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Cone complexes and PL transversality
Author:
Clint McCrory
Journal:
Trans. Amer. Math. Soc. 207 (1975), 269-291
MSC:
Primary 57C25
MathSciNet review:
0400243
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Abstract: A definition of PL transversality is given, using the orderreversing duality on partially ordered sets. David Stone's theory of stratified polyhedra is thereby simplified; in particular, the symmetry of blocktransversality is proved. Also, polyhedra satisfying Poincaré duality are characterized geometrically.
- [1]
Ethan
Akin, Manifold phenomena in the theory of
polyhedra, Trans. Amer. Math. Soc. 143 (1969), 413–473. MR 0253329
(40 #6544), http://dx.doi.org/10.1090/S0002-9947-1969-0253329-1
- [2]
Ethan
Akin, Transverse cellular mappings of
polyhedra, Trans. Amer. Math. Soc. 169 (1972), 401–438. MR 0326745
(48 #5088), http://dx.doi.org/10.1090/S0002-9947-1972-0326745-7
- [3]
M.
A. Armstrong, Transversality for polyhedra, Ann. of Math. (2)
86 (1967), 172–191. MR 0219075
(36 #2158)
- [4]
M.
A. Armstrong and E.
C. Zeeman, Transversality for piecewise linear manifolds,
Topology 6 (1967), 433–466. MR 0219074
(36 #2157)
- [5]
Thomas
Banchoff, Critical points and curvature for embedded
polyhedra, J. Differential Geometry 1 (1967),
245–256. MR 0225327
(37 #921)
- [6]
Garrett
Birkhoff, Lattice theory, Third edition. American Mathematical
Society Colloquium Publications, Vol. XXV, American Mathematical Society,
Providence, R.I., 1967. MR 0227053
(37 #2638)
- [7]
Marshall
M. Cohen, Simplicial structures and transverse cellularity,
Ann. of Math. (2) 85 (1967), 218–245. MR 0210143
(35 #1037)
- [8]
Marshall
M. Cohen, A general theory of relative regular
neighborhoods, Trans. Amer. Math. Soc. 136 (1969), 189–229.
MR
0248802 (40 #2052), http://dx.doi.org/10.1090/S0002-9947-1969-0248802-6
- [9]
M.
Cohen and D.
Sullivan, On the regular neighborhood of a two-sided
submanifold, Topology 9 (1970), 141–147. MR 0264671
(41 #9262)
- [10]
E. R. van Kampen, Die Kombinatorische Topologie und die Dualitätssätze, Den Haag (1929).
- [11]
A.
Kosiński, Singularities of piecewise linear
mappings. I. Mappings into the real line, Bull.
Amer. Math. Soc. 68
(1962), 110–114. MR 0132549
(24 #A2389), http://dx.doi.org/10.1090/S0002-9904-1962-10744-7
- [12]
S. Lefschetz, Topology, 2nd ed., Chelsea, New York, 1956.
- [13]
C. McCrory, Poincarè duality in spaces with singularities, Ph.D. Thesis, Brandeis University, 1972.
- [14]
H.
R. Morton, Joins of polyhedra, Topology 9
(1970), 243–249. MR 0261587
(41 #6200)
- [15]
C.
P. Rourke and B.
J. Sanderson, Block bundles. II. Transversality, Ann. of Math.
(2) 87 (1968), 256–278. MR 0226646
(37 #2234b)
- [16]
C.
P. Rourke and B.
J. Sanderson, An embedding without a normal microbundle,
Invent. Math. 3 (1967), 293–299. MR 0222904
(36 #5954)
- [17]
C.
P. Rourke and B.
J. Sanderson, Decompositions and the relative tubular neighbourhood
conjecture, Topology 9 (1970), 225–229. MR 0261611
(41 #6224)
- [18]
S.
Buoncristiano, C.
P. Rourke, and B.
J. Sanderson, A geometric approach to homology theory,
Cambridge University Press, Cambridge, 1976. London Mathematical Society
Lecture Note Series, No. 18. MR 0413113
(54 #1234)
- [19]
C.
P. Rourke and B.
J. Sanderson, Introduction to piecewise-linear topology,
Springer-Verlag, New York, 1972. Ergebnisse der Mathematik und ihrer
Grenzgebiete, Band 69. MR 0350744
(50 #3236)
- [20]
H. Seifert and W. Threlfall, Lehrbuch der Topologie, Teubner, Leipzig, 1934; reprint, Chelsea, New York, 1954.
- [21]
John
R. Stallings, Lectures on polyhedral topology, Notes by G.
Ananda Swarup. Tata Institute of Fundamental Research Lectures on
Mathematics, No. 43, Tata Institute of Fundamental Research, Bombay, 1967.
