## Morse Theory And Stokes' Theorem (0)

Citations: | 1 - 0 self |

### BibTeX

@MISC{Harvey_morsetheory,

author = {F. Reese Harvey and H. Blaine Lawson and Jr.},

title = {Morse Theory And Stokes' Theorem},

year = {}

}

### OpenURL

### Abstract

We present a new, intrinsic approach to Morse Theory which has interesting applications in geometry. We show that a Morse function f on a manifold determines a submanifold T of the product X \Theta X, and that (in the sense that Stokes theorem is valid) T has boundary consisting of the diagonal \Delta ae X \Theta X and a sum P = X p2Cr(f) Up \Theta Sp where Sp and Up are the stable and unstable manifolds at the critical point p. In the language of currents, @T = \Delta \Gamma P:(Stokes Theorem) This current (or kernel) equation on X \Theta X is equivalent to an operator equation d ffi T+T ffi d = I \Gamma P; ((Chain Homotopy)) where P is a chain map onto the finite complex of currents S f spanned by (integration over) the stable manifolds of f . The operator P can be defd on an exterior form ff by P(ff) = lim t!1 '

### Citations

765 |
Geometric Measure Theory
- Federer
- 1969
(Show Context)
Citation Context ...: R \Theta X \Theta X ! X \Theta X is the projection. Our finite-volume assumption on T implies that T is a rectifiable current and therefore that its current boundary is flat in the sense of Federer =-=[14]-=-. Applying the Federer Support Lemma, we conclude that P = X p2Cr(f) [U p ] \Theta [S p ]: (0.5) MORSE THEORY AND STOKES' THEOREM 5 Thus, T provides a homology between the diagonal and the sum of Kunn... |

393 | The Yang-Mills equations over Riemann surfaces - Atiyah, Bott - 1982 |

365 |
Theorie Des Distributions
- Schwartz
- 1997
(Show Context)
Citation Context ...n-form ~ ff 2 e E n (Z) the integral R Z ~ ff is well defined. (This generalizes the usual definition since, if Z is oriented, then e E k (Z) = E k (Z) are identified.) Following de Rham and Schwartz =-=[31]-=- we have the following. Definition A.1. The space of currents of degree k on Z is the topological dual space D 0 k (Z) def = e E n\Gammak (Z) 0 of the space of twisted (n \Gamma k)-forms on Z. Current... |

321 |
Differentiable dynamical systems
- Smale
- 1967
(Show Context)
Citation Context ...to find a more direct connection between the two theories. The idea of using the stable manifolds of a generic gradient flow to give a cell structure to a manifold goes back to R. Thom [33] (See also =-=[30]-=- and [29].) F. Laudenbach was the first to consider the stable currents as de Rham currents [23]. He studied Morse-Smale flows and computed the boundary operator in the Thom-Smale complex by using Sto... |

152 |
Feynman diagrams and low-dimensional topology, from: “First European Congress of Mathematics
- Kontsevich
- 1994
(Show Context)
Citation Context ...manifolds and knots to "Feynman graphs". Primary invariants of this type, such as those discussed in [3] and [4], can be constructed using operators P. The invariants of Kontsevich and Vasse=-=liev (cf. [24]-=-, [8]) involve the currents T. It should be remarked that while the Morse Theory due to Ed Witten [34] involves the de Rham complex, it is distinctly different from the approach presented here. Witten... |

135 |
Supersymmetry and Morse theory
- Witten
- 1982
(Show Context)
Citation Context ...3] and [4], can be constructed using operators P. The invariants of Kontsevich and Vasseliev (cf. [24], [8]) involve the currents T. It should be remarked that while the Morse Theory due to Ed Witten =-=[34]-=- involves the de Rham complex, it is distinctly different from the approach presented here. Witten considers the conjugates d t of exterior differentiation d by the function e \Gammatf for ts0 and exa... |

113 |
On the self-linking of knots
- Bott, Taubes
- 1994
(Show Context)
Citation Context ...lds and knots to "Feynman graphs". Primary invariants of this type, such as those discussed in [3] and [4], can be constructed using operators P. The invariants of Kontsevich and Vasseliev (=-=cf. [24], [8]-=-) involve the currents T. It should be remarked that while the Morse Theory due to Ed Witten [34] involves the de Rham complex, it is distinctly different from the approach presented here. Witten cons... |

102 |
Some theorems on actions of algebraic groups
- Bialynicki-Birula
- 1973
(Show Context)
Citation Context ...eneral results of Sommese imply that T and all of the stable and unstable manifolds of the flow are subvarieties of finite-volume. One retrieves, in particular, classical results of Bialynicki-Birula =-=[5]-=- and of Carrell-Lieberman-Sommese [9], [10]. The approach also fits directly into MacPherson's Grassmann graph construction and Gillet-Soul'e's construction of transgression classes appearing in the r... |

