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Introduction to the papers of R. Thom and J. Mather
Author:
Mark Goresky
Journal:
Bull. Amer. Math. Soc. 49 (2012), 469-474
Posted:
July 3, 2012
Link to article that is the subject of this commentary:
Bull. Amer. Math. Soc. 75 (1969), 240-284.
Related article:
Bull. Amer. Math. Soc. 49 (2012), 475-506.
Full-text PDF
References |
Additional Information
- 1.
Lou
van den Dries, Tame topology and o-minimal structures, London
Mathematical Society Lecture Note Series, vol. 248, Cambridge
University Press, Cambridge, 1998. MR 1633348
(99j:03001)
- 2.
Andrew
du Plessis and Terry
Wall, The geometry of topological stability, London
Mathematical Society Monographs. New Series, vol. 9, The Clarendon
Press Oxford University Press, New York, 1995. Oxford Science Publications.
MR
1408432 (97k:58024)
- 3.
Burghart
Giesecke, Simpliziale Zerlegung abzählbarer analytischer
Räume, Math. Z. 83 (1964), 177–213
(German). MR
0159346 (28 #2563)
- 4.
Christopher
G. Gibson, Klaus
Wirthmüller, Andrew
A. du Plessis, and Eduard
J. N. Looijenga, Topological stability of smooth mappings,
Lecture Notes in Mathematics, Vol. 552, Springer-Verlag, Berlin, 1976. MR 0436203
(55 #9151)
- 5.
Robert
M. Hardt, Slicing and intersection theory for chains associated
with real analytic varieties, Acta Math. 129 (1972),
75–136. MR
0315561 (47 #4110)
- 6.
Robert
M. Hardt, Homology and images of semianalytic
sets, Bull. Amer. Math. Soc. 80 (1974), 675–678. MR 0348140
(50 #638), http://dx.doi.org/10.1090/S0002-9904-1974-13544-5
- 7.
Robert
M. Hardt, Stratification of real analytic mappings and images,
Invent. Math. 28 (1975), 193–208. MR 0372237
(51 #8453)
- 8.
Robert
M. Hardt, Triangulation of subanalytic sets and proper light
subanalytic maps, Invent. Math. 38 (1976/77),
no. 3, 207–217. MR 0454051
(56 #12302)
- 9.
Heisuke
Hironaka, Subanalytic sets, Number theory, algebraic geometry
and commutative algebra, in honor of Yasuo Akizuki, Kinokuniya, Tokyo,
1973, pp. 453–493. MR 0377101
(51 #13275)
- 10.
Heisuke
Hironaka, Triangulations of algebraic sets, Algebraic geometry
(Proc. Sympos. Pure Math., Vol. 29, Humboldt State Univ., Arcata, Calif.,
1974), Amer. Math. Soc., Providence, R.I., 1975, pp. 165–185.
MR
0374131 (51 #10331)
- 11.
Heisuke
Hironaka, Introduction to real-analytic sets and real-analytic
maps, Istituto Matematico “L. Tonelli”
dell’Università di Pisa, Pisa, 1973. Quaderni dei Gruppi di
Ricerca Matematica del Consiglio Nazionale delle Ricerche. MR 0477121
(57 #16665)
- 12.
Heisuke
Hironaka, Stratification and flatness, Real and complex
singularities (Proc. Ninth Nordic Summer School/NAVF Sympos. Math., Oslo,
1976), Sijthoff and Noordhoff, Alphen aan den Rijn, 1977,
pp. 199–265. MR 0499286
(58 #17187)
- 13.
B.
O. Koopman and A.
B. Brown, On the covering of analytic loci by
complexes, Trans. Amer. Math. Soc.
34 (1932), no. 2,
231–251. MR
1501636, http://dx.doi.org/10.1090/S0002-9947-1932-1501636-9
- 14.
Ta
Lê Loi, Lecture 2: Stratifications in o-minimal
structures, The Japanese-Australian Workshop on Real and Complex
Singularities—JARCS III, Proc. Centre Math. Appl. Austral. Nat.
Univ., vol. 43, Austral. Nat. Univ., Canberra, 2010,
pp. 31–39. MR 2763234
(2012f:03059)
- 15.
S. Lefschetz, Topology, Amer. Math. Soc. Colloq. Publ., New York (1930).
- 16.
S.
Lefschetz and J.
H. C. Whitehead, On analytical complexes, Trans. Amer. Math. Soc. 35 (1933), no. 2, 510–517. MR
1501698, http://dx.doi.org/10.1090/S0002-9947-1933-1501698-X
- 17.
H. Levine, ``Singularities of differentiable mappings, notes of lectures by R. Thom'', University of Bonn, 1960; reprinted in Proceedings of Liverpool Singularities--Symposium I, Lecture Notes in Mathematics 192, Springer Verlag, New York, 1971, 1-89.
