W. Haken :
“Theorie der Normalflächen: Ein Isotopiekriterium für den Kreisknoten Theory of normal surfaces: An isotopic criterion for the circular knot ],
Acta Math.
105 : 3–4
(1961 ),
pp. 245–375 .
MR
141106 Zbl
0100.19402 article
BibTeX
@article {key141106m,
AUTHOR = {Haken, Wolfgang},
TITLE = {Theorie der {N}ormalfl\"achen: {E}in
{I}sotopiekriterium f\"ur den {K}reisknoten
[Theory of normal surfaces: {A}n isotopic
criterion for the circular knot]},
JOURNAL = {Acta Math.},
FJOURNAL = {Acta Mathematica},
VOLUME = {105},
NUMBER = {3--4},
YEAR = {1961},
PAGES = {245--375},
DOI = {10.1007/BF02559591},
NOTE = {MR:141106. Zbl:0100.19402.},
ISSN = {0001-5962},
}
W. Haken :
“Über das Homöomorphieproblem der 3-Mannigfaltigkeiten, I On the homomorphism problem for 3-manifolds, I ],
Math. Z.
80 : 1
(December 1962 ),
pp. 89–120 .
MR
160196 Zbl
0106.16605 article
BibTeX
@article {key160196m,
AUTHOR = {Haken, Wolfgang},
TITLE = {\"{U}ber das {H}om\"oomorphieproblem
der 3-{M}annigfaltigkeiten, {I} [On
the homomorphism problem for 3-manifolds,
{I}]},
JOURNAL = {Math. Z.},
FJOURNAL = {Mathematische Zeitschrift},
VOLUME = {80},
NUMBER = {1},
MONTH = {December},
YEAR = {1962},
PAGES = {89--120},
DOI = {10.1007/BF01162369},
NOTE = {MR:160196. Zbl:0106.16605.},
ISSN = {0025-5874},
}
W. Haken :
“On homotopy 3-spheres Ill. J. Math.
10 : 1
(1966 ),
pp. 159–178 .
Reprinted in Ill. J. Math. 60 :1 (2016)
MR
219072 Zbl
0131.20704 article
Abstract
BibTeX
A homotopy 3-sphere \( M^3 \) is a compact, simply connected 3-manifold without boundary. After the work of Moise [1952] and Bing [1959] \( M^3 \) possesses a triangulation. The Poincaré conjecture [1904] states that every homotopy 3-sphere \( M^3 \) is a 3-sphere. In this paper we prove three theorems, related to the Poincaré conjecture, about maps of a 3-sphere \( S^3 \) onto \( M^3 \) and about 1- and 2-spheres in \( M^3 \) .
@article {key219072m,
AUTHOR = {Haken, Wolfgang},
TITLE = {On homotopy 3-spheres},
JOURNAL = {Ill. J. Math.},
FJOURNAL = {Illinois Journal of Mathematics},
VOLUME = {10},
NUMBER = {1},
YEAR = {1966},
PAGES = {159--178},
URL = {http://projecteuclid.org/euclid.ijm/1256055210},
NOTE = {Reprinted in \textit{Ill. J. Math.}
\textbf{60}:1 (2016). MR:219072. Zbl:0131.20704.},
ISSN = {0019-2082},
}
W. Haken :
“Trivial loops in homotopy 3-spheres Ill. J. Math.
11 : 4
(1967 ),
pp. 547–554 .
MR
219073 Zbl
0153.25703 article
BibTeX
@article {key219073m,
AUTHOR = {Haken, Wolfgang},
TITLE = {Trivial loops in homotopy 3-spheres},
JOURNAL = {Ill. J. Math.},
FJOURNAL = {Illinois Journal of Mathematics},
VOLUME = {11},
NUMBER = {4},
YEAR = {1967},
PAGES = {547--554},
URL = {http://projecteuclid.org/euclid.ijm/1256054445},
NOTE = {MR:219073. Zbl:0153.25703.},
ISSN = {0019-2082},
}
W. Haken :
“Some results on surfaces in 3-manifolds ,”
pp. 39–98
in
Studies in modern topology .
Edited by P. J. Hilton .
MAA Studies in Mathematics 5 .
Prentice-Hall (Englewood Cliffs, NJ ),
1968 .
