全書分為四大部分：第一部分探討流形上的非退化光滑函數，先證明了 Morse 引理，說明此類函數在臨界點附近的性質便像歐氏空間中的非退化二次函數一般，並可依此定義一個臨界點的指標；以此為基礎在流形上做局部操作，證明了對任一在其上給定了一個非退化光滑函數的流形，其同倫型必等同於一 CW 複體，且一個指標為 m 的臨界點便對應了一個維度為 m 的原胞。接著對嵌入歐氏空間中的光滑流形探討了焦點與非退化光滑函數的關聯，利用 Whitney 嵌入定理及 Sard 定理，說明了對任意光滑流形其上均存在有非退化光滑函數；此外還包含了原始 Morse 不等式的討論，及在複多樣體上的應用：Lefschetz 超平面截痕定理。
第二部分為黎曼幾何的簡介，證明了黎曼流形上 Levi-Civita 聯絡的存在（黎曼幾何基本定理），討論了共變微分，曲率張量及其基本性質，測地線的延伸與流形完備性的關聯（Hopf-Rinow 定理）。
第三部分討論測地線"能量"的變分，試圖將第一部分理論中的光滑流形及其上的非退化光滑函數，分別代之以連結某流形上固定兩點的片段光滑路徑空間及路徑之能量泛函。探討了能量泛函的第一變分公式（相當於先前求函數臨界點，此時得出的"臨界點"即測地線），第二變分公式、Jacobi 向量場及定義共軛點的重數（相當於函數臨界點非退化條件及指標的討論），並證明了 Morse 指標定理，說明測地線作為能量泛函的臨界點，其指標即其上與起點共軛點之重數總合，搭配上一些有限維逼近的手法，證明了Morse 理論基本定理：對連接一完備黎曼流形上相對於任何測地線均不共軛之兩點的路徑空間，其同倫形等同於一可數 CW 複體，且任一連接該兩點具指標 m 之測地線對應了一個維度 m 原胞。
第四部份為 Morse 理論的應用，得出了對稱空間及李群的一些幾何性質，並對路徑空間中達能量最小的最短測地線流形的拓樸加以分析，搭配纖維化（fibration）導出的同倫群正則序列，到達本書的高潮，證明了關於酉群及正交群的 Bott 週期性定理。[齊震宇老師]
This book is devoted to an exposition of Morse theory. Starting from scratch, it goes through the proofs of the periodicity theorems of Bott for the unitary and orthogonal groups. The path taken to these theorems is by no means a minimal geodesic but the detours along the way serve to display the power of the theory as it is developed. The book is divided into four parts, the contents of which are outlined below. I. The book begins by showing how the topology of a manifold is related to the critical points of a smooth real-valued function on the manifold with no degenerate critical points. A point p is critical for a function f if the differential of f vanishes there and is degenerate if the Hessian of f has non-zero nullity there. The index of f at a critical point p is the dimension of the maximal sub-space of the tangent space at p on which the Hessian is negative definite. If f is a real-valued function on a manifold M with no degenerate critical points and if Ma=f−1(−∞,a) is compact for all real a, then M has the homotopy type of a CW-complex with one cell of dimension λ for each critical point of f of index λ. The results leading to this theorem are used to prove that a compact manifold which admits a function with just two critical points (non-degenerate) is homeomorphic to a sphere (Reeb's Theorem), to compute the integral homology groups of complex projective space, and to prove the Morse inequalities. In order to establish the existence of functions with no degenerate critical points whose preimages of compact sets are compact, manifolds embedded in Rn are considered. Let p∈Rn and let Lp be the function defined on M whose value at q is the square of the distance from p to q. For almost all p∈Rn it is shown that Lp has the desired properties. A point q∈M is a critical point for Lp if and only if the line joining p to q is perpendicular to M at q. As a preview of things to come the index theorem for Lp is proved. This states that if q is a non-degenerate critical point of Lp, the index of q is the sum of the nullities of the Hessian of Lp′ at q for all p′ on the segment from q to p. The final section of Part I presents the simple, elegant proof of the Lefschetz theorem on hyperplane sections due to Andreotti and Fraenkel. II. This part gives the reader a self-contained treatment of the topics in Riemannian geometry needed in the rest of the book. Affine connection, covariant derivative, parallel displacement along curves, curvature tensor, geodesics, completeness and the exponential map are defined and discussed briefly. III. Now a change of focus occurs and the main object of attention becomes the space of piecewise smooth paths from p to q on a manifold M, Ω(M;p,q). This is almost always written simply as Ω and no confusion results from this abbreviation. The tangent space to Ω at a path ω, TΩω, is the space of all piecewise smooth vector fields along ω which vanish at the end points. The energy function E is defined on Ω and if M is complete, E assumes its minimum d2 on the minimal geodesics from p to q, where the distance from p to q is d. In analogy with the finite-dimensional case of Part I, critical points of E are defined and the first variational formula is used to identify the critical points as the geodesics from p to q. Again motivated by the finite-dimensional case, γ∈Ω at a geodesic, the Hessian E∗∗ of E is defined to be a symmetric bilinear map of TΩγ×TΩγ into the reals whose index at γ is the dimension of the maximal sub-space of TΩγ on which E∗∗ is negative definite. The index of E∗∗ at a minimal geodesic is zero. A Jacobi field along a geodesic is a vector field which satisfies a certain second-order differential equation. Two points p=γ(a) and q=γ(b) are conjugate along a geodesic γ if there exists a non-zero Jacobi field J along γ such that J(a)=J(b)=0. The multiplicity of p and q as conjugate points along γ is the dimension of the space of all such Jacobi fields. The null space of E∗∗ at γ is the space of Jacobi fields along γ in TΩγ. Thus E∗∗ has non-zero nullity at γ if and only if p and q, the end points of γ, are conjugate along γ and the nullity of E∗∗ at γ is just the multiplicity of p and q as conjugate points along γ. A