb von trivialen, offenen, h-Kobordismus finden, der homöomorph zu ist. (de)
- In topology, an Akbulut cork is a structure that is frequently used to show that in 4-dimensions, the smooth h-cobordism theorem fails. It was named after Turkish mathematician Selman Akbulut. A compact contractible Stein 4-manifold with involution on its boundary is called an Akbulut cork, if extends to a self-homeomorphism but cannot extend to a self-diffeomorphism inside (hence a cork is an exotic copy of itself relative to its boundary). A cork is called a cork of a smooth 4-manifold , if removing from and re-gluing it via changes the smooth structure of (this operation is called "cork twisting"). Any exotic copy of a closed simply connected 4-manifold differs from by a single cork twist. The basic idea of the Akbulut cork is that when attempting to use the h-corbodism theorem in four dimensions, the cork is the sub-cobordism that contains all the exotic properties of the spaces connected with the cobordism, and when removed the two spaces become trivially h-cobordant and smooth. This shows that in four dimensions, although the theorem does not tell us that two manifolds are diffeomorphic (only homeomorphic), they are "not far" from being diffeomorphic. To illustrate this (without proof), consider a smooth h-cobordism between two 4-manifolds and . Then within there is a sub-cobordism between and and there is a diffeomorphism which is the content of the h-cobordism theorem for n ≥ 5 (here int X refers to the interior of a manifold X). In addition, A and B are diffeomorphic with a diffeomorphism that is an involution on the boundary ∂A = ∂B. Therefore, it can be seen that the h-corbordism K connects A with its "inverted" image B. This submanifold A is the Akbulut cork. (en)
|
| prov:wasDerivedFrom
| |
| page length (characters) of wiki page
| |
| foaf:isPrimaryTopicOf
| |
| is Link from a Wikipage to another Wikipage
of | |
| is Wikipage redirect
of | |
| is Wikipage disambiguates
of | |
| is known for
of | |
| is known for
of | |
| is foaf:primaryTopic
of | |
![[RDF Data]](/fct/images/sw-rdf-blue.png)
![[cxml]](/fct/images/cxml_doc.png)
![[csv]](/fct/images/csv_doc.png)
![[text]](/fct/images/ntriples_doc.png)
![[turtle]](/fct/images/n3turtle_doc.png)
![[ld+json]](/fct/images/jsonld_doc.png)
![[rdf+json]](/fct/images/json_doc.png)
![[rdf+xml]](/fct/images/xml_doc.png)
![[atom+xml]](/fct/images/atom_doc.png)
![[html]](/fct/images/html_doc.png)