## Finite Volume Flows And Morse Theory

Citations: | 7 - 1 self |

### BibTeX

@MISC{Harvey_finitevolume,

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

title = {Finite Volume Flows And Morse Theory},

year = {}

}

### OpenURL

### Abstract

this paper we present a new approach to Morse theory based on the de Rham - Federer theory of currents. The full classical theory is derived in a transparent way. The methods carry over uniformly to the equivariant and the holomorphic settings. Moreover, the methods are substantially stronger than the classical ones and have interesting applications to geometry. They lead, for example, to formulas relating characteristic forms and singularities of bundle maps. The ideas came from addressing the following. Question. Given a smooth flow ' t : X ! X on a manifold X, when does the limit P(ff) j lim t!1 '

### Citations

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(Show Context)
Citation Context ...evertheless a homomorphism \Theta j : Hs(F j ) \Gamma! Hsj + (S j ) which after 15 pushing forward to the one-point compactification of S j , is the Thom isomorphism. This leads to the following (cf. =-=[AB]-=-). Corollary 6.5. Suppose thatsp + n p + 1 !sq for all critical points p OE q and that X and all F j and S j are oriented. Then there is an isomorphism Hs(X) = M j H \Gamma j (F j ) One can drop the o... |

333 |
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Citation Context ...constructions of invariants of knots and 3-manifolds from certain "Feynman graphs" (cf. [K], [BT]). The cell decomposition of a manifold by the stable manifolds of a gradient flow is due to =-=Thom [T], [S]-=-. That these stable manifolds embed into the de Rham complex of currents was first observed by Laudenbach [La]. The authors are indebted to Janko Latschev for many useful comments. Notation. For an n-... |

293 |
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(Show Context)
Citation Context ... applies whenever ff is real analytic. Details appear in [HL 2;3 ]. x8. Equivariant Morse Theory. Our approach carries over to equivariant cohomology by using Cartan's equivariant de Rham theory (cf. =-=[BGV]-=-). Suppose G is a compact 16 Lie group with Lie algebra g acting on a compact n-manifold X. By an equivariant differential form on X we mean a G-equivariant polynomial map ff : g \Gamma! E (X). The se... |

152 | Feynman diagrams and low-dimensional topology, First European
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Citation Context ...l in X \Theta X \Theta X (x11). This idea can be further elaborated as in [BC]. Similarly, the method fits into constructions of invariants of knots and 3-manifolds from certain "Feynman graphs&q=-=uot; (cf. [K]-=-, [BT]). The cell decomposition of a manifold by the stable manifolds of a gradient flow is due to Thom [T], [S]. That these stable manifolds embed into the de Rham complex of currents was first obser... |

136 |
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Citation Context ...ication preserves orientations we set n fl = 1, and if not, n fl = \Gamma1. We then define (4.4) N p;q def = X fl2\Gamma p;q n fl : As in [La] Stokes' Theorem now directly gives us the following (cf. =-=[W]-=-). Proposition 4.5. When the gradient flow of f is Morse-Smale, the coefficients in (4.3) are given by n p;q = (\Gamma1) p N p;q 11 Proof. Given a form ff of degreesp \Gamma 1, we have (\Gamma1) p d[S... |

121 |
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Citation Context ...an be used to develop a combinatorial version of their study of knot invariants. 13. Differential characters The ring of differential characters on a smooth manifold, introduced by Cheeger and Simons =-=[ChS]-=- in 1973, has played an important role in geometry. In de Rham-Federer formulations of the theory (see [HL5]), differential characters are represented by sparks. These are currents T with the property... |

112 |
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Citation Context ...X \Theta X \Theta X (x11). This idea can be further elaborated as in [BC]. Similarly, the method fits into constructions of invariants of knots and 3-manifolds from certain "Feynman graphs" =-=(cf. [K], [BT]-=-). The cell decomposition of a manifold by the stable manifolds of a gradient flow is due to Thom [T], [S]. That these stable manifolds embed into the de Rham complex of currents was first observed by... |

