Homotopic maps

An important property of the homotopic maps is that they become identical once we pass to the algebraic invariants that we have defined. 

Corollary 7.12. The homotopy type of a CW complex does not change if the cell attachment maps are replaced by the homotopic maps. 

Proof. To start with, P is well-defined, since by Theorem 8.8 it takes homotopic maps f,g X —> BF to isomorphic principal F-bundles. 

In the general case, the classes of homotopic mappings of the line y threaded through a planar soliton form the relative homotopy group 7Ti(iR, 91), where 91 is the OP space far from the core of the soliton, shrunk (as compared to the complete OP space 91) by additional interactions (external field, boundary conditions, etc.). If 91 consists of a single point, as in Figure 5.18, 7Ti(9I, 91) coincides with the fundamental group 7Ti(91) [77], [78]. 

Most of the basic properties of standard triangles involve homotopy, and so are best stated in K( ). For example, the mapping cone C of the identity map A —> A is homotopically equivalent to zero, a homotopy between the identity map of C and the zero map being as indicated ... 

First we prove that I is /F-injective. It suffices to show that for any exact complex F, any chain map F —> I is null-homotopic. Let C = Cone(95), where Cone denotes the mapping cone. Consider the exact sequence... 

A further intuitive statement is that the homotopy tjrpe of spaces obtained by gluing over a map should not change if the map is replaced by a homotopic one. 

Proposition 7.11. Let (X, X) be a CW pair. LetY be an arbitrary topological space, and assume that continuous maps f,g A Y are homotopic. Then the spaces X UfY and X UgY are homotopy equivalent. 

Proof. Assume that a = E,B,p) is a fiber bundle, and assume that B is contractible. Let q B B he map that takes the whole space B to some point b B. By our assumptions, the maps q and idg are homotopic. It follows by Theorem 8.8 that the pullbacks q a and idgo are isomorphic. On the other hand, we see that (7 0 is a trivial brmdle, and idgo = E. ... 

Step 1. The identity map of ET into itself is homotopic to the map y> under which the ith factor is mapped identically to the (2i)th factor, i.e.,... 

Theorem 12.9. Consider a one-parameter family of dynamical systems which has a saddle-node periodic orbit L at = 0 such that all orbits in the global unstable set tend to L as t -foo, but do not lie in W[. Let the essential map satisfy m = 0 and fo (p) < 1 for all (p. Then after disappearance of the saddle-node for /i > 0, the system has a stable periodic orbit non-homotopic to L in U) which is the only attractor for all trajectories in U. 

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