By Liviu Nicolaescu

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

This is the 1st textbook to incorporate issues akin to Morse-Smale flows, Floer homology, min-max thought, second maps and equivariant cohomology, and intricate Morse thought. The exposition is improved with examples, difficulties, and illustrations, and may be of curiosity to graduate scholars in addition to researchers. The reader is anticipated to have a few familiarity with cohomology concept and with the differential and crucial calculus on delicate manifolds.

Some good points of the second one version contain additional functions, corresponding to Morse concept and the curvature of knots, the cohomology of the moduli house of planar polygons, and the Duistermaat-Heckman formulation. the second one version additionally incorporates a new bankruptcy on Morse-Smale flows and Whitney stratifications, many new routines, and diverse corrections from the 1st version.

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This self-contained therapy of Morse thought specializes in purposes and is meant for a graduate direction on differential or algebraic topology. The publication is split into 3 conceptually precise elements. the 1st half comprises the rules of Morse conception. the second one half includes purposes of Morse conception over the reals, whereas the final half describes the fundamentals and a few purposes of advanced Morse conception, a.

Traditionally, functions of algebraic topology to the learn of topological transformation teams have been originated within the paintings of L. E. 1. Brouwer on periodic modifications and, a bit later, within the attractive fastened aspect theorem ofP. A. Smith for top periodic maps on homology spheres. Upon evaluating the mounted element theorem of Smith with its predecessors, the mounted element theorems of Brouwer and Lefschetz, one reveals that it's attainable, not less than for the case of homology spheres, to improve the realization of mere lifestyles (or non-existence) to the particular decision of the homology form of the mounted aspect set, if the map is believed to be best periodic.

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Show that f uniformly continuous implies f is continuous, but construct an example to show that the converse does not hold. 5. Let f : X → Y be a continuous map of the compact metric space (X, dX ) to the metric space (Y, dY ). Show that f is uniformly continuous. 6 Connectedness We next want to discuss the concept of connectedness. The definition is given in terms of its negation, as it is easier to say what we mean by a space not being connected. 1. A topological space X is called separated if it is the union of two disjoint, nonempty open sets.

But this means Ank ⊂ Up , which is a contradiction. 7. In a metric space, compactness. sequential compactness implies Proof. A metric space is totally bounded if given ǫ > 0, we can cover X by a finite number of balls of radius ǫ. We first show that X sequentially compact implies that it is totally bounded. We show this by proving the contrapositive. Suppose X cannot be covered by a finite number of balls of radius ǫ. Let x1 ∈ X. Since B(x1 , ǫ) does not cover X, choose x2 ∈ B(x1 , ǫ). Since B(x1 , ǫ) ∪ B(x2 , ǫ) does not cover X, we may choose x2 ∈ B(x1 , ǫ) ∪ B(x2 , ǫ).

The homeomorphism may be described geometrically as follows. Each ray from the origin intersects D2 and S in a line segment. The intersection with D2 is sent linearly to the intersection with S. We can verify that this is a homeomorphism by deriving a formula for it. This is somewhat tedious, however, so we will give a geometrical explanation, leaving the verification based on this as an exercise. We describe some corresponding open sets from our construction. Given a point x inside the disk which is not the center, we get f (x) by first forming the circle about the center on which x lies, then forming the square which circumscribes this circle, and then sending x to the the point f (x) on the intersection of the perimeter of this square and the ray through x.