By A. T. Fomenko

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Functors are used to transform problems in one category to analogous problems in another. Shortly, we will introduce a functor, the fundamental group, from the category of pointed spaces to the category of groups. A proof of the Brouwer fixed point theorem will be given to illustrate the usefulness of this functor. Actually, what we have just defined is called a covariant functor. Later in the book, we will have occasion to consider contravariant functors. These reverse all morphism arrows and, consequently, satisfy F ( f o g) = F(9) o F ( f ) .

Let M be a manifold, U _c M an open subset, K C U a set that is closed in M, and let f : U --~ R be continuous. 14. Let K C S n be a closed subset, U D K an open neighborhood of K, v a vector field defined on U. Prove that v l K extends to a vector field on all of S ~. 5. Imbeddings and Immersions We will prove that compact manifolds can always be imbedded in Euclidean spaces of suitably large dimensions. 1. Let N and M be topological manifolds of respective dimensions n < m. A topological imbedding of N in M is a continuous map i : N --~ M that carries N homeomorphically onto its image i(N).

Intuitively, we have glued together the two ends of the interval [0, 1] to obtain a circle. 22. Consider the map p : S 1 x[0,1]---*D 2 defined by viewing S 1 C D 2 C C and writing p(z,t) = (1 - t)z. This is one-to-one on S 1 x [0, 1) and collapses S 1 x {1} to the single point 0 C C. Arguing as in the previous example, we see that the quotient space (S a x [0, 1 ] ) / ( S 1 x {0}) is canonically homeomorphic to D 2. Intuitively, we have collapsed the top of the cylinder S 1 x [0, 1] to a point, obtaining a cone that can then be flattened to a disk.