MR
224071 Zbl
0194.24902 incollection
People
BibTeX
@incollection {key224071m,
AUTHOR = {Haken, Wolfgang},
TITLE = {Some results on surfaces in 3-manifolds},
BOOKTITLE = {Studies in modern topology},
EDITOR = {Hilton, P. J.},
SERIES = {MAA Studies in Mathematics},
NUMBER = {5},
PUBLISHER = {Prentice-Hall},
ADDRESS = {Englewood Cliffs, NJ},
YEAR = {1968},
PAGES = {39--98},
NOTE = {MR:224071. Zbl:0194.24902.},
ISSN = {0081-8208},
}
W. W. Boone, W. Haken, and V. Poénaru :
“On recursively unsolvable problems in topology and their classification 37–74
in
Contributions to mathematical logic
(Hannover, Germany, August 1966 ).
Edited by H. A. Schmidt, K. Schütte, and H.-J. Thiele .
Studies in Logic and the Foundations of Mathematics 50 .
North-Holland (Amsterdam ),
1968 .
MR
263090 Zbl
0246.57015 incollection
Abstract
People
BibTeX
The present paper has been inspired by Markov [1958] on the unsolvability of the homeomorphism problem. We shall show that certain familiar decision problems in topology, in dimension \( \geq 4 \) , are recursively unsolvable, in the strong sense that these problems can be taken to be of any preassigned, recursively enumerable degree of unsolvability.
We have tried to make this paper accessible to both logicians and topologists. For this reason, we have recalled many familiar definitions, and we have carried out some of the proofs in more detail than would be necessary for a reader familiar with the techniques applied.
@incollection {key263090m,
AUTHOR = {Boone, W. W. and Haken, W. and Po\'enaru,
V.},
TITLE = {On recursively unsolvable problems in
topology and their classification},
BOOKTITLE = {Contributions to mathematical logic},
EDITOR = {Schmidt, H. Arnold and Sch\"utte, Kurt
and Thiele, H.-J.},
SERIES = {Studies in Logic and the Foundations
of Mathematics},
NUMBER = {50},
PUBLISHER = {North-Holland},
ADDRESS = {Amsterdam},
YEAR = {1968},
PAGES = {37--74},
DOI = {10.1016/S0049-237X(08)70518-4},
NOTE = {(Hannover, Germany, August 1966). MR:263090.
Zbl:0246.57015.},
ISSN = {0049-237X},
ISBN = {9780444534149},
}
W. Haken :
“Algebraically trivial decompositions of homotopy 3-spheres Ill. J. Math.
12 : 1
(1968 ),
pp. 133–170 .
MR
222902 Zbl
0171.22302 article
BibTeX
@article {key222902m,
AUTHOR = {Haken, Wolfgang},
TITLE = {Algebraically trivial decompositions
of homotopy 3-spheres},
JOURNAL = {Ill. J. Math.},
FJOURNAL = {Illinois Journal of Mathematics},
VOLUME = {12},
NUMBER = {1},
YEAR = {1968},
PAGES = {133--170},
URL = {http://projecteuclid.org/euclid.ijm/1256054324},
NOTE = {MR:222902. Zbl:0171.22302.},
ISSN = {0019-2082},
}
W. Haken :
“Various aspects of the three-dimensional Poincaré problem ,”
pp. 140–152
in
Topology of manifolds
(Athens, GA, 11–22 August 1969 ).
Edited by J. C. Cantrell and C. H. Edwards .
Markham (Chicago ),
1970 .
MR
273624 Zbl
0298.55002 incollection
People
BibTeX
@incollection {key273624m,
AUTHOR = {Haken, Wolfgang},
TITLE = {Various aspects of the three-dimensional
{P}oincar\'e problem},
BOOKTITLE = {Topology of manifolds},
EDITOR = {Cantrell, J. C. and Edwards, C. H.},
PUBLISHER = {Markham},
ADDRESS = {Chicago},
YEAR = {1970},
PAGES = {140--152},
NOTE = {(Athens, GA, 11--22 August 1969). MR:273624.
Zbl:0298.55002.},
ISBN = {9780841010185},
}
W. Haken :
“Connections between topological and group theoretical decision problems 427–441
in
Word problems: Decision problems and the Burnside problem in group theory
(Irvine, CA, September 1969 ).
Edited by W. W. Boone, R. C. Lyndon, and F. B. Cannonito .
Studies in Logic and the Foundations of Mathematics 71 .
North-Holland (Amsterdam ),
1973 .
Conference dedicated to Hanna Neumann.