53 |
Morse-Bott theory and equivariant cohomology. The Floer memorial volume
- Austin, Braam
- 1995
(Show Context)
Citation Context ... carries a metric as in 14.3. Let a ! b be regular values of f and consider the compact manifold with boundary Z j f \Gamma1 ([a; b]): On Z we define a vector field V = (/ ffi f)rf where / : [a; b] ! =-=[0; 1] is a-=- smooth function satisfying: (i) / \Gamma1 (0) = fa; bg, (ii) / is linear on [a; a+ffl] and [b\Gammaffl; b], (iii) / j 1 on [a+2ffl; b\Gamma2ffl], (iv) f(Cr(f))"Z ae (a+2ffl; b\Gamma2ffl) for som... |

40 |
Soulé: Characteristic classes for algebraic vector bundles with hermitian metric II
- Gillet, C
- 1990
(Show Context)
Citation Context ...mmese [9], [10]. The approach also fits directly into MacPherson's Grassmann graph construction and Gillet-Soul'e's construction of transgression classes appearing in the refined Riemann-Roch Theorem =-=[17]-=-. The arguments apply literally without change to the case of equivariant cohomology. It yields rapid calculations in certain cases and has been used by J. Latschev to derive a spectral sequence assoc... |

38 |
Fixed points and torsion on Ka¨ hler manifolds
- Frankel
- 1959
(Show Context)
Citation Context ... and that all cohomology theories on X (eg. algebraic cycles modulo rational equivalence, algebraic cycles modulo algebraic equivalence, singular cohomology) are naturally isomorphic. (See [5], [13], =-=[15]-=-). When the fixed-point set has positive dimension, one can recover results of Carrell-Lieberman-Sommese for C -actions ([9], [10]), which assert among other things that if dim(X C ) = k, then H p;q (... |

37 | A theory of characteristic currents associated with a singular connection. Astérisque 213
- Harvey, Lawson
- 1993
(Show Context)
Citation Context ...ristic forms of E and F . At the level of cohomology this retrieves a formula of MacPherson [25], [26]. This work, which is discussed 6 F. REESE HARVEY & H. BLAINE LAWSON, JR. briefly in x9, began in =-=[18] and then -=-inspired the Morse Theory presented here. Much has been written about assigning topological invariants of manifolds and knots to "Feynman graphs". Primary invariants of this type, such as th... |

36 |
Rham, Variétés differentiables
- de
- 1955
(Show Context)
Citation Context ...or is chain homotopic to the inclusion I : E (X) ,! D 0s(X), that is, there exists a continuous operator T : E (X) \Gamma! D 0s(X) of degree-1 such that d ffi T+T ffi d = I \Gamma P: (0.2) By de Rham =-=[12]-=-, I induces an isomorphism in cohomology. Hence so does P. The existence of P and T satisfying (0.1) and (0.2) is established for any flow of finite volume. This concept, which is central to our paper... |

19 | R.Cohen Graph moduli spaces and cohomology operations
- Betz
- 1994
(Show Context)
Citation Context ...rse Theory presented here. Much has been written about assigning topological invariants of manifolds and knots to "Feynman graphs". Primary invariants of this type, such as those discussed i=-=n [3] and [4]-=-, can be constructed using operators P. The invariants of Kontsevich and Vasseliev (cf. [24], [8]) involve the currents T. It should be remarked that while the Morse Theory due to Ed Witten [34] invol... |

19 | Intersection theory on spherical varieties - Fulton, MacPherson, et al. - 1995 |

13 | on Vanishing Theorems - Lectures - 1992 |

12 |
Fundamental solutions in complex analysis
- Harvey, Polking
(Show Context)
Citation Context ...mma P (0.4) on X \Theta X, where \Delta denotes the diagonal. There is a general correspondence between operators K : E (X) ! D 0s(X) of degree ` and currents K of dimension n \Gamma ` on X \Theta X (=-=[21]-=-, See Appendix A.) Under this transformation: I corresponds to [\Delta], the pull-back of forms by ' t corresponds to the graph of ' t , and the chain homotopy d ffi K + K ffi d corresponds to the cur... |

