David E. Hurtubise

• Ph.D. in Mathematics
• Professor at Pennsylvania State University

This book presents in great detail all the results one needs to prove the Morse Homology Theorem using classical techniques from algebraic topology and homotopy theory. Most of these results can be found scattered throughout the literature dating from the mid 1900's in some form or another, but often the results are proved in different contexts wit...

We define the symplectic displacement energy of a non-empty subset of a compact symplectic manifold as the infimum of the Hoferlike norm [4] of symplectic diffeomorphisms that displace the set. We show that this energy (like the usual displacement energy defined using Hamiltonian diffeomorphisms) is a strictly positive number on sets with non-empty...

In this paper we survey three approaches to computing the homology of a finite dimensional compact smooth closed manifold using a Morse-Bott function and discuss relationships among the three approaches. The first approach is to perturb the function to a Morse function, the second approach is to use moduli spaces of cascades, and the third approach...

Let $f:M \rightarrow \mathbb{R}$ be a Morse-Bott function on a finite dimensional closed smooth manifold $M$. Choosing an appropriate Riemannian metric on $M$ and Morse-Smale functions $f_j:C_j \rightarrow \mathbb{R}$ on the critical submanifolds $C_j$, one can construct a Morse chain complex whose boundary operator is defined by counting cascades...

We give a new proof of the Morse Homology Theorem by constructing a chain complex associated to a Morse-Bott-Smale function that reduces to the Morse-Smale-Witten chain complex when the function is Morse-Smale and to the chain complex of smooth singular $N$-cube chains when the function is constant. We show that the homology of the chain complex is...

In this chapter we prove the Twisted Morse Homology Theorem, which says that a twisted Morse chain complex on a closed finite dimensional smooth manifold M with coefficients in a bundle of abelian groups G computes the singular homology of M with coefficients in G. This result is not surprising, but the proof is more technical than one might expect...

The focus of this chapter is on proving that the \(\eta \)-twisted Morse cohomology groups are isomorphic to the Lichnerowicz cohomology groups associated to \(-\eta \). We define the twisted Morse-Smale-Witten cochain complex for a general bundle of abelian groups G, and then we restrict to the case where G is defined by a closed 1-form. In this c...

In this chapter we demonstrate the effectiveness of twisted Morse complexes by collecting and expanding on some applications of homology and cohomology with local coefficients. We discuss relationships between parallel 1-forms and Lichnerowicz cohomology, and we show how to use twisted Morse cohomology to explicitly compute the Lichnerowicz cohomol...

In this chapter we prove an invariance theorem which shows that on a closed finite dimensional smooth Riemannian manifold the homology of the twisted Morse-Smale-Witten chain complex is independent of the Morse-Smale pair and depends only on the isomorphism class of the bundle of abelian groups G. The proof of the twisted invariance theorem follows...

In this chapter the Morse-Smale-Witten chain complex with coefficients in a bundle of abelian groups over a finite dimensional smooth Riemannian manifold is constructed. The twisted Morse chain complex is described in detail for \(S^1\) and real projective space. Relationships with Morse complexes on covering spaces are discussed, and a Morse theor...

In this paper we study Morse homology and cohomology with local coefficients, i.e. "twisted" Morse homology and cohomology, on closed finite dimensional smooth manifolds. We prove a Morse theoretic version of Eilenberg's Theorem, and we prove isomorphisms between twisted Morse homology, Steenrod's CW-homology with local coefficients for regular CW-...

Corrected bibliography for Lectures on Morse Homology.

In this note, we present some algebraic examples of multicomplexes whose differentials differ from those in the spectral sequences associated to the multicomplexes. The motivation for constructing examples showing the algebraic distinction between a multicomplex and its associated spectral sequence comes from the author's work on Morse–Bott homolog...

Let f:M→ be a Morse–Bott function on a compact smooth finite-dimensional manifold M. The polynomial Morse inequalities and an explicit perturbation of f defined using Morse functions fj on the critical submanifolds Cj of f show immediately that MBt(f)=Pt(M)+(1+t)R(t), where MBt(f) is the Morse–Bott polynomial of f and Pt(M) is the Poincaré polynomi...

Let f : M → R be a Morse-Bott function on a compact finite dimen-sional manifold M . The polynomial Morse inequalities and an explicit perturbation of f defined using Morse functions on the critical submanifolds of f show imme-diately that M B t (f) = P t (M) + (1 + t)R(t), where M B t (f) is the Morse-Bott polynomial of f and P t (M) is the Poinca...

Let f: M → R be a Morse-Bott function on a compact smooth finite dimensional manifold M. The polynomial Morse inequalities and an explicit perturbation of f defined using Morse functions fj on the critical submanifolds Cj of f show immediately that MBt(f) = Pt(M) + (1 + t)R(t), where MBt(f) is the Morse-Bott polynomial of f and Pt(M) is the Poincar...