MR
397736 Zbl
0265.02033 incollection
Abstract
People
BibTeX
This chapter discusses connections between topological and group theoretical decision problems. The chapter provides the logician with a brief survey concerning results and open questions on decision problems in topology and their relations to group theoretic decision problems. Usually topological decision problems are concerned with finite simplicial complexes. Such complexes can be described (up to isomorphism) by their incidence matrices that are finite matrices with integral entries. The chapter also discusses homeomorphism problems, isomorphism problems of fundamental groups, unsolvability results, solvability results, and some open decision problems. The chapter discusses the sufficiently largeness problem for 3-manifolds.
@incollection {key397736m,
AUTHOR = {Haken, Wolfgang},
TITLE = {Connections between topological and
group theoretical decision problems},
BOOKTITLE = {Word problems: {D}ecision problems and
the {B}urnside problem in group theory},
EDITOR = {Boone, W. W. and Lyndon, R. C. and Cannonito,
F. B.},
SERIES = {Studies in Logic and the Foundations
of Mathematics},
NUMBER = {71},
PUBLISHER = {North-Holland},
ADDRESS = {Amsterdam},
YEAR = {1973},
PAGES = {427--441},
DOI = {10.1016/S0049-237X(08)71911-6},
NOTE = {(Irvine, CA, September 1969). Conference
dedicated to Hanna Neumann. MR:397736.
Zbl:0265.02033.},
ISSN = {0049-237X},
ISBN = {9780720422719},
}
W. Haken :
“Some special presentations of homotopy 3-spheres 97–107
in
Topology conference
(Blacksburg, VA, 22–24 March 1973 ).
Edited by H. F. Dickman, Jr. and P. Fletcher .
Lecture Notes in Mathematics 375 .
Springer (Berlin ),
1974 .
MR
356054 Zbl
0289.55003 incollection
People
BibTeX
@incollection {key356054m,
AUTHOR = {Haken, Wolfgang},
TITLE = {Some special presentations of homotopy
3-spheres},
BOOKTITLE = {Topology conference},
EDITOR = {Dickman, Jr., H. F. and Fletcher, P.},
SERIES = {Lecture Notes in Mathematics},
NUMBER = {375},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {1974},
PAGES = {97--107},
DOI = {10.1007/BFb0064015},
NOTE = {(Blacksburg, VA, 22--24 March 1973).
MR:356054. Zbl:0289.55003.},
ISSN = {0075-8434},
ISBN = {9783540066842},
}
K. Appel and W. Haken :
“Every planar map is four colorable Bull. Am. Math. Soc.
82 : 5
(September 1976 ),
pp. 711–712 .
MR
424602 Zbl
0331.05106 article
Abstract
People
BibTeX
@article {key424602m,
AUTHOR = {Appel, K. and Haken, W.},
TITLE = {Every planar map is four colorable},
JOURNAL = {Bull. Am. Math. Soc.},
FJOURNAL = {Bulletin of the American Mathematical
Society},
VOLUME = {82},
NUMBER = {5},
MONTH = {September},
YEAR = {1976},
PAGES = {711--712},
DOI = {10.1090/S0002-9904-1976-14122-5},
NOTE = {MR:424602. Zbl:0331.05106.},
ISSN = {0002-9904},
}
K. Appel and W. Haken :
“Every planar map is four colorable, I: Discharging Ill. J. Math.
21 : 3
(1977 ),
pp. 429–490 .
A microfiche supplement to both parts was published in Ill. J. Math. 21 :3 (1977)
MR
543792 Zbl
0387.05009 article
Abstract
People
BibTeX
We begin by describing, in chronological order, the earlier results which led to the work of this paper. The proof of the Four Color Theorem requires the results of Sections 2 and 3 of this paper and the reducibility results of Part II. Sections 4 and 5 will be devoted to an attempt to explain the difficulties of the Four Color Problem and the unusual nature of the proof.
@article {key543792m,
AUTHOR = {Appel, K. and Haken, W.},
TITLE = {Every planar map is four colorable,
{I}: {D}ischarging},
JOURNAL = {Ill. J. Math.},
FJOURNAL = {Illinois Journal of Mathematics},
VOLUME = {21},
NUMBER = {3},
YEAR = {1977},
PAGES = {429--490},
URL = {http://projecteuclid.org/euclid.ijm/1256049011},
NOTE = {A microfiche supplement to both parts
was published in \textit{Ill. J. Math.}
\textbf{21}:3 (1977). MR:543792. Zbl:0387.05009.},
ISSN = {0019-2082},
}
K. Appel, W. Haken, and J. Koch :
“Every planar map is four colorable, II: Reducibility Ill. J. Math.
