By Liviu Nicolaescu

This self-contained remedy of Morse conception makes a speciality of purposes and is meant for a graduate path on differential or algebraic topology. The ebook is split into 3 conceptually detailed components. the 1st half comprises the principles of Morse thought. the second one half includes functions of Morse concept over the reals, whereas the final half describes the fundamentals and a few functions of complicated Morse concept, a.k.a. Picard-Lefschetz theory.

This is the 1st textbook to incorporate issues equivalent to Morse-Smale flows, Floer homology, min-max thought, second maps and equivariant cohomology, and complicated Morse idea. The exposition is more suitable with examples, difficulties, and illustrations, and should be of curiosity to graduate scholars in addition to researchers. The reader is predicted to have a few familiarity with cohomology concept and with the differential and crucial calculus on gentle manifolds.

Some good points of the second one variation comprise extra functions, equivalent to Morse thought and the curvature of knots, the cohomology of the moduli house of planar polygons, and the Duistermaat-Heckman formulation. the second one variation additionally contains a new bankruptcy on Morse-Smale flows and Whitney stratifications, many new workouts, and numerous corrections from the 1st version.

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**Extra resources for An Invitation to Morse Theory (2nd Edition) (Universitext)**

**Example text**

3 We like to think of R and R2 as metric subspaces of R3 , although, strictly speaking, they are not even subsets of R3 . 1) a ∈ R3 a2 = a3 = 0 of R in R3 , popularly known as the x-axis, is isometric to R, and the isomorphic copy a ∈ R3 a3 = 0 of R2 in R3 , popularly known as the xy-plane, is isometric to R2 , all spaces being endowed with their Euclidean metrics. Of course, the various other lines and planes of R3 are also isometric to R and R2 , respectively. 4 Suppose Z is a set, (X, d) is a metric space and f : Z → X is an injective function.

1 that dist(z , Y \S) = 0, so that z ∈ ∂Y S. 2 Suppose Y is a metric superspace of a metric space X and S ⊆ X. 1 that ∂X S ⊆ ∂Y S. 5. But some may be in X and some in Y \X; compare, for example, ∂Q Q, which is empty, with ∂R Q, which is R. It may also happen that all are in Y \X; consider the interval (0 , 1) and compare ∂(0 ,1) (0 , 1), which is empty, with ∂R (0 , 1), which is {0, 1}. 5 Boundaries of Unions and Intersections Is the boundary of a union or an intersection related to the boundary of the individual sets that are being united or intersected?

18. 1). 8 Nearest Points The distance from a point x to a subset S of a given metric space is dist(x , S). Under what conditions is there an element of S that is distant exactly dist(x , S) from x? We are interested particularly in knowing what property of S will ensure that such a nearest point of S exists irrespective of what metric superspace X enfolds S and what point x of X we are considering. 1. 1 Suppose (X, d) is a metric space, S is a subset of X, and z ∈ X. A member s of S is called a nearest point of S to z in X if, and only if, d(z, s) = dist(z , S).