- 18.
S.
Lojasiewicz, Triangulation of semi-analytic sets, Ann. Scuola
Norm. Sup. Pisa (3) 18 (1964), 449–474. MR 0173265
(30 #3478)
- 19.
S. Lojasiewiz, Ensembles semi-analytiques (mimeographed notes), I.H.E.S., Bures-sur-Yvette, France, 1965.
- 20.
John
N. Mather, Stability of 𝐶^{∞} mappings. I. The
division theorem, Ann. of Math. (2) 87 (1968),
89–104. MR
0232401 (38 #726)
- 21.
John
N. Mather, Stability of 𝐶^{∞} mappings. II.
Infinitesimal stability implies stability, Ann. of Math. (2)
89 (1969), 254–291. MR 0259953
(41 #4582)
- 22.
J. Mather, Stability of
mappings: finitely determined map-germs, Publ. Math. I.H.E.S. 35 (1969), 127-156.
- 23.
John
N. Mather, Stability of 𝐶^{∞} mappings. IV.
Classification of stable germs by 𝑅-algebras, Inst. Hautes
Études Sci. Publ. Math. 37 (1969), 223–248.
MR
0275460 (43 #1215b)
- 24.
John
N. Mather, Stability of 𝐶^{∞} mappings. V.
Transversality, Advances in Math. 4 (1970),
301–336 (1970). MR 0275461
(43 #1215c)
- 25.
J.
N. Mather, Stability of 𝐶^{∞} mappings. VI: The nice
dimensions, Proceedings of Liverpool Singularities-Symposium, I
(1969/70), Springer, Berlin, 1971, pp. 207–253. Lecture Notes
in Math., Vol. 192. MR 0293670
(45 #2747)
- 26.
J. Mather, Notes on Topological Stability, Harvard University, 1970.
- 27.
John
N. Mather, Stratifications and mappings, Dynamical systems
(Proc. Sympos., Univ. Bahia, Salvador, 1971) Academic Press, New York,
1973, pp. 195–232. MR 0368064
(51 #4306)
- 28.
John
N. Mather, How to stratify mappings and jet spaces,
Singularités d’applications différentiables
(Sém., Plans-sur-Bex, 1975), Springer, Berlin, 1976,
pp. 128–176. Lecture Notes in Math., Vol. 535. MR 0455018
(56 #13259)
- 29.
Markus
J. Pflaum, Analytic and geometric study of stratified spaces,
Lecture Notes in Mathematics, vol. 1768, Springer-Verlag, Berlin,
2001. MR
1869601 (2002m:58007)
- 30.
Kenkichi
Satō, Local triangulation of real analytic varieties,
Osaka Math. J. 15 (1963), 109–129. MR 0156355
(27 #6279)
- 31.
B. Teissier, Théorèmes de finitude en géometrie analytique, Sem. Bourbaki, 1974.
- 32.
Bernard
Teissier, Variétés polaires locales et conditions de
Whitney, C. R. Acad. Sci. Paris Sér. A-B 290
(1980), no. 17, A799–A802 (French, with English summary). MR 580569
(82m:32006)
- 33.
R.
Thom, Local topological properties of differentiable mappings,
Differential Analysis, Bombay Colloq., Oxford Univ. Press, London, 1964,
pp. 191–202. MR 0195102
(33 #3307)
- 34.
René
Thom, La stabilité topologique des applications
polynomiales, Enseignement Math. (2) 8 (1962),
24–33 (French). MR 0148079
(26 #5588)
- 35.
R.
Thom, Ensembles et morphismes
stratifiés, Bull. Amer. Math. Soc.
75 (1969), 240–284
(French). MR
0239613 (39 #970), http://dx.doi.org/10.1090/S0002-9904-1969-12138-5
- 36.
R.
Thom, The bifurcation subset of a space of maps,
Manifolds—Amsterdam 1970 (Proc. Nuffic Summer School), Lecture Notes
in Mathematics, Vol. 197, Springer, Berlin, 1971, pp. 202–208.
MR
0281222 (43 #6941)
- 37.
Andrei
Verona, Stratified mappings—structure and
triangulability, Lecture Notes in Mathematics, vol. 1102,
Springer-Verlag, Berlin, 1984. MR 771120
(86k:58010)
- 38.
B. L. van der Waerden, Topologische Begründung des Kalküls der abzählenden Geometrie, Math. Ann. 102 (1929), 337-362.
- 39.
Hassler
Whitney, On singularities of mappings of euclidean spaces. I.
Mappings of the plane into the plane, Ann. of Math. (2)
62 (1955), 374–410. MR 0073980
(17,518d)
- 40.
Hassler
Whitney, Complexes of manifolds, Proc. Nat. Acad. Sci. U. S.