pair of antipodal points on an n-sphere is shown to be conjugate along any great circle with multiplicity n−1. The index of E∗∗ at a geodesic γ is finite and is equal to the number of points γ(t), with 0<t<1, such that γ(0) and γ(t) are conjugate along γ and each such point is counted with its multiplicity (Morse's Index Theorem). If M is a complete Riemannian manifold and p and q are non-conjugate along any geodesic, Ω has the homotopy type of a countable CW-complex with one cell of dimension λ for each geodesic from p to q at which E∗∗ has index λ. This is proved by applying the theory of Part I to E restricted to a finite-dimensional compact manifold with boundary, which is a deformation retract of Ωa=E−1[0,a], and then taking the union of Ωai for an unbounded monotone sequence {ai} of non-critical values of E. As an application, the loop space of the n-sphere is shown to have the homotopy type of a CW-complex with one cell in dimension k(n−1) for all integers k≥0. The assumption that p and q are non-conjugate along any geodesic is not restrictive since in a complete Riemannian manifold M, given any p∈M, for almost any q∈M, p and q are non-conjugate along any geodesic. Next the relation between curvature and geodesics is investigated. In particular, the results just described are applied to prove Cartan's theorem that a simply connected complete Riemannian manifold with all sectional curvatures ≤0 is diffeomorphic to Rn and that any two points on such a manifold are joined by a unique geodesic. The important results for the last part of the book are about manifolds on which the Ricci curvature K is positive definite. The key result here is S. B. Myers's theorem which states that if K(U,U)≥(n−1)/r2 for every unit vector U, then every geodesic of length >πr contains conjugate points and so is not minimal. This result yields that Ω has the homotopy type of a CW-complex with finitely many cells in each dimension if M is compact and K is positive definite everywhere. IV. The object of this last part of the book is the proof of Bott's periodicity theorems for the unitary and orthogonal groups. It begins with a short discussion of symmetric spaces in which the conjugate points to a given point p=γ(0) along a geodesic γ are located. They are the points γ(πk/ei), where k is any non-zero integer and ei is any positive eigenvalue of the linear endomorphism of the tangent space at p which takes a vector W to the vector R(V,W)V, where R is the curvature tensor and V is the velocity vector of γ at p. Next a Lie group with a left- and right-invariant metric is shown to be a symmetric space, and if it is simply connected it is seen via Myers's theorem to be a Cartesian product of a Euclidean space by a compact group on which all sectional curvatures are positive. Thus the Ricci tensor of a compact simply connected Lie group is everywhere positive definite, and so its loop space has the homotopy of a CW-complex with finitely many cells in each dimension. By a direct computation using the location of conjugate points described above, it is shown that this CW-complex has no odd-dimensional cells (Bott). Now a special situation is considered for a complete Riemannian manifold M. Suppose that the distance between p and q is d, and that the space of minimal geodesics Ωd is a topological manifold. If every non-minimal geodesic from p to q has index ≥λ0, then πi(Ω,Ωd)=0 for 0≤i<λ0. Letting M=Sn+1 and p and q be antipodal, this theorem has the Freudenthal suspension theorem as corollary, that πi(Sn)=πi+1(Sn+1) for i≤2n−2. Letting M=SU(2m) it is shown that the space of minimal geodesics from I to −I is homeomorphic to the complex Grassmann manifold of m-dimensional vector subspaces of C2m, Gm(C2m) and that every non-minimal geodesic from I to −I in SU(2m) has index ≥2m+2. Thus πi(Gm(C2m))=πi+1(SU(2m)) for i≤2m. This set of isomorphisms is the contribution of Morse theory to the periodicity theorem for the unitary group. The remaining isomorphisms needed to obtain the periodicity for the stable groups, πi−1U=πi+1U for i≥1, are all consequences of the exactness of the homotopy sequences of bundles. The proof of the periodicity for the infinite orthogonal group O is somewhat longer since the period here is eight rather than two. Thus iterated loop spaces of O(n) must be studied. To this end, for n divisible by a high power of 2, subsets Ωk(n) of O(n) are defined such that Ωk(n)⊂Ωk−1(n)⊂⋯⊂Ω0(n)=O(n). Each Ωj(n) is a smooth totally geodesic submanifold of O(n) and each component of Ωj(n) is a symmetric space. For each 0≤j<k, an element Jj∈Ωj(n) is distinguished and the space of minimal geodesics from Jj to −Jj in Ωj(n) is homeomorphic to Ωj+1(n). Further, in each component of the space of paths from Jj to −Jj in Ωj(n), the index of a non-minimal geodesic is bounded from below by a function of n which goes to infinity with n. Thus πi+1Ωj=πiΩj+1 for all i, where Ωj is the direct limit of Ωj(n) as n goes to infinity. The periodicity πiO=πi+8O is completed now by showing that Ω8 is homeomorphic to O.
Reviewer: Levine, H. I. [form MathSciNet]