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Homotopy classification of maps between simply connected four manifolds

Xu-an Zhao, Hongzhu Gao, Xiaole Su
<span title="">2005</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/ezljl2d3lzga5efenbxdvvfcpa" style="color: black;">Journal of symbolic computation</a> </i> &nbsp;
In this paper, we give a homotopy classification of continuous maps between two simply connected four manifolds M, N and design an algorithm and program to give explicit computations.  ...  Some calculations for self-maps of a rational surface and manifolds of type E 8 are given as examples.  ...  Introduction Suppose M, N be two simply connected four dimensional topological manifolds with base points and [M, N] be the homotopy set of all homotopy classes of continuous maps from M to N preserving  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.jsc.2004.12.009">doi:10.1016/j.jsc.2004.12.009</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/xufawjin2vdp5n642ryd6ljv6y">fatcat:xufawjin2vdp5n642ryd6ljv6y</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20190318142805/https://core.ac.uk/download/pdf/81948764.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/59/41/5941db10ae8b6fa96e22f5c73923c037dad44166.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.jsc.2004.12.009"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> elsevier.com </button> </a>

A diffeomorphism classification of manifolds which are like projective planes

L. Kramer, S. Stolz
<span title="">2007</span> <i title="International Press of Boston"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/gdzmim3p7fehxgwa2izhc64xfy" style="color: black;">Journal of differential geometry</a> </i> &nbsp;
If M is simply connected and its homology has minimal size (i.e., H * (M ) ∼ = Z ⊕ Z), then M is a homotopy sphere (i.e., M is homotopy equivalent to a sphere).  ...  We give a complete diffeomorphism classification of 1-connected closed manifolds M with integral homology H * (M ) ∼ = Z ⊕ Z ⊕ Z, provided that dim(M ) = 4.  ...  Any 1-connected projective plane like manifold of dimension 4 is homeomorphic to the complex projective plane by Freedman's homeomorphism classification of simply connected smooth 4-manifolds [5] .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.4310/jdg/1191860392">doi:10.4310/jdg/1191860392</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/fazpz4nwp5ck3a4w34pw7uhofq">fatcat:fazpz4nwp5ck3a4w34pw7uhofq</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170813184252/https://www3.nd.edu/~stolz/proj_planes.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/5f/2e/5f2e0b9fa6c5b7f885b306c93db6879bf8d6140b.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.4310/jdg/1191860392"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="unlock alternate icon" style="background-color: #fb971f;"></i> Publisher / doi.org </button> </a>

A diffeomorphism classification of manifolds which are like projective planes [article]

Linus Kramer, Stephan Stolz
<span title="2007-02-12">2007</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We give a complete diffeomorphism classification of 1-connected manifolds (of dimension different from 4) whose integral homology is H(M)=Z+Z+Z.  ...  A direct consequence of this result (see Corollary 2.5) is that the connected sum M #Σ of a closed simply connected manifold M of dimension 2m = 4 with a homotopy sphere Σ is diffeomorphic to M provided  ...  In Section 2 we use Kreck's modified surgery approach [11] to show that for a closed simply connected manifold M of dimension 2m = 4 the connected sum M #Σ with a homotopy sphere Σ is diffeomorphic to  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/math/0505621v3">arXiv:math/0505621v3</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/dfuitflxzfflji5jhwjagq7t3e">fatcat:dfuitflxzfflji5jhwjagq7t3e</a> </span>
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Page 492 of Mathematical Reviews Vol. , Issue 2000a [page]

<span title="">2000</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
The authors complete the homotopy classification of “sufficiently connectedmanifolds.  ...  Taylor (1-NDM; Notre Dame, IN) 2000a:55017 55P15 57R19 Kobayakawa, Norio (J-KANASS; Kanazawa); Ishimoto, Hiroyasu (J-KANASS; Kanazawa) Homotopy classification of sufficiently connected manifolds.  ... 
<span class="external-identifiers"> </span>
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The work of Tom Farrell and Lowell Jones in topology and geometry [article]

James F. Davis
<span title="2010-08-08">2010</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
This is a survey of some of the work of Tom Farrell and Lowell Jones. This is the lead article of a special issue of the Pure and Applied Mathematics Quarterly.  ...  Although, I have only met Lowell a handful of times, Tom has become a great friend and a mathematical inspiration.  ...  classification of four-manifolds. ideas.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1006.1489v2">arXiv:1006.1489v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/hjlwoau6vzckldlcbqcqyqjal4">fatcat:hjlwoau6vzckldlcbqcqyqjal4</a> </span>
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The Work of Tom Farrell and Lowell Jones in Topology and Geometry

