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and are required to fit together in a certain way. Thom–Mather stratified spaces provide a purely topological setting for the study of singularities analogous to the more differential-geometric theory of
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88:(top dimension, codimension 1 boundary, codimension 2 corners), real or complex analytic varieties, or orbit spaces of smooth transformation groups.
1245:
One of the original motivations for stratified spaces were decomposing singular spaces into smooth chunks. For example, given a singular variety
17:
1008:
1517:{\displaystyle {\text{Spec}}\left(\mathbb {C} /\left(x^{4}+y^{4}+z^{4}\right)\right)\xleftarrow {(0,0,0)} {\text{Spec}}(\mathbb {C} )}
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will eventually give a natural stratification. A simple algebreo-geometric example is the singular
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was also a topologically stratified space, with the same strata. Another proof was given by
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is a continuous function. These data need to satisfy the following conditions.
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1665:. Chicago Lectures in Mathematics. Chicago, IL: University of Chicago Press.
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1090:{\displaystyle (\pi _{Y},\rho _{Y}):T_{Y}\cap X\to Y\times (0,+\infty )}
1580:
1652:, Bulletin of the American Mathematical Society 75 (1969), pp.240-284.
1474:
1693:
57:
31:
1685:
1236:(both over the common domain of both sides of the equation).
390:{\displaystyle \{(T_{X}),(\pi _{X}),(\rho _{X})\ |X\in S\}}
80:
Basic examples of Thom–Mather stratified spaces include
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690:{\displaystyle Y\cap {\overline {X}}\neq \emptyset }
266:{\displaystyle V=\bigsqcup _{X\in {\mathcal {S}}}X,}
162:
is a topological space (often we require that it is
1662:
The topological classification of stratified spaces
1358:{\displaystyle \mathrm {Sing} (\mathrm {Sing} (X))}
918:{\displaystyle X=\{v\in T_{X}\ |\ \rho _{X}(v)=0\}}
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135:{\displaystyle (V,{\mathcal {S}},{\mathfrak {J}})}
134:
1229:{\displaystyle \rho _{Y}\circ \pi _{X}=\rho _{Y}}
575:is a locally closed subset and the decomposition
1738:
1176:{\displaystyle \pi _{Y}\circ \pi _{X}=\pi _{Y}}
544:{\displaystyle \rho _{X}:T_{X}\to [0,+\infty )}
84:(top dimension and codimension 1 boundary) and
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52:that has been decomposed into pieces called
1265:, there is a naturally defined subvariety,
96:A Thom–Mather stratified space is a triple
1728:
1714:
1655:
1630:, Invent. Math. 72 (1983), no. 1, 77--129.
723:{\displaystyle Y\subseteq {\overline {X}}}
1507:
1389:
730:. This condition implies that there is a
786:{\displaystyle Y\subset {\overline {X}}}
622:satisfies the axiom of the frontier: if
424:is an open neighborhood of the stratum
14:
1739:
651:{\displaystyle X,Y\in {\mathcal {S}}}
1680:
483:{\displaystyle \pi _{X}:T_{X}\to X}
444:(called the tubular neighborhood),
285:
124:
77:in 1970, inspired by Thom's proof.
24:
1649:Ensembles et morphismes stratifiés
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1298:{\displaystyle \mathrm {Sing} (X)}
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250:
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25:
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1617:, Springer-Verlag, Berlin, 1988.
1550:{\displaystyle {\text{Spec}}(-)}
490:is a continuous retraction, and
293:{\displaystyle {\mathfrak {J}}}
1638:Notes on topological stability
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191:{\displaystyle {\mathcal {S}}}
129:
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34:, a branch of mathematics, an
18:Topologically stratified space
13:
1:
1601:
91:
1700:. You can help Knowledge by
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715:
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43:Thom–Mather stratified space
7:
1641:, Harvard University, 1970.
1564:
1240:
65:. They were introduced by
10:
1778:
1679:
1591:Thom's first isotopy lemma
945:{\displaystyle \rho _{X}}
1628:Intersection homology II
1104:For each pair of strata
979:For each pair of strata
71:Whitney stratified space
69:, who showed that every
1615:Stratified Morse theory
812:{\displaystyle Y\neq X}
82:manifolds with boundary
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1123:{\displaystyle Y<X}
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998:{\displaystyle Y<X}
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753:{\displaystyle Y<X}
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198:is a decomposition of
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86:manifolds with corners
1747:Generalized manifolds
1586:Intersection homology
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952:can be viewed as the
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417:{\displaystyle T_{X}}
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56:; these strata are
1752:Singularity theory
1657:Weinberger, Shmuel
1625:MacPherson, Robert
1612:MacPherson, Robert
1576:Whitney conditions
1571:Singularity theory
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1005:, the restriction
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602:The decomposition
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1258:{\displaystyle X}
969:{\displaystyle X}
956:from the stratum
954:distance function
886:
878:
835:{\displaystyle X}
781:
718:
679:
615:{\displaystyle S}
588:{\displaystyle S}
568:{\displaystyle X}
437:{\displaystyle X}
369:
237:
211:{\displaystyle V}
155:{\displaystyle V}
47:topological space
27:Topological space
16:(Redirected from
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1694:topology-related
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1596:stratified space
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172:second countable
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38:stratified space
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1757:Stratifications
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164:locally compact
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1762:Topology stubs
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1559:prime spectrum
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734:among strata:
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597:locally finite
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300:is the set of
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1696:article is a
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1672:9780226885667
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1626:
1622:
1621:Goresky, Mark
1619:
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1613:
1609:
1608:Goresky, Mark
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829:
822:Each stratum
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741:
733:
732:partial order
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681:
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632:
629:
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582:
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555:Each stratum
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19:
1702:expanding it
1691:
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1367:hypersurface
1244:
953:
302:control data
301:
275:
219:
95:
79:
53:
49:
42:
35:
29:
75:John Mather
1741:Categories
1634:Mather, J.
1602:References
1581:Stratifold
1099:submersion
92:Definition
1542:−
1218:ρ
1205:π
1201:∘
1192:ρ
1165:π
1152:π
1148:∘
1139:π
1082:∞
1067:×
1061:→
1055:∩
1030:ρ
1017:π
934:ρ
889:ρ
864:∈
804:≠
779:¯
771:⊂
716:¯
708:⊆
685:∅
682:≠
677:¯
669:∩
639:∈
536:∞
521:→
499:ρ
475:→
453:π
379:∈
356:ρ
337:π
246:∈
239:⨆
168:Hausdorff
67:René Thom
58:manifolds
36:abstract
1659:(1994).
1645:Thom, R.
1565:See also
1472:←
1241:Examples
32:topology
1557:is the
697:, then
220:strata,
63:Whitney
41:, or a
1669:
1526:where
885:
877:
397:where
368:
170:, and
142:where
54:strata
1692:This
1097:is a
925:. So
218:into
45:is a
1698:stub
1667:ISBN
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