[1] D. L. Johnson :
The homology of locally finite CW complexes .
Ph.D. thesis ,
UCLA ,
1968 .
Advised by N. Grossman .
phdthesis
People
BibTeX
@phdthesis {key63473411,
AUTHOR = {Johnson, Dennis L.},
TITLE = {The homology of locally finite {CW}
complexes},
SCHOOL = {UCLA},
YEAR = {1968},
PAGES = {xi+315},
NOTE = {Advised by N. Grossman.},
}
[2] D. L. Johnson :
“The diophantine problem \( Y^2 - X^3 = A \) in a polynomial ring ,”
Pac. J. Math.
43 : 1
(March 1972 ),
pp. 151–155 .
MR
0316382
Zbl
0273.10044
article
Abstract
BibTeX
Let \( C[z] \) be the ring of polynomials in \( z \) with complex coefficients; we consider the equation
\[ Y^2 - X^3 = A ,\]
with \( A \in C[z] \) given, and seek solutions of this with \( X,Y \in C[z] \) , i.e. we treat the equation as a “polynomial diophantine” problem. We show that when \( A \) is of degree 5 or 6 and has no multiple roots, then there are exactly 240 solutions \( (X,Y) \) to the problem with \( \deg X \leq 2 \) and \( \deg Y \leq 3 \) .
@article {key0316382m,
AUTHOR = {Johnson, Dennis L.},
TITLE = {The diophantine problem \$Y^2 - X^3 =
A\$ in a polynomial ring},
JOURNAL = {Pac. J. Math.},
FJOURNAL = {Pacific Journal of Mathematics},
VOLUME = {43},
NUMBER = {1},
MONTH = {March},
YEAR = {1972},
PAGES = {151--155},
DOI = {10.2140/pjm.1972.43.151},
NOTE = {MR:0316382. Zbl:0273.10044.},
ISSN = {0030-8730},
}
[3] D. L. Johnson :
“Homeomorphisms of a surface which act trivially on homology ,”
Proc. Am. Math. Soc.
75 : 1
(1979 ),
pp. 119–125 .
MR
529227
Zbl
0407.57003
article
Abstract
BibTeX
Let \( \mathfrak{M} \) be the mapping class group of a surface of genus \( g \geq 3 \) , and \( \mathscr{I} \) the subgroup of those classes acting trivially on homology. An infinite set of generators for \( \mathscr{I} \) , involving three conjugacy classes, was obtained by Powell. In this paper we improve Powell’s result to show that \( \mathscr{I} \) is generated by a single conjugacy class and that \( [\mathfrak{M},\mathscr{I}] = \mathscr{I} \) .
@article {key529227m,
AUTHOR = {Johnson, Dennis L.},
TITLE = {Homeomorphisms of a surface which act
trivially on homology},
JOURNAL = {Proc. Am. Math. Soc.},
FJOURNAL = {Proceedings of the American Mathematical
Society},
VOLUME = {75},
NUMBER = {1},
YEAR = {1979},
PAGES = {119--125},
DOI = {10.2307/2042686},
NOTE = {MR:529227. Zbl:0407.57003.},
ISSN = {0002-9939},
CODEN = {PAMYAR},
}
[4] D. Johnson :
“Conjugacy relations in subgroups of the mapping class group and a group-theoretic description of the Rochlin invariant ,”
Math. Ann.
249 : 3
(1980 ),
pp. 243–263 .
MR
579104
Zbl
0409.57010
article
BibTeX
@article {key579104m,
AUTHOR = {Johnson, Dennis},
TITLE = {Conjugacy relations in subgroups of
the mapping class group and a group-theoretic
description of the {R}ochlin invariant},
JOURNAL = {Math. Ann.},
FJOURNAL = {Mathematische Annalen},
VOLUME = {249},
NUMBER = {3},
YEAR = {1980},
PAGES = {243--263},
DOI = {10.1007/BF01363898},
NOTE = {MR:579104. Zbl:0409.57010.},
ISSN = {0025-5831},
CODEN = {MAANA3},
}
[5] D. Johnson :
“An abelian quotient of the mapping class group \( \mathscr{I}_g \) ,”
Math. Ann.
249 : 3
(1980 ),
pp. 225–242 .