James F. Davis
<span title="">2012</span> <i title="International Press of Boston"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/zqclzy5yebar3oqqmppyjia4le" style="color: black;">Pure and Applied Mathematics Quarterly</a> </i> &nbsp;
classification of four-manifolds. ideas.  ...  Wall wrote his magnificent tome Surgery on compact manifolds, the foundational work on the surgery classification of non-simply connected manifolds.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.4310/pamq.2012.v8.n1.a3">doi:10.4310/pamq.2012.v8.n1.a3</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/hjebol6yjvdbnoyzkqcbawqaua">fatcat:hjebol6yjvdbnoyzkqcbawqaua</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20190429194453/https://www.intlpress.com/site/pub/files/_fulltext/journals/pamq/2012/0008/0001/PAMQ-2012-0008-0001-a003.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/8d/77/8d77abcc0b9e867b9581ab891aaacf3dfeaddd06.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.4310/pamq.2012.v8.n1.a3"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="unlock alternate icon" style="background-color: #fb971f;"></i> Publisher / doi.org </button> </a>

The Loop Space Homotopy Type of Simply-connected Four-manifolds and their Generalizations [article]

Piotr Beben, Stephen Theriault
<span title="2014-06-03">2014</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We determine loop space decompositions of simply-connected four-manifolds, (n-1)-connected 2n-dimensional manifolds provided n∉{4,8}, and connected sums of products of two spheres.  ...  These are obtained as special cases of a more general loop space decomposition of certain torsion-free CW-complexes with well-behaved skeleta and some Poincaré duality features.  ...  The homotopy theory of simply-connected four-manifolds has continued to attract considerable attention since Milnor's classification.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1406.0651v1">arXiv:1406.0651v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/m4dkha2rofhojpewuavvghxyn4">fatcat:m4dkha2rofhojpewuavvghxyn4</a> </span>
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Topological 4-manifolds with 4-dimensional fundamental group [article]

Daniel Kasprowski, Markus Land
<span title="2021-06-28">2021</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
This shows rigidity in many cases that lie between aspherical 4-manifolds, where rigidity is expected by Borel's conjecture, and simply connected manifolds where rigidity is a consequence of Freedman's  ...  We consider topological, closed, connected manifolds with fundamental group π whose canonical map to Bπ has degree 1 and show that two such manifolds are s-cobordant if and only if their equivariant intersection  ...  This builds a bridge between the rigidity phenomena envisioned by Borel for aspherical manifolds and the rigidity present in simply connected topological 4-manifolds by Freedman's results.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2007.03399v3">arXiv:2007.03399v3</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/zd2q2qvpqjbwlhek2j46evtry4">fatcat:zd2q2qvpqjbwlhek2j46evtry4</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20201031214413/https://arxiv.org/pdf/2007.03399v2.pdf" title="fulltext PDF download [not primary version]" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <span style="color: #f43e3e;">&#10033;</span> <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/ab/f3/abf348bdc02f726443e795cb4b46e0f6a381892a.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2007.03399v3" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

The loop space homotopy type of simply-connected four-manifolds and their generalizations

Piotr Beben, Stephen Theriault
<span title="">2014</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/37jjomjmvfhrzf3di2gjnrrfuu" style="color: black;">Advances in Mathematics</a> </i> &nbsp;
We determine loop space decompositions of simply-connected four-manifolds, (n − 1)-connected 2n-dimensional manifolds provided n / ∈ {4, 8}, and connected sums of products of two spheres.  ...  These are obtained as special cases of a more general loop space decomposition of certain torsion-free CW -complexes with well-behaved skeleta and some Poincaré duality features.  ...  The homotopy theory of simply-connected four-manifolds has continued to attract considerable attention since Milnor's classification.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.aim.2014.05.015">doi:10.1016/j.aim.2014.05.015</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/zbwkbct5v5atdl555aryvfh75a">fatcat:zbwkbct5v5atdl555aryvfh75a</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200306051149/https://eprints.soton.ac.uk/365615/1/1_s2.0_S0001870814001911_main.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/99/9b/999b4521e30ef907c2fb48ffa88e0955a2f46261.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.aim.2014.05.015"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> elsevier.com </button> </a>

