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Algorithmic Solvability of the Lifting-Extension Problem
Let X and Y be finite simplicial sets (e.g. finite simplicial complexes), both equipped with a free simplicial action of a finite group G . Assuming that Y is d -connected and dim X ≤ 2 d , for some d ≥ 1 , we provide an algorithm that computes the set of all equivariant homotopy classes of equivari...
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| Published in: | Discrete & computational geometry 2017-06, Vol.57 (4), p.915-965 |
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| Main Authors: | , , |
| Format: | Article |
| Language: | English |
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| cites | cdi_FETCH-LOGICAL-c349t-43553964bb291f7bb6fe10ccc01c2d8fdce8db7ad7e156d95380fdc04c91afe83 |
| container_end_page | 965 |
| container_issue | 4 |
| container_start_page | 915 |
| container_title | Discrete & computational geometry |
| container_volume | 57 |
| creator | Cadek, Martin KrAeal, Marek Vokrinek, Lukas |
| description | Let
X
and
Y
be finite simplicial sets (e.g. finite simplicial complexes), both equipped with a free simplicial action of a finite group
G
. Assuming that
Y
is
d
-connected and
dim
X
≤
2
d
, for some
d
≥
1
, we provide an algorithm that computes the set of all equivariant homotopy classes of equivariant continuous maps
|
X
|
→
|
Y
|
; the existence of such a map can be decided even for
dim
X
≤
2
d
+
1
. This yields the first algorithm for deciding topological embeddability of a
k
-dimensional finite simplicial complex into
R
n
under the condition
k
≤
2
3
n
-
1
. More generally, we present an algorithm that, given a lifting-extension problem satisfying an appropriate stability assumption, computes the set of all homotopy classes of solutions. This result is new even in the non-equivariant situation. |
| doi_str_mv | 10.1007/s00454-016-9855-6 |
| format | article |
| fullrecord | <record><control><sourceid>proquest_cross</sourceid><recordid>TN_cdi_proquest_miscellaneous_1904209600</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>4321957197</sourcerecordid><originalsourceid>FETCH-LOGICAL-c349t-43553964bb291f7bb6fe10ccc01c2d8fdce8db7ad7e156d95380fdc04c91afe83</originalsourceid><addsrcrecordid>eNp1kEtLAzEUhYMoWKs_wN2AGzfRm8ljkmUpvqCgoK7DTCZpU2YmNZmK_fem1IUIri4cvnO4fAhdErghANVtAmCcYSACK8k5FkdoQhgtMTDGjtEESKUwp5U4RWcprSHjCuQEyVm3DNGPq96b4jV0n3XjOz_uiuCKcWWLhXejH5b47mu0Q_JhKF5iaDrbn6MTV3fJXvzcKXq_v3ubP-LF88PTfLbAhjI1YkY5p0qwpikVcVXTCGcJGGOAmLKVrjVWtk1Vt5UlXLSKUwk5BGYUqZ2VdIquD7ubGD62No2698nYrqsHG7ZJEwWsBCUAMnr1B12HbRzyd5pIRWjJKkEyRQ6UiSGlaJ3eRN_XcacJ6L1LfXCps0u9d6lF7pSHTsrssLTx1_K_pW_DdnZF</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>1891324761</pqid></control><display><type>article</type><title>Algorithmic Solvability of the Lifting-Extension Problem</title><source>Springer Nature:Jisc Collections:Springer Nature Read and Publish 2023-2025: Springer Reading List</source><creator>Cadek, Martin ; KrAeal, Marek ; Vokrinek, Lukas</creator><creatorcontrib>Cadek, Martin ; KrAeal, Marek ; Vokrinek, Lukas</creatorcontrib><description>Let
X
and
Y
be finite simplicial sets (e.g. finite simplicial complexes), both equipped with a free simplicial action of a finite group
G
. Assuming that
Y
is
d
-connected and
dim
X
≤
2
d
, for some
d
≥
1
, we provide an algorithm that computes the set of all equivariant homotopy classes of equivariant continuous maps
|
X
|
→
|
Y
|
; the existence of such a map can be decided even for
dim
X
≤
2
d
+
1
. This yields the first algorithm for deciding topological embeddability of a
k
-dimensional finite simplicial complex into
R
n
under the condition
k
≤
2
3
n
-
1
. More generally, we present an algorithm that, given a lifting-extension problem satisfying an appropriate stability assumption, computes the set of all homotopy classes of solutions. This result is new even in the non-equivariant situation.</description><identifier>ISSN: 0179-5376</identifier><identifier>EISSN: 1432-0444</identifier><identifier>DOI: 10.1007/s00454-016-9855-6</identifier><identifier>CODEN: DCGEER</identifier><language>eng</language><publisher>New York: Springer US</publisher><subject>Algorithms ; Combinatorics ; Computational geometry ; Computational Mathematics and Numerical Analysis ; Mathematical analysis ; Mathematical models ; Mathematical problems ; Mathematics ; Mathematics and Statistics ; Stability ; Texts ; Topology</subject><ispartof>Discrete & computational geometry, 2017-06, Vol.57 (4), p.915-965</ispartof><rights>Springer Science+Business Media New York 2017</rights><rights>Discrete & Computational Geometry is a copyright of Springer, 2017.</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c349t-43553964bb291f7bb6fe10ccc01c2d8fdce8db7ad7e156d95380fdc04c91afe83</citedby><cites>FETCH-LOGICAL-c349t-43553964bb291f7bb6fe10ccc01c2d8fdce8db7ad7e156d95380fdc04c91afe83</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>314,776,780,27903,27904</link.rule.ids></links><search><creatorcontrib>Cadek, Martin</creatorcontrib><creatorcontrib>KrAeal, Marek</creatorcontrib><creatorcontrib>Vokrinek, Lukas</creatorcontrib><title>Algorithmic Solvability of the Lifting-Extension Problem</title><title>Discrete & computational geometry</title><addtitle>Discrete Comput Geom</addtitle><description>Let
