M. GORESKY AND R. MACPHEARSON. Stratified Morse Theory. Springer-Verlag, Heidelberg, Germany, 1988. Home/Search Document Not in Database Summary Related Articles Check |
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On Representations and K-Theory of the Braid Groups - Adem, Cohen, Cohen (2001) (Correct)
...., so is homologically toroidal by induction. Since W is transverse to A and is q dimensional, the inclusion i : W M(A) ## M(A) induces an isomorphism : H j (W H j (M(A) Z) in integral homology for each j, 1 1, and a surjection i H q (M(A) Z) by a Lefschetz type theorem (cf. [19]) Now, as is well known, the homology of the complement of an arrangement A is torsion free. Furthermore, the Betti numbers are determined by the intersection poset L(A) the partially ordered set of multi intersections of of elements of A, typically) ordered by reverse inclusion, with rank ....
....the intersection B i M(A) is homeomorphic to M(A i ) the complement of the central subarrangement A i , so is homologically toroidal. Finally, it is known that the top homology of M(A) is isomorphic to the direct sum H # (M(A) Z) H # (B i H # (M(A i ) Z) see [23] or [19]. Since M(A i ) is homologically toroidal for each i, the result follows. # In particular, the pure braid group P n , the fundamental group of the complement of the braid arrangement A = ker(z z j ) 1 i j n in C , is homologically toroidal. Corollary 2.7. Let A be a complex ....
M. Goresky, R. MacPherson, Stratified Morse Theory, Ergeb. Math. Grenzgeb., vol. 14, SpringerVerlag, Berlin-New York, 1988.
Motion Planning for Kinematic Stratified Systems with.. - Goodwine, Burdick (2002) (1 citation) (Correct)
....The dimension of the strata varies between zero, which are isolated point manifolds, and m, which are open subsets of R m . The Whitney condition requires that the tangent spaces of two neighboring strata meet nicely, and for our purposes this condition is generically satisfied (see Ref. [12] for details) In the classical definition of a stratification [12] stratum X i consists of the submanifold S i with all lower dimensional strata (that arise from intersections of S i 2 Note that the terms stratification and strata are also used in other contexts to describe the topology ....
....point manifolds, and m, which are open subsets of R m . The Whitney condition requires that the tangent spaces of two neighboring strata meet nicely, and for our purposes this condition is generically satisfied (see Ref. 12] for details) In the classical definition of a stratification [12], stratum X i consists of the submanifold S i with all lower dimensional strata (that arise from intersections of S i 2 Note that the terms stratification and strata are also used in other contexts to describe the topology of orbit spaces of Lie group actions, and are a slight generalization ....
Goresky and Macpherson. Stratified Morse Theory. Springer--Verlag, New York, 1980.
Arrangements and Local Systems - Cohen, Orlik (2000) (4 citations) (Correct)
....This filtration is well suited for the study of local system cohomology in the sense of the following. Proposition 2.4. For each q, 0 # q # #, we have H i (M q ,M q 1 ; L) 0if i #= q, and dim H q (M q ,M q 1 ; L) b q (A) This Proposition may be proved using stratified Morse theory [GM]. For details, the reader is referred to [C1, Sections 2, 3, and 5] For each q, let K q (A) H q (M q ,M q 1 ; L) and then denote by # q the boundary homomorphism H q (M q ,M q 1 ; L) # H q 1 (M q 1 ,M q ; L) of the triple (M q 1 ,M q ,M q 1 ) It is readily checked that the ....
M. Goresky and R. MacPherson, Stratified Morse Theory, Ergeb. Math. Grenzgeb., vol. 14, Springer-Verlag, Berlin-New York, 1988.
Breaking of linear symmetries and Michel theory: degeneracy.. - Gaeta (Correct)
....particular, there is a unique generic stratum, open and dense, and strata of dimension k are in the frontier of strata of dimension m k; both of these features are always true for stratified manifolds. An analysis, or even a very sketchy summary, of stratified geometry would be out of place here [20]. To avoid any confusion, we will call these strata geometric strata, or also Whitney strata; the strata previously defined in terms of isotropy types will be called isotropy strata, or also Michel strata. As we will mainly deal with the latter kind, whenever I write just stratum it is ....
M. Goresky and R. MacPherson, Stratified Morse theory, Springer 1988
Geometric Topology Of Stratified Spaces - Hughes (1996) (2 citations) (Correct)
....= such that the strata X i = X i n X i Gamma1 are manifolds with neighborhoods in X i [X k (for k i) which have the local homotopy properties of mapping cylinders of fibrations. These spaces include the smoothly stratified spaces of Whitney [28] Thom [24] and Mather [16] see e.g. [9]) as well as the locally conelike stratified spaces of Siebenmann [21] and, hence, orbit spaces of finite groups acting locally linearly on manifolds. Smoothly stratified spaces have the property that strata have neighborhoods which are mapping cylinders of fibre bundles, a fact which is often ....