21 : 3
(1977 ),
pp. 491–567 .
A microfiche supplement to both parts was published in Ill. J. Math. 21 :3 (1977)
MR
543793 Zbl
0387.05010 article
Abstract
People
BibTeX
In Part I of this paper, a discharging procedure is defined which yields the unavoidability (in planar triangulations) of a set \( \mathscr{U} \) of configurations of ring size fourteen or less. In this part, \( \mathscr{U} \) is presented (as Table \( \mathscr{U} \) consisting of Figures 1–63) together with a discussion of the reducibility proofs of its members.
@article {key543793m,
AUTHOR = {Appel, K. and Haken, W. and Koch, J.},
TITLE = {Every planar map is four colorable,
{II}: {R}educibility},
JOURNAL = {Ill. J. Math.},
FJOURNAL = {Illinois Journal of Mathematics},
VOLUME = {21},
NUMBER = {3},
YEAR = {1977},
PAGES = {491--567},
URL = {http://projecteuclid.org/euclid.ijm/1256049012},
NOTE = {A microfiche supplement to both parts
was published in \textit{Ill. J. Math.}
\textbf{21}:3 (1977). MR:543793. Zbl:0387.05010.},
ISSN = {0019-2082},
}
K. Appel and W. Haken :
“The class check lists corresponding to the supplement to ‘Every planar map is four colorable. Part I and Part II’ ,”
Ill. J. Math.
21 : 3
(1977 ),
pp. C1–C210 .
Microfiche supplement.
Extra material to accompany the supplement published in Ill. J. Math. 21 :3 (1977)
MR
543794 article
People
BibTeX
@article {key543794m,
AUTHOR = {Appel, K. and Haken, W.},
TITLE = {The class check lists corresponding
to the supplement to ``{E}very planar
map is four colorable. {P}art {I} and
{P}art {II}''},
JOURNAL = {Ill. J. Math.},
FJOURNAL = {Illinois Journal of Mathematics},
VOLUME = {21},
NUMBER = {3},
YEAR = {1977},
PAGES = {C1--C210},
NOTE = {Microfiche supplement. Extra material
to accompany the supplement published
in \textit{Ill. J. Math.} \textbf{21}:3
(1977). MR:543794.},
ISSN = {0019-2082},
}
K. Appel and W. Haken :
“Microfiche supplement to ‘Every planar map is four colorable. Part I and Part II’ Ill. J. Math.
21 : 3
(1977 ),
pp. 1–251 .
Microfiche supplement.
Supplement to the two part article published as Ill. J. Math. 21 :3 (1977)Ill. J. Math. 21 :3 (1977)Ill. J. Math. 21 :3 (1977)
MR
543795 article
People
BibTeX
@article {key543795m,
AUTHOR = {Appel, K. and Haken, W.},
TITLE = {Microfiche supplement to ``{E}very planar
map is four colorable. {P}art {I} and
{P}art {II}''},
JOURNAL = {Ill. J. Math.},
FJOURNAL = {Illinois Journal of Mathematics},
VOLUME = {21},
NUMBER = {3},
YEAR = {1977},
PAGES = {1--251},
URL = {https://projecteuclid.org/euclid.ijm/1256049023},
NOTE = {Microfiche supplement. Supplement to
the two part article published as \textit{Ill.
J. Math.} \textbf{21}:3 (1977) and \textit{Ill.
J. Math.} \textbf{21}:3 (1977). A class
check list was also published as \textit{Ill.
J. Math.} \textbf{21}:3 (1977). MR:543795.},
ISSN = {0019-2082},
}
K. Appel and W. Haken :
Every planar map is four colorable Contemporary Mathematics 98 .
American Mathematical Society (Providence, RI ),
1989 .
With the collaboration of J. Koch.
MR
1025335 Zbl
0681.05027 book
People
BibTeX
@book {key1025335m,
AUTHOR = {Appel, Kenneth and Haken, Wolfgang},
TITLE = {Every planar map is four colorable},
SERIES = {Contemporary Mathematics},
NUMBER = {98},
PUBLISHER = {American Mathematical Society},
ADDRESS = {Providence, RI},
YEAR = {1989},
PAGES = {xvi+741},
DOI = {10.1090/conm/098},
NOTE = {With the collaboration of J. Koch. MR:1025335.
Zbl:0681.05027.},
ISSN = {0271-4132},
ISBN = {9780821851036},
}