MR
579103
Zbl
0409.57009
article
BibTeX
@article {key579103m,
AUTHOR = {Johnson, Dennis},
TITLE = {An abelian quotient of the mapping class
group \$\mathscr{I}_g\$},
JOURNAL = {Math. Ann.},
FJOURNAL = {Mathematische Annalen},
VOLUME = {249},
NUMBER = {3},
YEAR = {1980},
PAGES = {225--242},
DOI = {10.1007/BF01363897},
NOTE = {MR:579103. Zbl:0409.57009.},
ISSN = {0025-5831},
CODEN = {MAANA3},
}
[6] D. Johnson :
“Quadratic forms and the Birman–Craggs homomorphisms ,”
Trans. Am. Math. Soc.
261 : 1
(September 1980 ),
pp. 235–254 .
MR
576873
Zbl
0457.57006
article
Abstract
BibTeX
Let \( \mathfrak{M}_g \) be the mapping class group of a genus \( g \) orientable surface \( M \) , and \( \mathscr{I}_g \) the subgroup of those maps acting trivially on the homology group \( H_1(M,Z) \) . Birman and Craggs produced homomorphisms from \( \mathscr{I}_g \) to \( Z_2 \) via the Rochlin invariant and raised the question of enumerating them; in this paper we answer their question. It is shown that the homomorphisms are closely related to the quadratic forms on \( H_1(M,Z_2) \) which induce the intersection form; in fact, they are in 1-1 correspondence with those quadratic forms of Arf invariant zero. Furthermore, the methods give a description of the quotient of \( \mathscr{I}_g \) by the intersection of the kernels of all these homomorphisms. It is a \( Z_2 \) -vector space isomorphic to a certain space of cubic polynomials over \( H_1(M,Z_2) \) . The dimension is then computed and found to be
\[ \begin{pmatrix} 2g \\ 3 \end{pmatrix} + \begin{pmatrix} 2g \\ 2 \end{pmatrix} .\]
These results are also extended to the case of a surface with one boundary component, and in this situation the linear relations among the various homomorphisms are also determined.
@article {key576873m,
AUTHOR = {Johnson, Dennis},
TITLE = {Quadratic forms and the {B}irman--{C}raggs
homomorphisms},
JOURNAL = {Trans. Am. Math. Soc.},
FJOURNAL = {Transactions of the American Mathematical
Society},
VOLUME = {261},
NUMBER = {1},
MONTH = {September},
YEAR = {1980},
PAGES = {235--254},
DOI = {10.2307/1998327},
NOTE = {MR:576873. Zbl:0457.57006.},
ISSN = {0002-9947},
CODEN = {TAMTAM},
}
[7] D. Johnson :
“Spin structures and quadratic forms on surfaces ,”
J. London Math. Soc. (2)
22 : 2
(1980 ),
pp. 365–373 .
MR
588283
Zbl
0454.57011
article
BibTeX
@article {key588283m,
AUTHOR = {Johnson, Dennis},
TITLE = {Spin structures and quadratic forms
on surfaces},
JOURNAL = {J. London Math. Soc. (2)},
FJOURNAL = {The Journal of the London Mathematical
Society. Second Series},
VOLUME = {22},
NUMBER = {2},
YEAR = {1980},
PAGES = {365--373},
DOI = {10.1112/jlms/s2-22.2.365},
NOTE = {MR:588283. Zbl:0454.57011.},
ISSN = {0024-6107},
CODEN = {JLMSAK},
}
[8] D. Johnson :
“Homomorphs of knot groups ,”
Proc. Am. Math. Soc.
78 : 1
(1980 ),
pp. 135–138 .
MR
548101
Zbl
0435.57003
article
Abstract
BibTeX
@article {key548101m,
AUTHOR = {Johnson, Dennis},
TITLE = {Homomorphs of knot groups},
JOURNAL = {Proc. Am. Math. Soc.},
FJOURNAL = {Proceedings of the American Mathematical
Society},
VOLUME = {78},
NUMBER = {1},
YEAR = {1980},
PAGES = {135--138},
DOI = {10.2307/2043056},
NOTE = {MR:548101. Zbl:0435.57003.},
ISSN = {0002-9939},
CODEN = {PAMYAR},
}
[9] K. Johannson and D. Johnson :
Non-bording diffeomorphisms of surfaces which act trivially on homology .
Preprint ,
1980 .
techreport
People
BibTeX
Read PDF
@techreport {key30621128,
AUTHOR = {Johannson, Klaus and Johnson, Dennis},
TITLE = {Non-bording diffeomorphisms of surfaces
which act trivially on homology},
TYPE = {preprint},
YEAR = {1980},
}
[10] D. Johnson :
“The structure of the Torelli group, I: A finite set of generators for \( \mathscr{I} \) ,”
Ann. Math. (2)
118 : 3
(November 1983 ),
pp. 423–442 .