MR
0238329 (38 #6605)
- [22]
David
A. Stone, Stratified polyhedra, Lecture Notes in Mathematics,
Vol. 252, Springer-Verlag, Berlin, 1972. MR 0372870
(51 #9074)
- [23]
E. Hendricks, Triangulation of stratified sets, Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge, Mass., 1973
- [1]
- E. Akin, Manifold phenomena in the theory of polyhedra, Trans. Amer. Math. Soc. 143 (1969), 413-473. MR 40 #6544. MR 0253329 (40:6544)
- [2]
- -, Transverse cellular mappings of polyhedra, Trans. Amer. Math. Soc. 169 (1972), 401-438. MR 0326745 (48:5088)
- [3]
- M. A. Armstrong, Transversality for polyhedra, Ann. of Math. (2) 86 (1967), 172-191. MR 36 #2158. MR 0219075 (36:2158)
- [4]
- M. A. Armstrong and E. C. Zeeman, Transversality for piecewise linear manifolds, Topology 6 (1967), 433-466. MR 36 #2157. MR 0219074 (36:2157)
- [5]
- T. Banchoff, Critical points and curvature for embedded polyhedra, J. Differential Geometry 1 (1967), 245-256. MR 37 #921. MR 0225327 (37:921)
- [6]
- G. Birkhoff, Lattice theory, 3rd ed., Amer. Math. Soc. Colloq. Publ., Vol. 25, Amer. Math. Soc., Providence, R.L, 1967. MR 37 #2638. MR 0227053 (37:2638)
- [7]
- M. M. Cohen, Simplicial structures and transverse cellularity, Ann. of Math. (2) 85 (1967), 218-245. MR 35 #1037. MR 0210143 (35:1037)
- [8]
- -, A general theory of relative regular neighborhoods, Trans. Amer. Math. Soc. 136 (1969), 189-229. MR 40 #2052. MR 0248802 (40:2052)
- [9]
- M. Cohen and D. Sullivan, On the regular neighborhood of a two-sided submanifold, Topology 9 (1970), 141-147. MR 41 #9262. MR 0264671 (41:9262)
- [10]
- E. R. van Kampen, Die Kombinatorische Topologie und die Dualitätssätze, Den Haag (1929).
- [11]
- A. Kosiński, Singularities of piecewise linear mappings. I: Mappings into the real line, Bull. Amer. Math. Soc. 68 (1962), 110-114. MR 24 #A2389. MR 0132549 (24:A2389)
- [12]
- S. Lefschetz, Topology, 2nd ed., Chelsea, New York, 1956.
- [13]
- C. McCrory, Poincarè duality in spaces with singularities, Ph.D. Thesis, Brandeis University, 1972.
- [14]
- H. R. Morton, Joins of polyhedra, Topology 9 (1970), 243-249. MR 41 #6200. MR 0261587 (41:6200)
- [15]
- C. P. Rourke and B. J. Sanderson, Block bundles. II: Transversality, Ann. of Math. (2) 87 (1968), 256-278. MR 37 #2234b. MR 0226646 (37:2234b)
- [16]
- -, An embedding without a normal microbundle, Invent. Math 3 (1967), 293-299. MR 36 #5954. MR 0222904 (36:5954)
- [17]
- -, Decompositions and the relative tubular neighborhood conjecture, Topology 9 (1970), 225-229. MR 41 #6224. MR 0261611 (41:6224)
- [18]
- S. Buoncristiano, C. P. Rourke and B. J. Sanderson, A geometric approach to homology theory, London Math. Soc. Lecture Notes Ser., Cambridge (to appear). MR 0413113 (54:1234)
- [19]
- -, Introduction to piecewise linear topology, Ergebnisse der Mathematik 69, Springer-Verlag, Heidelberg, 1972. MR 0350744 (50:3236)
- [20]
- H. Seifert and W. Threlfall, Lehrbuch der Topologie, Teubner, Leipzig, 1934; reprint, Chelsea, New York, 1954.
- [21]
- J. R. Stallings, Lectures on polyhedral topology. Tata Inst. Fund. Res. Lec. Math., no. 43, Tata Institute of Fundamental Research, Bombay, 1967. MR 38 #6605. MR 0238329 (38:6605)
- [22]
- D. A. Stone, Stratified polyhedra, Lecture Notes in Math., no. 252, Springer-Verlag, New York, 1972. MR 0372870 (51:9074)
- [23]
- E. Hendricks, Triangulation of stratified sets, Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge, Mass., 1973
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0002-9947-1975-0400243-7
PII:
S 0002-9947(1975)0400243-7
Keywords:
Piecewise linear transversality,
blocktransversality,
cone complex,
poset,
piecewise linear stratification,
intrinsic dimension,
Poincaré duality
Article copyright:
© Copyright 1975 American Mathematical Society
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