Page 4136 of Mathematical Reviews Vol. , Issue 81J [page]

<span title="">1981</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
XS7) diffeomorphism of homotopic simply connected 4-manifolds, the Cappell-Shaneson counterexample to this theorem in a non-simply connected case (fake RP“s), classification of integral bilinear unimodular  ...  forms, Rohlin’s theorem on the relation between the signature of a 4 manifold and the Arf invariant of 2-submanifolds, Matsumoto’s proof of it, Rohlin’s homotopy-theoretic proof of the special case of  ... 
<span class="external-identifiers"> </span>
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Periodic maps on simply connected four-manifolds

Dariusz M. Wilczyński
<span title="">1991</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/jinclcgdtbhstao347on5p4j2m" style="color: black;">Topology</a> </i> &nbsp;
IN THIS paper we describe certain invariants of periodic maps on closed, simply connected four-manifolds with the purpose of classifying such maps up to homeomorphism.  ...  In Section 2 we discuss the homotopy classification of locally linear, pseudofree actions of finite groups on simply connected fourmanifolds.  ...  tIO\IOTOPY TYPES OF PSEL.DOFREE AC-TIOXS Let ,U be a closed. oriented. simply connected four-manifold. and let G be a finite group acting locally linearly on ,\I.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/0040-9383(91)90033-z">doi:10.1016/0040-9383(91)90033-z</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/r2luczvnhrf2xambliec3jdjzu">fatcat:r2luczvnhrf2xambliec3jdjzu</a> </span>
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Topology of homology manifolds [article]

John L. Bryant, Steven C. Ferry, Washington Mio, Shmuel Weinberger
<span title="1993-04-01">1993</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We construct examples of nonresolvable generalized n-manifolds, n≥ 6, with arbitrary resolution obstruction, homotopy equivalent to any simply connected, closed n-manifold.  ...  We further investigate the structure of generalized manifolds and present a program for understanding their topology.  ...  Thus, one knows that if X is simply connected, S(X) contains generalized manifolds of every index and there is a one-to-one correspondence between s-cobordism classes of generalized manifolds homotopy  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/math/9304210v1">arXiv:math/9304210v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/slo2niktvncuxkcics4tcpz3pm">fatcat:slo2niktvncuxkcics4tcpz3pm</a> </span>
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Branched coverings of simply connected manifolds [article]

Christoforos Neofytidis
<span title="2014-11-24">2014</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
map; (2) every simply connected, closed five-manifold admits a branched double covering by a product of the circle with a connected sum of copies of S^3 × S^1, followed by a map whose degree is determined  ...  More precisely, we show that: (1) every simply connected, closed four-manifold admits a branched double covering by a product of the circle with a connected sum of copies of S^2 × S^1, followed by a collapsing  ...  It is well-known that the homotopy classification of simply connected four-manifolds alone implies the existence of such a degree one map between the homotopy types of CP 2 #CP 2 and M.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1210.1555v3">arXiv:1210.1555v3</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/37thj3mv6faj3j6sh3y56aphi4">fatcat:37thj3mv6faj3j6sh3y56aphi4</a> </span>
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Page 641 of Mathematical Reviews Vol. 39, Issue 3 [page]

<span title="">1970</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
The connection between stable and non-stable re- sults follows from another application of Lees’ immersion theorem [J. A.  ...  The above result for simply connected manifolds, and for reductions of the actual tangent bundle rather than the stable tangent bundle, was given previously by the re- viewer [Bull. Amer. Math.  ... 
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Rational homotopy theory and nonnegative curvature [article]

Jianzhong Pan
<span title="2001-06-29">2001</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
In this note, we answer positively a question by Belegradek and Kapovitch about the relation between rational homotopy theory and a problem in Riemannian geometry which asks that total spaces of which  ...  vector bundles over compact nonnegative curved manifolds admit (complete) metrics with nonnegative curvature.  ...  [X, Y ] will be the based homotopy classes of based maps between them. map(X, Y ) is the space of maps from X to Y and map(X, Y ) f is the connected component of map(X, Y ) which contains the map f : X  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/math/0106262v1">arXiv:math/0106262v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/lo36lgyfqvezfn5oegwy2fzhry">fatcat:lo36lgyfqvezfn5oegwy2fzhry</a> </span>
<a target="_blank" rel="noopener" href="https://archive.org/download/arxiv-math0106262/math0106262.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> File Archive [PDF] </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/math/0106262v1" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>
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