104 |
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Citation Context ...lex analogue of the current T . General results of Sommese imply that all the stable and unstable manifolds of the flow are analytic subvarieties. One retrieves classical results of Bialynicki-Birula =-=[BB]-=- and Carrell-Lieberman-Sommese 2 [CL],[CS]. The approach also fits into MacPherson's Grassmann graph construction and the construction of transgression classes in the refined Riemann-Roch Theorem [GS]... |

40 |
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Citation Context ...[BB] and Carrell-Lieberman-Sommese 2 [CL],[CS]. The approach also fits into MacPherson's Grassmann graph construction and the construction of transgression classes in the refined Riemann-Roch Theorem =-=[GS]-=-. The method has many other extensions. It applies to the multiplication and comultiplication operators in cohomology whose kernel is the triple diagonal in X \Theta X \Theta X (x11). This idea can be... |

38 |
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(Show Context)
Citation Context ... relation of C -actions to Morse-Theory is classical. The action ' t decomposes into an "angular" S 1 -action and a radial flow. Averaging a Kahler metric over S 1 and applying an argument o=-=f Frankel [Fr]-=-, we find a function f : X ! R of Bott-Morse type whose gradient generates the radial action. Theorem 9.1 implies that the associated current T on X \Theta X, which is a real slice of Tl, is real anal... |

36 |
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(Show Context)
Citation Context ...nclusion I : E (X) ,! D 0s(X). More precisely using the graph T we shall construct a continuous operator T : E (X) \Gamma! D 0s(X) of degree-1 such that (FME) d ffi T+T ffi d = I \Gamma P: By de Rham =-=[deR]-=-, I induces an isomorphism in cohomology. Hence so does P. Now let f : X ! R be a Morse function with critical set Cr(f ), and suppose there is a riemannian metric on X for which the gradient flow ' t... |

36 | A Theory of Characteristic Currents Associated with a Singular Connection - Harvey, Lawson - 1993 |

20 |
une partition en cellules associée à une fonction sur une variété, Comptes Rendus
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(Show Context)
Citation Context ...into constructions of invariants of knots and 3-manifolds from certain "Feynman graphs" (cf. [K], [BT]). The cell decomposition of a manifold by the stable manifolds of a gradient flow is du=-=e to Thom [T]-=-, [S]. That these stable manifolds embed into the de Rham complex of currents was first observed by Laudenbach [La]. The authors are indebted to Janko Latschev for many useful comments. Notation. For ... |

19 | R.Cohen Graph moduli spaces and cohomology operations
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- 1994
(Show Context)
Citation Context ...r extensions. It applies to the multiplication and comultiplication operators in cohomology whose kernel is the triple diagonal in X \Theta X \Theta X (x11). This idea can be further elaborated as in =-=[BC]. Similarl-=-y, the method fits into constructions of invariants of knots and 3-manifolds from certain "Feynman graphs" (cf. [K], [BT]). The cell decomposition of a manifold by the stable manifolds of a ... |

18 |
A.: Towards the Chow ring of the Hilbert scheme of P 2
- Ellingsrud, Strømme
- 1993
(Show Context)
Citation Context ...braic and that all cohomology theories on X (eg. algebraic cycles modulo rational equivalence, algebraic cycles modulo algebraic equivalence, singular cohomology) are naturally isomorphic. (See [BB], =-=[ES]-=-, [Fr].)22 F. REESE HARVEY AND H. BLAINE LAWSON, JR. When the fixed-point set has positive dimension, one can recover results of Carrell-Lieberman-Sommese for C∗-actions ([CL], [CS]), which assert am... |