Parts II and III were published in Topology 24 :2 (1985) and Topology 24 :2 (1985) .
MR
727699
Zbl
0549.57006
article
Abstract
BibTeX
This is the first of three papers concerning the so-called Torelli group. Let \( M = M_{g,n} \) be a compact orientable surface of genus \( g \) having \( n \) boundary components and let \( \mathfrak{M} = \mathfrak{M}_{g,n} \) be its mapping class group, that is, the group of orientation preserving diffeomorphisms of \( M \) which are 1 on the boundary \( \partial M \) modulo isotopies which fix \( \partial M \) pointwise. This group is also known to the complex analysts as the Teichmüller group or modular group . If \( n = 0 \) or 1, let further \( \mathscr{I} = \mathscr{I}_{g,n} \) be the subgroup of \( \mathfrak{M} \) which acts trivially on \( H_1(M,Z) \) . The topologists have no traditional name for \( \mathscr{I} \) , but the analysts tell me it was known classically and is called the Torelli group . Several interesting problems and conjectures exists concerning \( \mathscr{I} \) . The principal one can be found in Kirby’s problem list [1975] and asks if \( \mathscr{I}_{g,0} \) is finitely generated. In this first paper we shall answer the question affirmatively for both \( \mathscr{I}_{g,0} \) and \( \mathscr{I}_{g,1} \) when \( g\geq 3 \) and shall give a fairly simple set of generators.
@article {key727699m,
AUTHOR = {Johnson, Dennis},
TITLE = {The structure of the {T}orelli group,
{I}: {A} finite set of generators for
\$\mathscr{I}\$},
JOURNAL = {Ann. Math. (2)},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {118},
NUMBER = {3},
MONTH = {November},
YEAR = {1983},
PAGES = {423--442},
DOI = {10.2307/2006977},
NOTE = {Parts II and III were published in \textit{Topology}
\textbf{24}:2 (1985) and \textit{Topology}
\textbf{24}:2 (1985). MR:727699. Zbl:0549.57006.},
ISSN = {0003-486X},
CODEN = {ANMAAH},
}
[11] D. Johnson :
“A survey of the Torelli group ,”
pp. 165–179
in
Low-dimensional topology
(San Francisco, CA, 7–11 January 1981 ).
Edited by S. J. Lomonaco, Jr.
Contemporary Mathematics 20 .
American Mathematical Society (Providence, RI ),
1983 .
MR
718141
Zbl
0553.57002
incollection
People
BibTeX
@incollection {key718141m,
AUTHOR = {Johnson, Dennis},
TITLE = {A survey of the {T}orelli group},
BOOKTITLE = {Low-dimensional topology},
EDITOR = {Lomonaco, Jr., Samuel J.},
SERIES = {Contemporary Mathematics},
NUMBER = {20},
PUBLISHER = {American Mathematical Society},
ADDRESS = {Providence, RI},
YEAR = {1983},
PAGES = {165--179},
DOI = {10.1090/conm/020/718141},
NOTE = {(San Francisco, CA, 7--11 January 1981).
MR:718141. Zbl:0553.57002.},
ISSN = {0271-4132},
ISBN = {9780821850169},
}
[12] D. Johnson :
“The structure of the Torelli group, II: A characterization of the group generated by twists on bounding curves ,”
Topology
24 : 2
(1985 ),
pp. 113–126 .
Part I was published in Ann. Math. 118 :3 (1983) .
MR
793178
Zbl
0571.57009
article
BibTeX
@article {key793178m,
AUTHOR = {Johnson, Dennis},
TITLE = {The structure of the {T}orelli group,
{II}: {A} characterization of the group
generated by twists on bounding curves},
JOURNAL = {Topology},
FJOURNAL = {Topology. An International Journal of
Mathematics},
VOLUME = {24},
NUMBER = {2},
YEAR = {1985},
PAGES = {113--126},
DOI = {10.1016/0040-9383(85)90049-7},
NOTE = {Part I was published in \textit{Ann.
Math.} \textbf{118}:3 (1983). MR:793178.
Zbl:0571.57009.},
ISSN = {0040-9383},
CODEN = {TPLGAF},
}
[13] D. Johnson :
“The structure of the Torelli group, III: The abelianization of \( \mathscr{I} \) ,”
Topology
24 : 2
(1985 ),
pp. 127–144 .