X
and
Y
be finite simplicial sets (e.g. finite simplicial complexes), both equipped with a free simplicial action of a finite group
G
. Assuming that
Y
is
d
-connected and
dim
X
≤
2
d
, for some
d
≥
1
, we provide an algorithm that computes the set of all equivariant homotopy classes of equivariant continuous maps
|
X
|
→
|
Y
|
; the existence of such a map can be decided even for
dim
X
≤
2
d
+
1
. This yields the first algorithm for deciding topological embeddability of a
k
-dimensional finite simplicial complex into
R
n
under the condition
k
≤
2
3
n
-
1
. More generally, we present an algorithm that, given a lifting-extension problem satisfying an appropriate stability assumption, computes the set of all homotopy classes of solutions. This result is new even in the non-equivariant situation.</description><subject>Algorithms</subject><subject>Combinatorics</subject><subject>Computational geometry</subject><subject>Computational Mathematics and Numerical Analysis</subject><subject>Mathematical analysis</subject><subject>Mathematical models</subject><subject>Mathematical problems</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Stability</subject><subject>Texts</subject><subject>Topology</subject><issn>0179-5376</issn><issn>1432-0444</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2017</creationdate><recordtype>article</recordtype><recordid>eNp1kEtLAzEUhYMoWKs_wN2AGzfRm8ljkmUpvqCgoK7DTCZpU2YmNZmK_fem1IUIri4cvnO4fAhdErghANVtAmCcYSACK8k5FkdoQhgtMTDGjtEESKUwp5U4RWcprSHjCuQEyVm3DNGPq96b4jV0n3XjOz_uiuCKcWWLhXejH5b47mu0Q_JhKF5iaDrbn6MTV3fJXvzcKXq_v3ubP-LF88PTfLbAhjI1YkY5p0qwpikVcVXTCGcJGGOAmLKVrjVWtk1Vt5UlXLSKUwk5BGYUqZ2VdIquD7ubGD62No2698nYrqsHG7ZJEwWsBCUAMnr1B12HbRzyd5pIRWjJKkEyRQ6UiSGlaJ3eRN_XcacJ6L1LfXCps0u9d6lF7pSHTsrssLTx1_K_pW_DdnZF</recordid><startdate>20170601</startdate><enddate>20170601</enddate><creator>Cadek, Martin</creator><creator>KrAeal, Marek</creator><creator>Vokrinek, Lukas</creator><general>Springer US</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope><scope>3V.</scope><scope>7SC</scope><scope>7TB</scope><scope>7XB</scope><scope>88I</scope><scope>8AL</scope><scope>8AO</scope><scope>8FD</scope><scope>8FE</scope><scope>8FG</scope><scope>8FK</scope><scope>8G5</scope><scope>ABJCF</scope><scope>ABUWG</scope><scope>AFKRA</scope><scope>ARAPS</scope><scope>AZQEC</scope><scope>BENPR</scope><scope>BGLVJ</scope><scope>CCPQU</scope><scope>DWQXO</scope><scope>FR3</scope><scope>GNUQQ</scope><scope>GUQSH</scope><scope>HCIFZ</scope><scope>JQ2</scope><scope>K7-</scope><scope>KR7</scope><scope>L6V</scope><scope>L7M</scope><scope>L~C</scope><scope>L~D</scope><scope>M0N</scope><scope>M2O</scope><scope>M2P</scope><scope>M7S</scope><scope>MBDVC</scope><scope>P5Z</scope><scope>P62</scope><scope>PADUT</scope><scope>PQEST</scope><scope>PQQKQ</scope><scope>PQUKI</scope><scope>PRINS</scope><scope>PTHSS</scope><scope>Q9U</scope></search><sort><creationdate>20170601</creationdate><title>Algorithmic Solvability of the Lifting-Extension Problem</title><author>Cadek, Martin ; KrAeal, Marek ; Vokrinek, Lukas</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c349t-43553964bb291f7bb6fe10ccc01c2d8fdce8db7ad7e156d95380fdc04c91afe83</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2017</creationdate><topic>Algorithms</topic><topic>Combinatorics</topic><topic>Computational geometry</topic><topic>Computational Mathematics and Numerical Analysis</topic><topic>Mathematical analysis</topic><topic>Mathematical models</topic><topic>Mathematical problems</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Stability</topic><topic>Texts</topic><topic>Topology</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Cadek, Martin</creatorcontrib><creatorcontrib>KrAeal, Marek</creatorcontrib><creatorcontrib>Vokrinek, Lukas</creatorcontrib><collection>CrossRef</collection><collection>ProQuest Central (Corporate)</collection><collection>Computer and Information Systems Abstracts</collection><collection>Mechanical & Transportation