12 |
Fundamental solutions in complex analysis
- Harvey, Polking
(Show Context)
Citation Context ...ental Morse equation. As noted above it naturally implies the two homology isomorphisms (over R and Z) which together encapsule Morse Theory. The proof of this equation employs the kernel calculus of =-=[HP]-=- to convert current equations on X \Theta X to operator equations. The method introduced here has many applications. It was used in [HL 2 ] to derive a local version of a formula of MacPherson [Mac 1;... |

10 | G.B.Segal, Morse theory and classifying spacesPreprint
- Cohen
(Show Context)
Citation Context ...ommute with d. Each mapsj : 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 (cf. =-=[CJS]-=-). There is nevertheless a homomorphism \Theta j : Hs(F j ) \Gamma! Hsj + (S j ) which after 15 pushing forward to the one-point compactification of S j , is the Thom isomorphism. This leads to the fo... |

10 |
of Vector Bundle Maps
- MacPherson
(Show Context)
Citation Context ...lculus of [HP] to convert current equations on X × X to operator equations. The method introduced here has many applications. It was used in [HL2] to derive a local version of a formula of MacPherson =-=[Mac1]-=-, [Mac2] which relates the singularities of a generic bundle map A : E → F to characteristic forms of E and F. However, the techniques apply as well to a significantly larger class of bundle maps call... |

8 |
Some topological aspects of C· actions on compact Kahler manifolds
- Carrell, Sommese
- 1979
(Show Context)
Citation Context ...sults of Sommese imply that all the stable and unstable manifolds of the flow are analytic subvarieties. One retrieves classical results of Bialynicki-Birula [BB] and Carrell-Lieberman-Sommese 2 [CL],=-=[CS]-=-. The approach also fits into MacPherson's Grassmann graph construction and the construction of transgression classes in the refined Riemann-Roch Theorem [GS]. The method has many other extensions. It... |

7 |
Differential Characters and Geometric
- Cheeger, Simons
(Show Context)
Citation Context ...sults of Sommese imply that all the stable and unstable manifolds of the flow are analytic subvarieties. One retrieves classical results of Bialynicki-Birula [BB] and Carrell-Lieberman-Sommese 2 [CL],=-=[CS]-=-. The approach also fits into MacPherson's Grassmann graph construction and the construction of transgression classes in the refined Riemann-Roch Theorem [GS]. The method has many other extensions. It... |

6 |
Towards the Chow ring of the Hilbert scheme on P
- Ellingsrud, Strømme
- 1993
(Show Context)
Citation Context ...braic and that all cohomology theories on X (eg. algebraic cycles modulo rational equivalence, algebraic cycles modulo algebraic equivalence, singular cohomology) are naturally isomorphic. (See [BB], =-=[ES]-=-, [Fr]). When the fixed-point set has positive dimension, one can recover results of CarrellLieberman -Sommese for C -actions ([CL], [CS]), which assert among other things that if dim(X C ) = k, then ... |

6 |
Extension theorems for reductive group actions on compact Kaehler manifolds
- Sommese
(Show Context)
Citation Context ...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. The following is a result of Sommese =-=[So]-=-. Theorem 9.1. 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. Th... |

4 |
On the Thom-Smale complex. In An extension of a theorem by Cheeger and Müller by J.-M.Bismuth and
- Laudenbach
- 1992
(Show Context)
Citation Context ... induces an isomorphism H(S Z f ) = Hs(X; Z): In the Morse-Smale case the constants n pq can be computed explicitly by counting flow lines from p to q. This fact follows directly from Stokes' Theorem =-=[La]-=-. The operator T above can be thought of as the fundamental solution for the de Rham complex provided by the Morse flow. It is the Morse-theoretic analogue of the Hodge operator d ffiG where G is the ... |

4 |
and Chern-Weil theory, I - The local MacPherson formula
- Singularities
(Show Context)
Citation Context ...se Theory. The proof of this equation employs the kernel calculus of [HP] to convert current equations on X × X to operator equations. The method introduced here has many applications. It was used in =-=[HL2]-=- to derive a local version of a formula of MacPherson [Mac1], [Mac2] which relates the singularities of a generic bundle map A : E → F to characteristic forms of E and F. However, the techniques apply... |