Part I was published in Ann. Math. 118 :3 (1983) .
MR
793179
Zbl
0571.57010
article
BibTeX
@article {key793179m,
AUTHOR = {Johnson, Dennis},
TITLE = {The structure of the {T}orelli group,
{III}: {T}he abelianization of \$\mathscr{I}\$},
JOURNAL = {Topology},
FJOURNAL = {Topology. An International Journal of
Mathematics},
VOLUME = {24},
NUMBER = {2},
YEAR = {1985},
PAGES = {127--144},
DOI = {10.1016/0040-9383(85)90050-3},
NOTE = {Part I was published in \textit{Ann.
Math.} \textbf{118}:3 (1983). MR:793179.
Zbl:0571.57010.},
ISSN = {0040-9383},
CODEN = {TPLGAF},
}
[14] D. Johnson and J. J. Millson :
“Deformation spaces associated to compact hyperbolic manifolds ,”
Bull. Am. Math. Soc. (N.S.)
14 : 1
(January 1986 ),
pp. 99–102 .
Preliminary research announcement for an article published in Discrete groups in geometry and analysis (1987) .
MR
818061
Zbl
0613.53032
article
People
BibTeX
@article {key818061m,
AUTHOR = {Johnson, Dennis and Millson, John J.},
TITLE = {Deformation spaces associated to compact
hyperbolic manifolds},
JOURNAL = {Bull. Am. Math. Soc. (N.S.)},
FJOURNAL = {Bulletin of the American Mathematical
Society. New Series},
VOLUME = {14},
NUMBER = {1},
MONTH = {January},
YEAR = {1986},
PAGES = {99--102},
DOI = {10.1090/S0273-0979-1986-15404-2},
NOTE = {Preliminary research announcement for
an article published in \textit{Discrete
groups in geometry and analysis} (1987).
MR:818061. Zbl:0613.53032.},
ISSN = {0273-0979},
CODEN = {BAMOAD},
}
[15] D. Johnson and J. J. Millson :
“Deformation spaces associated to compact hyperbolic manifolds ,”
pp. 48–106
in
Discrete groups in geometry and analysis: Papers in honor of G. D. Mostow on his sixtieth birthday
(New Haven, CT, 23–25 March 1984 ).
Edited by R. Howe .
Progress in Mathematics 67 .
Birkhäuser (Boston ),
1987 .
A preliminary research announcement was published in Bull. Am. Math. Soc. 14 :1 (1986) .
MR
900823
Zbl
0664.53023
incollection
Abstract
People
BibTeX
@incollection {key900823m,
AUTHOR = {Johnson, Dennis and Millson, John J.},
TITLE = {Deformation spaces associated to compact
hyperbolic manifolds},
BOOKTITLE = {Discrete groups in geometry and analysis:
{P}apers in honor of {G}.~{D}. {M}ostow
on his sixtieth birthday},
EDITOR = {Howe, Roger},
SERIES = {Progress in Mathematics},
NUMBER = {67},
PUBLISHER = {Birkh\"auser},
ADDRESS = {Boston},
YEAR = {1987},
PAGES = {48--106},
DOI = {10.1007/978-1-4899-6664-3_3},
NOTE = {(New Haven, CT, 23--25 March 1984).
A preliminary research announcement
was published in \textit{Bull. Am. Math.
Soc.} \textbf{14}:1 (1986). MR:900823.
Zbl:0664.53023.},
ISSN = {0743-1643},
ISBN = {9780817633011},
}
[16] D. Johnson :
A geometric form of Casson’s invariant and its connection to Reidemeister torsion .
Preprint ,
1988 .
techreport
BibTeX
Read PDF
@techreport {key41112690,
AUTHOR = {Johnson, Dennis},
TITLE = {A geometric form of Casson's invariant
and its connection to Reidemeister torsion},
TYPE = {preprint},
YEAR = {1988},
}
[17] D. Johnson and C. Livingston :
“Peripherally specified homomorphs of knot groups ,”
Trans. Am. Math. Soc.
311 : 1
(January 1989 ),
pp. 135–146 .
