By J. F. Adams, G. C. Shepherd

This set of notes, for graduate scholars who're focusing on algebraic topology, adopts a unique method of the educating of the topic. It starts with a survey of the main useful parts for learn, with innovations in regards to the top written debts of every subject. simply because many of the assets are quite inaccessible to scholars, the second one a part of the publication contains a set of a few of those vintage expositions, from journals, lecture notes, theses and convention complaints. they're hooked up by means of brief explanatory passages written through Professor Adams, whose personal contributions to this department of arithmetic are represented within the reprinted articles.

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**Example text**

Let (X, ϕ) and (Y, ψ) be Smale spaces and let π : (Y, ψ) → (X, ϕ) be a map. We say that π is s-bijective (or u-bijective) if, for any y in Y , its restriction to Y s (y) (or Y u (y), respectively) is a bijection to X s (π(y)) (or X u (π(y)), respectively). It is relatively easy to ﬁnd an example of a map which is s-resolving, but not s-bijective and we will give one in a moment. However, one important distinction between the two cases should be pointed out at once. The image of a Smale space under an s-resolving map is not necessarily a Smale space.

Let π : Y → X be a continuous map and let x0 be in X with π −1 {x0 } = {y1 , y2 , . . , yN } ﬁnite. For any > 0, there exists δ > 0 such that π −1 (X(x0 , δ)) ⊂ ∪N n=1 Y (yn , ). Proof. If there is no such δ, we may construct a sequence xk , k ≥ 1 in X converging to x0 and a sequence y k , k ≥ 1 with π(y k ) = xk and y k not in ∪N n=1 Y (yn , ). Passing to a convergent subsequence of the y k , let y be the limit point. Then y is k k not in ∪N n=1 Y (yn , ), since that set is open, while π(y) = limk π(y ) = limk x = x0 .

Similarly, X s (x0 ) = ∪l≥0 ϕ−l (X s (x0 , δ)) and the topology is the inductive limit topology. It follows at once that π is a homeomorphism from the former to the latter. Now we turn to arbitrary point y in Y and x = π(y) in X and show that π : Y s (y) → X s (x) is onto. We choose x0 and {y1 , . . , yN } to be periodic points as above so that π : Y s (yn ) → X s (x0 ) are homeomorphisms. By replacing x0 by another point in its orbit (which will satisfy the same condition), we may assume that x is in the closure of X s (x0 ).