A. 33 (1947), 10–11. MR 0019306
(8,398a)
- 41.
Hassler
Whitney, Elementary structure of real algebraic varieties,
Ann. of Math. (2) 66 (1957), 545–556. MR 0095844
(20 #2342)
- 42.
Hassler
Whitney, Local properties of analytic varieties, Differential
and Combinatorial Topology (A Symposium in Honor of Marston Morse),
Princeton Univ. Press, Princeton, N. J., 1965, pp. 205–244. MR 0188486
(32 #5924)
- 43.
Hassler
Whitney, Tangents to an analytic variety, Ann. of Math. (2)
81 (1965), 496–549. MR 0192520
(33 #745)
- 1.
- L. van den Dries, Tame Topology and
-minimal Structures, Cambridge Univ. Press, Cambridge, 1998. MR 1633348 (99j:03001)
- 2.
- A. du Plessis and T. Wall, The Geometry of Topological Stability, London Math. Soc. Monographs 9, Oxford University Press, Oxford, 1995. MR 1408432 (97k:58024)
- 3.
- B. Giesecke. Simpliziale Zerlegung abzählbarer analytische Räume, Math. Zeit. 83 (1964), 177-213. MR 0159346 (28:2563)
- 4.
- C. Gibson, K. Wirthmüller, A. du Plessis, and E. Looijenga, Topological Stability of Smooth Mappings, Lecture Notes in Mathematics 552, Springer Verlag, 1976. MR 0436203 (55:9151)
- 5.
- Slicing and intersection theory for chains associated with real analytic varieties, Acta Math. 129 (1972), 57-136. MR 0315561 (47:4110)
- 6.
- R. Hardt, Homology and images of semianalytic sets, Bull. Amer. Math. Soc. 80 (1974), 675-678. MR 0348140 (50:638)
- 7.
- R. Hardt, Stratification of real analytic mappings and images, Invent. Math. 28 (1975), 193-208. MR 0372237 (51:8453)
- 8.
- R. Hardt, Triangulation of subanalytic sets and proper light subanalytic maps, Invent. Math. 38 (1976/77), 207-217. MR 0454051 (56:12302)
- 9.
- H. Hironaka, ``Subanalytic sets'', in Number Theory, Algebraic Geometry, and Commutative Algebra (Dedicated to Akizuki), Kinokunia, Tokyo Japan, 1973, 453-493. MR 0377101 (51:13275)
- 10.
- H. Hironaka, ``Triangulations of algebraic sets'', in Algebraic Geometry (Proc. Sympos. Pure Math., 29, Amer. Math. Soc., Providence, Rhode Island, 1974, pp. 165-185. MR 0374131 (51:10331)
- 11.
- H. Hironaka, Introduction to real-analytic sets and real-analytic maps. Quaderni dei Gruppi di Ricerca Matematica del Consiglio Nazionale delle Ricerche. Istituto Matematico ``L. Tonelli'' dell'Università di Pisa, Pisa, 1973. iii+162 pp. MR 0477121 (57:16665)
- 12.
- H. Hironaka, ``Stratification and flatness'', in Real and Complex Singularities, Proc. Ninth Nordic Summer School/NAVF Symp. Math. Oslo, Sijthoff and Noordhoff, Alphen aan den Rijn, Netherlands, 1977, 199-265. MR 0499286 (58:17187)
- 13.
- B. C. Koopman and A. B. Brown, On the covering of analytic loci by complexes, Trans. Amer. Math. Soc. 34 (1932), 231-251. MR 1501636
- 14.
- T. Le Loi, ``Lecture 2: Stratifications in
-minimal structures'' in The Japanese-Australian Workshop on Real and Complex Singularities-JARCS III, 31-39, Proc. Centre Math. Appl. Austral. Nat. Univ., 43, Austral. Nat. Univ., Canberra, 2010. MR 2763234 (2012f:03059)
- 15.
- S. Lefschetz, Topology, Amer. Math. Soc. Colloq. Publ., New York (1930).
- 16.
- S. Lefschetz and J. H. C. Whitehead, On analytical complexes, Trans. Amer. Math. Soc. 35 (1933), 510-517. MR 1501698
- 17.
- H. Levine, ``Singularities of differentiable mappings, notes of lectures by R. Thom'', University of Bonn, 1960; reprinted in Proceedings of Liverpool Singularities--Symposium I, Lecture Notes in Mathematics 192, Springer Verlag, New York, 1971, 1-89.
- 18.
- S. Lojasiewicz, Triangulation of semi-analytic sets, Ann. Scuola Norm. Pisa 18 (1964), 449-474. MR 0173265 (30:3478)
- 19.
- S. Lojasiewiz, Ensembles semi-analytiques (mimeographed notes), I.H.E.S., Bures-sur-Yvette, France, 1965.
- 20.