MR
942427
Zbl
0662.57003
article
Abstract
People
BibTeX
Let \( G \) be a group and let \( \mu \) and \( \lambda \) be elements of \( G \) . Necessary and sufficient conditions are presented for the solution of the following problem: Is there a knot \( K \) in \( S^3 \) and a representation
\[ \rho: \pi_1(S^3 - K) \to G \]
such that \( \rho(m) = \mu \) and \( \rho(l) = \lambda \) , where \( m \) and \( l \) are the meridian and longitude of \( K \) ?
@article {key942427m,
AUTHOR = {Johnson, Dennis and Livingston, Charles},
TITLE = {Peripherally specified homomorphs of
knot groups},
JOURNAL = {Trans. Am. Math. Soc.},
FJOURNAL = {Transactions of the American Mathematical
Society},
VOLUME = {311},
NUMBER = {1},
MONTH = {January},
YEAR = {1989},
PAGES = {135--146},
DOI = {10.2307/2001020},
NOTE = {MR:942427. Zbl:0662.57003.},
ISSN = {0002-9947},
CODEN = {TAMTAM},
}
[18] S. P. Humphries and D. Johnson :
“A generalization of winding number functions on surfaces ,”
Proc. London Math. Soc. (3)
58 : 2
(1989 ),
pp. 366–386 .
MR
977482
Zbl
0681.57003
article
People
BibTeX
@article {key977482m,
AUTHOR = {Humphries, Stephen P. and Johnson, Dennis},
TITLE = {A generalization of winding number functions
on surfaces},
JOURNAL = {Proc. London Math. Soc. (3)},
FJOURNAL = {Proceedings of the London Mathematical
Society. Third Series},
VOLUME = {58},
NUMBER = {2},
YEAR = {1989},
PAGES = {366--386},
DOI = {10.1112/plms/s3-58.2.366},
NOTE = {MR:977482. Zbl:0681.57003.},
ISSN = {0024-6115},
CODEN = {PLMTAL},
}
[19] D. Johnson and J. J. Millson :
“Modular Lagrangians and the theta multiplier ,”
Invent. Math.
100 : 1
(1990 ),
pp. 143–165 .
Dedicated to Armand Borel.
MR
1037145
Zbl
0699.10042
article
People
BibTeX
@article {key1037145m,
AUTHOR = {Johnson, Dennis and Millson, John J.},
TITLE = {Modular {L}agrangians and the theta
multiplier},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {100},
NUMBER = {1},
YEAR = {1990},
PAGES = {143--165},
DOI = {10.1007/BF01231183},
NOTE = {Dedicated to Armand Borel. MR:1037145.
Zbl:0699.10042.},
ISSN = {0020-9910},
CODEN = {INVMBH},
}
[20]
J. S. Birman, D. Johnson, and A. Putman :
“Symplectic Heegaard splittings and linked abelian groups ,”
pp. 135–220
in
Groups of diffeomorphisms: In honor of Shigeyuki Morita on the occasion of his 60th birthday
(Tokyo, 11–15 September 2006 ).
Edited by R. C. Penner .
Advanced Studies in Pure Mathematics 52 .
Mathematical Society of Japan (Tokyo ),
2008 .
MR
2509710
Zbl
1170.57018
ArXiv
0712.2104
incollection
Abstract
People
BibTeX
Let \( f \) be the gluing map of a Heegaard splitting of a 3-manifold \( W \) . The goal of this paper is to determine the information about \( W \) contained in the image of \( f \) under the symplectic representation of the mapping class group. We prove three main results. First, we show that the first homology group of the three manifold together with Seifert’s linking form provides a complete set of stable invariants. Second, we give a complete, computable set of invariants for these linking forms. Third, we show that a slight augmentation of Birman’s determinantal invariant for a Heegaard splitting gives a complete set of unstable invariants.
@incollection {key2509710m,
AUTHOR = {Birman, Joan S. and Johnson, Dennis
and Putman, Andrew},
TITLE = {Symplectic {H}eegaard splittings and
linked abelian groups},
BOOKTITLE = {Groups of diffeomorphisms: {I}n honor
of {S}higeyuki {M}orita on the occasion
of his 60th birthday},
EDITOR = {Penner, R. C.},
SERIES = {Advanced Studies in Pure Mathematics},
NUMBER = {52},
PUBLISHER = {Mathematical Society of Japan},
ADDRESS = {Tokyo},
YEAR = {2008},
PAGES = {135--220},
NOTE = {(Tokyo, 11--15 September 2006). ArXiv:0712.2104.
MR:2509710. Zbl:1170.57018.},
ISSN = {0920-1971},
ISBN = {9784931469488},
}