Engineering Abstracts</collection><collection>ProQuest Central (purchase pre-March 2016)</collection><collection>Science Database (Alumni Edition)</collection><collection>Computing Database (Alumni Edition)</collection><collection>ProQuest Pharma Collection</collection><collection>Technology Research Database</collection><collection>ProQuest SciTech Collection</collection><collection>ProQuest Technology Collection</collection><collection>ProQuest Central (Alumni) (purchase pre-March 2016)</collection><collection>Research Library (Alumni Edition)</collection><collection>Materials Science & Engineering Collection</collection><collection>ProQuest Central (Alumni Edition)</collection><collection>ProQuest Central UK/Ireland</collection><collection>Advanced Technologies & Aerospace Collection</collection><collection>ProQuest Central Essentials</collection><collection>ProQuest Central</collection><collection>Technology Collection (ProQuest)</collection><collection>ProQuest One Community College</collection><collection>ProQuest Central Korea</collection><collection>Engineering Research Database</collection><collection>ProQuest Central Student</collection><collection>Research Library Prep</collection><collection>SciTech Premium Collection</collection><collection>ProQuest Computer Science Collection</collection><collection>Computer Science Database</collection><collection>Civil Engineering Abstracts</collection><collection>ProQuest Engineering Collection</collection><collection>Advanced Technologies Database with Aerospace</collection><collection>Computer and Information Systems Abstracts Academic</collection><collection>Computer and Information Systems Abstracts Professional</collection><collection>Computing Database</collection><collection>Research Library</collection><collection>Science Database</collection><collection>Engineering Database</collection><collection>Research Library (Corporate)</collection><collection>Advanced Technologies & Aerospace Database</collection><collection>ProQuest Advanced Technologies & Aerospace Collection</collection><collection>Research Library China</collection><collection>ProQuest One Academic Eastern Edition (DO NOT USE)</collection><collection>ProQuest One Academic</collection><collection>ProQuest One Academic UKI Edition</collection><collection>ProQuest Central China</collection><collection>Engineering Collection</collection><collection>ProQuest Central Basic</collection><jtitle>Discrete & computational geometry</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Cadek, Martin</au><au>KrAeal, Marek</au><au>Vokrinek, Lukas</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Algorithmic Solvability of the Lifting-Extension Problem</atitle><jtitle>Discrete & computational geometry</jtitle><stitle>Discrete Comput Geom</stitle><date>2017-06-01</date><risdate>2017</risdate><volume>57</volume><issue>4</issue><spage>915</spage><epage>965</epage><pages>915-965</pages><issn>0179-5376</issn><eissn>1432-0444</eissn><coden>DCGEER</coden><abstract>Let
X
and
Y
be finite simplicial sets (e.g. finite simplicial complexes), both equipped with a free simplicial action of a finite group
G
. Assuming that
Y
is
d
-connected and
dim
X
≤
2
d
, for some
d
≥
1
, we provide an algorithm that computes the set of all equivariant homotopy classes of equivariant continuous maps
|
X
|
→
|
Y
|
; the existence of such a map can be decided even for
dim
X
≤
2
d
+
1
. This yields the first algorithm for deciding topological embeddability of a
k
-dimensional finite simplicial complex into
R
n
under the condition
k
≤
2
3
n
-
1
. More generally, we present an algorithm that, given a lifting-extension problem satisfying an appropriate stability assumption, computes the set of all homotopy classes of solutions. This result is new even in the non-equivariant situation.</abstract><cop>New York</cop><pub>Springer US</pub><doi>10.1007/s00454-016-9855-6</doi><tpages>51</tpages></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0179-5376 |
| ispartof | Discrete & computational geometry, 2017-06, Vol.57 (4), p.915-965 |
| issn | 0179-5376 1432-0444 |
| language | eng |
| recordid | cdi_proquest_miscellaneous_1904209600 |
| source | Springer Nature:Jisc Collections:Springer Nature Read and Publish 2023-2025: Springer Reading List |
| subjects | Algorithms Combinatorics Computational geometry Computational Mathematics and Numerical Analysis Mathematical analysis Mathematical models Mathematical problems Mathematics Mathematics and Statistics Stability Texts Topology |
| title | Algorithmic Solvability of the Lifting-Extension Problem |
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