10 | G.B.Segal, Morse theory and classifying spacesPreprint
- Cohen
(Show Context)
Citation Context ...re precise. Note thatsj : S j ! F j can be given the structure of a vector bundle of ranksj . The closure S j ae X is a compactification of this bundle with a complicated structure at infinity. ( See =-=[11]-=- for example.) There is nevertheless a homomorphism \Theta j : Hs(F j ) \Gamma! Hsj + (S j ) which after pushing forward to the onepoint compactification of S j , is the Thom isomorphism. This leads t... |

10 |
of Vector Bundle Maps
- MacPherson
(Show Context)
Citation Context ... equation of forms and currents which relates the singularities of a smooth bundle map A : E ! F to characteristic forms of E and F . At the level of cohomology this retrieves a formula of MacPherson =-=[25]-=-, [26]. This work, which is discussed 6 F. REESE HARVEY & H. BLAINE LAWSON, JR. briefly in x9, began in [18] and then inspired the Morse Theory presented here. Much has been written about assigning to... |

8 |
Some topological aspects of C· actions on compact Kahler manifolds
- Carrell, Sommese
- 1979
(Show Context)
Citation Context ... all of the stable and unstable manifolds of the flow are subvarieties of finite-volume. One retrieves, in particular, classical results of Bialynicki-Birula [5] and of Carrell-Lieberman-Sommese [9], =-=[10]-=-. The approach also fits directly into MacPherson's Grassmann graph construction and Gillet-Soul'e's construction of transgression classes appearing in the refined Riemann-Roch Theorem [17]. The argum... |

6 |
Extension theorems for reductive group actions on compact Kaehler manifolds
- Sommese
(Show Context)
Citation Context ...act Kahler manifold X, there is a complex graph Tl def = f(t; ' t (x); x) 2 C \Theta X \Theta X : t 2 C and x 2 Xg ae P 1 (C) \Theta X \Theta X analogous to the graphs considered above. Theorem 8.1. (=-=[32]-=-) If ' t has fixed-points, then Tl has finite volume and its closure Tl in P 1 (C) \Theta X \Theta X is an analytic subvariety. The relation of C -actions to Morse-Theory is classical. One can decompo... |

3 |
Holomorphic vector manifolds and compact Kähler manifolds, Invent
- Carrell, Lieberman
- 1973
(Show Context)
Citation Context ...T and all of the stable and unstable manifolds of the flow are subvarieties of finite-volume. One retrieves, in particular, classical results of Bialynicki-Birula [5] and of Carrell-Lieberman-Sommese =-=[9]-=-, [10]. The approach also fits directly into MacPherson's Grassmann graph construction and Gillet-Soul'e's construction of transgression classes appearing in the refined Riemann-Roch Theorem [17]. The... |

1 |
Operad representations in Morse theory and Floer homology
- Betz
- 1996
(Show Context)
Citation Context ...d the Morse Theory presented here. Much has been written about assigning topological invariants of manifolds and knots to "Feynman graphs". Primary invariants of this type, such as those dis=-=cussed in [3]-=- and [4], can be constructed using operators P. The invariants of Kontsevich and Vasseliev (cf. [24], [8]) involve the currents T. It should be remarked that while the Morse Theory due to Ed Witten [3... |

1 |
On the Thom-Smale complex, Ast'erisque, An Extension of a Theorem of Cheeger and
- Laudenbach
- 1992
(Show Context)
Citation Context ...s of a generic gradient flow to give a cell structure to a manifold goes back to R. Thom [33] (See also [30] and [29].) F. Laudenbach was the first to consider the stable currents as de Rham currents =-=[23]-=-. He studied Morse-Smale flows and computed the boundary operator in the Thom-Smale complex by using Stokes' Theorem as we do here. The authors would like to thank Janko Latschev for many useful comme... |

1 |
A generalization of Morse-Smale inequalities
- Rosenberg
- 1964
(Show Context)
Citation Context ... more direct connection between the two theories. The idea of using the stable manifolds of a generic gradient flow to give a cell structure to a manifold goes back to R. Thom [33] (See also [30] and =-=[29]-=-.) F. Laudenbach was the first to consider the stable currents as de Rham currents [23]. He studied Morse-Smale flows and computed the boundary operator in the Thom-Smale complex by using Stokes' Theo... |

1 |
Sur une partition en cellules assici'ees `a une fonction sur une vari'et'e, CRAS 228
- Thom
- 1949
(Show Context)
Citation Context ...be interesting to find a more direct connection between the two theories. The idea of using the stable manifolds of a generic gradient flow to give a cell structure to a manifold goes back to R. Thom =-=[33]-=- (See also [30] and [29].) F. Laudenbach was the first to consider the stable currents as de Rham currents [23]. He studied Morse-Smale flows and computed the boundary operator in the Thom-Smale compl... |