- J. Mather, Stability of
mappings: the division theorem, Ann. Math. 87 (1968), 89-104. MR 0232401 (38:726)
- 21.
- J. Mather, Stability of
mappings: infinitesimal stability implies stability, Ann. Math. 89 (1969), 254-291. MR 0259953 (41:4582)
- 22.
- J. Mather, Stability of
mappings: finitely determined map-germs, Publ. Math. I.H.E.S. 35 (1969), 127-156.
- 23.
- J. Mather, Stability of
mappings: classification of stable germs by -algebras, Publ. Math. I.H.E.S. 37 (1970), 223-248. MR 0275460 (43:1215b)
- 24.
- J. Mather, Stability of
mappings: transversality, Adv. Math. 4 (1970), 301-335. MR 0275461 (43:1215c)
- 25.
- J. Mather, ``Stability of
mappings: the nice dimensions'', in Proceedings of Liverpool Singularities--Symposium 1, Lecture Notes in Mathematics 192, Springer Verlag, New York (1971), 207-253. MR 0293670 (45:2747)
- 26.
- J. Mather, Notes on Topological Stability, Harvard University, 1970.
- 27.
- J. Mather, ``Stratifications and mappings'', in Dynamical Systems, Proc. Sympos., Univ. Bahia, Salvador, 1971, pp. 195-232. Academic Press, New York, 1973. MR 0368064 (51:4306)
- 28.
- J. Mather, ``How to stratify manifolds and jet spaces'', in Singularités d'Applications Différentiables, Lecture Notes in Mathematics 135 (1976), Springer Verlag, Berlin, pp. 128-176. MR 0455018 (56:13259)
- 29.
- M. Pflaum, Analytic and Geometric Study of Stratified Spaces, Lecture Notes in Mathematics 1768, Springer Verlag, 2001. MR 1869601 (2002m:58007)
- 30.
- K. Sato, Local triangulation of real analytic varieties, Osaka Math. J. 15 (1963), 109-125. MR 0156355 (27:6279)
- 31.
- B. Teissier, Théorèmes de finitude en géometrie analytique, Sem. Bourbaki, 1974.
- 32.
- B. Teissier, Variétés polaires locales et conditions de Whitney, C. R. Acad. Sci. Paris Sér. A-B 290 (1980), A799-A802. MR 580569 (82m:32006)
- 33.
- R. Thom, ``Local topological properties of differentiable mappings'', in Colloquium on Differential Analysis (Tata Institute, Bombay, 1964), Oxford University Press, 1964, pp. 191-202. MR 0195102 (33:3307)
- 34.
- R. Thom, La stabilité topologique des applications polynomiales, L'Enseignement Math. (2) 8 (1962), 24-33. MR 0148079 (26:5588)
- 35.
- R. Thom, Ensembles et morphismes stratifiés, Bull. Amer. Math. Soc. 75 (1969), no. 2, 240-284. MR 0239613 (39:970)
- 36.
- R. Thom, ``The bifurcation subset of a space of maps'', in Manifolds--Amsterdam 1970, Lecture Notes in Mathematics, vol. 197, Springer Verlag, New York, 1971. MR 0281222 (43:6941)
- 37.
- A. Verona, Stratified Mappings--Structure and Triangulability, Lecture Notes in Mathematics 1102, Springer Verlag, 1984. MR 771120 (86k:58010)
- 38.
- B. L. van der Waerden, Topologische Begründung des Kalküls der abzählenden Geometrie, Math. Ann. 102 (1929), 337-362.
- 39.
- H. Whitney, On singularities of mappings of euclidean spaces. I. Mappings of the plane into the plane, Ann. of Math. (2) 62 (1955), 374-410. MR 0073980 (17:518d)
- 40.
- H. Whitney, Complexes of manifolds, Proc. Nat. Acad. Sci. U.S.A. 33, (1947), 10-11. MR 0019306 (8:398a)
- 41.
- H. Whitney, Elementary structure of real algebraic varieties, Ann. Math. 66 (1957), 545-556. MR 0095844 (20:2342)
- 42.
- H. Whitney, ``Local properties of analytic varieties'', in Differential and Combinatorial Topology, Princeton Univ. Press, Princeton, New Jersey, 1965. MR 0188486 (32:5924)
- 43.
- H. Whitney, Tangents to an analytic variety. Ann. of Math. (2) 81 (1965), 496-549. MR 0192520 (33:745)
Additional Information
Mark Goresky
Affiliation:
Institute for Advanced Study, Princeton, New Jersey
Email:
goresky@ias.edu
DOI:
http://dx.doi.org/10.1090/S0273-0979-2012-01382-4
PII:
S 0273-0979(2012)01382-4
Posted:
July 3, 2012
Article copyright:
© Copyright 2012 American Mathematical Society
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