It is often said that the Morse-Bott Lemma can be viewed as a “parameterized” Morse Lemma, and its proof should follow from the differentiability of the methods used to prove the Morse Lemma. The goal of this expository paper is to fill in the details. We present Palais' proof of the Morse Lemma using Moser's path method, which yields the necessary...

In this chapter we construct the Morse-Smale-Witten chain complex and prove that its homology coincides with the singular homology. For an interesting history of this complex we refer the reader to Bott’s colorful paper [26]. The story started with a Comptes Rendus Note of the French Academy of Sciences by René Thom in 1949 [145] and culminated wit...

In this chapter we prove some results on transversality, general position, orientations, and intersection numbers that will be used in later chapters, including the Inverse Image Theorem (Theorem 5.11) and the Homotopy Transversality Theorems (Theorem 5.17 and Theorem 5.19). As an application of these results we show that the class of Morse functio...

In this chapter we introduce singular homology, and we prove the CWHomology Theorem. The CW-Homology Theorem (Theorem 2.15) states that the singular homology H * (X,A; Λ) is isomorphic to the homology of the CWchain complex (C * (X, A; A), a), and it gives a formula for computing the boundary operator ∂ * in the CW-chain complex in terms of the deg...

The main goal of this chapter is to show how to construct a CW-complex that is homotopy equivalent to a given smooth manifold M using some special functions on M called “Morse” functions (Theorem 3.28). The CW-homology of the resulting CW-complex is isomorphic to the singular homology of M by Theorem 2.15, and hence it is independent of the choice...

Floer homology theories are attempts to build in infinite dimensions the equivalent of the Morse-Smale-Witten chain complex (C * (f ), ∂ * ) The finite dimensional manifold M is replaced by an infinite dimensional manifold M and the Morse-Smale function f : M → ℝ is replaced by some ”functional” on M.

In this chapter we show how Bott’s perfect Morse functions (discussed in Example 3.7) are examples of a more general class of Morse-Smale functions defined on the complex Grassmann manifolds. The Morse-Smale functions, f A : G n,n+k (ℂ) → ℝ, are defined analogous to the Morse functions constructed in Theorem 3.8.

In this chapter we introduce the Morse-Smale transversality condition for gradient vector fields, and we prove the Kupka-Smale Theorem (Theorem 6.6) which says that the space of smooth Morse-Smale gradient vector fields is a dense subspace of the space of all smooth gradient vector fields on a finite dimensional compact smooth Riemannian manifold (...

The main goal of this chapter is to prove the Stable/Unstable Manifold Theorem for a Morse Function (Theorem 4.2). To do this, we first show that a non-degenerate critical point of a smooth function f : M → ℝ on a finite dimensional smooth Riemannian manifold (M, g) is a hyperbolic fixed point of the diffeomorphism φ t coming from the gradient flow...

We construct a compactification of the space of holo-morphic curves of fixed degree in a finite dimensional complex Grassmann manifold using basic algebra. The algebraic compact-ification is defined as the quotient of n-tuples of linearly indepen-dent elements in a C[z]-module. The complex analytic structure on the space of holomorphic curves of fi...

In this paper we study the topology of a compactifi-cation of the space of holomorphic maps of fixed degree from CP^1 into a finite dimensional complex Grassmann manifold. We show that there is a homotopy equivalence through a range, increasing with the degree, between these compact spaces and an infinite di-mensional complex Grassmann manifold. Th...

The flow category of a Morse-Bott-Smale function f_A:G_n(C∞)→R is shown to be related to the flow category of the action functional on the universal cover of LG_{n,n+k}(C) via a group action. The Floer homotopy type and the associated cohomology ring of f_A:G_n(C)→R are computed. When n=1 this cohomology ring is the Floer cohomology of G_{1,1+k}(C)...

In this note we provide a detailed proof of a ``well-known folk theorem." This theorem has been used by many authors who study the topology of spaces of holomorphic maps [1] [7] [5]. The theorem gives a description of the space of holomorphic maps from $\mathbb{C}P^1$ to the complex Grassmann manifold $G_{n,n+k}(\mathbb{C})$ in terms of equivalence...

A family of Floer functions on the infinite dimensional complex Grassmann manifold is defined by taking direct limits of height functions on adjoint orbits of unitary groups. The Floer cohomology of a generic function in the family is computed using the Schubert calculus. The Floer homotopy type of this function is computed and the Floer cohomology...

Thesis (Ph. D.)--Stanford University, 1995. Submitted to the Department of Mathematics. Copyright by the author.

A stochastic-mechanical lattice model is introduced and extensive calculation are performed to assess the role of multipolar correlations and surface imperfections in influencing the efficiacy of encounter-controlled reactive processes on the surface of a molecular organizate or colloidal catalyst particle. The role of repulsive potentials in influ...

At the molecular level, the surface of a colloidal catalyst particle or molecular organizate (such as a cell or vesicle) is not smooth and continuous, but rather differentiated by the geometry of the constituents and, if the surface composition is not homogeneous, often organized into clusters or domains. The authors develop a model to study the in...