M. Goresky and R. MacPherson, Stratified Morse theory, Ergeb. Math. Grenzgeb. (3) 14, Springer-Verlag, New York, 1988. MR 90d:57039
Theory and Experiments in Autonomous Sensor-Based Motion Planning .. - Laubach (1999) (7 citations) (Correct)
....field V # associated with f T and the stratification of F given in the text. Note that V # has zeroes at T and where V is normal to the obstacle boundary. Since the obstacle boundaries are smooth, we have a situation applicable for the machinery of stratified Morse theory (described in detail in [17]) We will not discuss this theory in detail, but rather note that in this case, it can be shown that f T is a Morse function on #O i , given a generic environment # [55] 51] One nice property of Morse functions is that their critical points are non degenerate , and therefore isolated [18] ....
M. Goresky and R. MacPherson. Stratified Morse Theory. Springer-Verlag, Berlin, 1988. 154
Lie Algebras Associated To Fiber-Type Arrangements - Cohen, Cohen.. (2000) (1 citation) (Correct)
....= A[1] Proposition 2.2 may be used to determine the cohomology of M (A k ) for k 1 in terms of that of M (A) Let P (A k ; t) P q0 b q (M (A k ) Delta t q be the Poincar e polynomial of M (A k ) where b q (X) is the q th Betti number of X. Results of Goresky and MacPherson [12], and Yuzvinsky [23] see also Feichtner and Ziegler [11] together with Proposition 2.2, yield the following. Corollary 2.3. Let A be a hyperplane arrangement in C . 1) For each k, the integral (co)homology of M (A k ) is torsion free, and we have P (A k ; t) P (A; t 2k Gamma1 ) ....
M. Goresky, R. MacPherson, Stratified Morse Theory, Ergeb. Math. Grenzgeb., vol. 14, SpringerVerlag, Berlin, 1988.
Lower Bound on Testing Membership to a Polyhedron by.. - Grigoriev.. (1994) (1 citation) (Correct)
....on C) of such points w 1 , but since one can take an arbitrary , we get a contradiction. This implies that dim(K 0 ) 0 and completes the proof of the lemma. 4. Faces of P and Whitney stratification of K i Recall that K i , as any semialgebraic set, admits a Whitney stratification (see, e.g. GM 88] Namely, K i can be represented as a disjoint union K i = S j S j of a finite number of semialgebraic sets, called strata, which are smooth manifolds and such that: 1) frontier condition) S j 1 cl(S j 2 ) 6= if and only if S j 1 ae cl(S j 2 ) this defines a partial order S j 1 OE S j 2 ....
M. Goresky, and R. MacPherson "Stratified Morse Theory," Springer-Verlag, Berlin, 1988.
Using the CW-Complex to Represent the Topological Structure of.. - Hart (1999) (1 citation) (Correct)
....M [Milnor, 1963] For example, let 0 be a regular value of g # # # # #: Then by the implicit function theorem, its inverse image g ## ### is a manifold and is called the implicit surface of g. # Note that with special care, Morse theory can be applied to manifolds of continuity as low as # # [Goresky MacPherson, 1988] and for functions of continuity as low as # # [Hart et al. 1998] Let f be a height function on g ## ### such that f#x; y; z##y: Then f is a Morse function. Using a classic example of Bott, let g ## ### be a torus encircling the z axis. Then there exist four critical points such that #f ....
Goresky, M. and MacPherson, R. Stratified Morse Theory. Springer, April 1988.
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M. Goresky, R. MacPherson, Stratified Morse Theory, Ergeb. Math. 14, Springer, Berlin, 1988
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M. GORESKY AND R. MACPHEARSON. Stratified Morse Theory. Springer-Verlag, Heidelberg, Germany, 1988.
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M. Goresky, R. MacPherson, Stratified Morse Theory, Ergeb. Math. Grenzgeb., vol. 14, Springer-Verlag, Berlin-New York, 1988. MR 90d:57039
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M. GORESKY AND R. MACPHEARSON. Stratified Morse Theory. Springer-Verlag, Heidelberg, Germany, 1988.
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M. Goresky, R. MacPherson, Stratified Morse Theory, Ergeb. Math. Grenzgeb., vol. 14, Springer-Verlag, Berlin-New York, 1988. MR 90d:57039
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M. Goresky & R. MacPherson, Stratified Morse Theory, Springer, Berlin, 1988.
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