3 |
Holomorphic vector manifolds and compact Kähler manifolds, Invent
- Carrell, Lieberman
- 1973
(Show Context)
Citation Context ...al results of Sommese imply that all the stable and unstable manifolds of the flow are analytic subvarieties. One retrieves classical results of Bialynicki-Birula [BB] and Carrell-Lieberman-Sommese 2 =-=[CL]-=-,[CS]. The approach also fits into MacPherson's Grassmann graph construction and the construction of transgression classes in the refined Riemann-Roch Theorem [GS]. The method has many other extension... |

3 |
Gradient flows of Morse-Bott functions
- Latschev
(Show Context)
Citation Context ...ical set is a finite disjoint union Cr(f) = a j=1 F j 13 of compact submanifolds F j in X and that Hess(f) is non-degenerate on the normal spaces to Cr(f ). Then for any f-tame gradient flow ' t (cf. =-=[L]-=-) there are stable and unstable manifolds S j = fx 2 X : lim t!1 ' t (x) 2 F j g and U j = fx 2 X : lim t!\Gamma1 ' t (x) 2 F j g for each j, with projections (6.1) S jsj \Gamma! F j oe j /\Gamma U j ... |

3 |
Morse theory and classifying spaces, Stanford preprint
- Cohen, Jones, et al.
- 1993
(Show Context)
Citation Context ...t commute with d. Each map τj : Sj → Fj can be given the structure of a vector bundle of rank λj. The closure Sj ⊂ X is a compactification of this bundle with a complicated structure at infinity (cf. =-=[CJS]-=-). There is nevertheless a homomorphism Θj : H∗(Fj) −→ Hλj+∗(Sj) which after pushing forward to the one-point compactification of Sj, is the Thom isomorphism. This leads to the following (cf. [AB]). C... |

3 |
and Chern-Weil theory, II - Geometric atomicity
- Singularities
(Show Context)
Citation Context ...ic. These include every real analytic bundle map as well as generic direct sums and tensor products of bundle maps. New explicit formulas relating curvature and singularities are derived in each case =-=[HL3]-=-. This extends and simplifies previous work on singular connections and characteristic currents [HL1]. The approach also works for holomorphic C ∗ -actions with fixed-points on Kähler manifolds. One f... |

3 |
theory and Stokes’ theorem
- Morse
(Show Context)
Citation Context ...p and where p ≺ q means there is a piecewise flow line connecting p in forward time to q. Metrics yielding such gradient flows always exist. In fact Morse-Smale gradient systems have these properties =-=[HL4]-=-. Under the hypotheses (1)–(3) we prove that the operator P has the following simple form P(α) = ∑ ( ∫ Up α ) [Sp] where ∫ p∈Cr(f) Up α = 0 if deg α ̸= λp. Thus P gives a retraction P : E ∗ { } def (X... |

2 |
Graph moduli spaces and cohomology operations, Stanford preprint
- Betz, Cohen
- 1993
(Show Context)
Citation Context ... many other extensions. It applies to the multiplication and comultiplication operators in cohomology whose kernel is the triple diagonal in X × X × X (§11). This idea can be further elaborated as in =-=[BC]-=-. Similarly, the method fits into constructions of invariants of knots and 3-manifolds from certain “Feynman graphs” (cf. [K], [BT]). The cell decomposition of a manifold by the stable manifolds of a ... |

2 |
duality for differential characters
- Poincaré-Pontrjagin
(Show Context)
Citation Context ... The ring of differential characters on a smooth manifold, introduced by Cheeger and Simons [ChS] in 1973, has played an important role in geometry. In de Rham-Federer formulations of the theory (see =-=[HL5]-=-), differential characters are represented by sparks. These are currents T with the property that dT = α − R where α is smooth and R is integrally flat. Our Morse Theory systematically produces such c... |