Download e-book for kindle: An Introduction to Algebraic Topology by Andrew H. Wallace

By Andrew H. Wallace

ISBN-10: 0486457869

ISBN-13: 9780486457864

This self-contained therapy assumes just some wisdom of genuine numbers and actual research. the 1st 3 chapters specialise in the fundamentals of point-set topology, and then the textual content proceeds to homology teams and non-stop mapping, barycentric subdivision, and simplicial complexes. routines shape a vital part of the textual content. 1961 version.

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An Introduction to Algebraic Topology - download pdf or read online

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Extra resources for An Introduction to Algebraic Topology

Example text

We often call T the underlying space of the c-stratifold. As for manifolds, we allow ∂T to be empty. Then, of course, a cstratifold is nothing but a stratifold without boundary (or better with an empty boundary). In this way stratifolds are incorporated into the world of c-stratifolds as those c-stratifolds T with ∂T = ∅. The simplest examples of c-stratifolds are given by c-manifolds W . Here ◦ we define T = W and ∂T = ∂W and attach to T and ∂T the stratifold and collar structures given by the smooth manifolds.

Show that if the action is free the quotient space S/G has a unique structure of a k-dimensional stratifold such that the quotient map is a local isomorphism. (9) Let (S, C) be a k-dimensional stratifold. Show that the inclusion map of each stratum f : Si → S is a morphism and the differential dfx is an isomorphism for all x ∈ Si . (10) Show that the composition of morphisms is again a morphism. (11) Prove the statement from the second section that for a stratifold (S, C) a map f is in C if and only if f |Si ∈ C(Si ) for all i and it commutes with local retractions.

Proof: By definition f : Y → R is in C(Y ) if and only if for each y ∈ Y there is a function gy ∈ C and an open neighbourhood Uy of y in S such that f |Uy ∩Y = g|Uy ∩Y . Since Y is closed, the subsets Uy for y ∈ Y and S − Y form an open covering of S. Let {ρi : S → R} be a subordinate smooth partition of unity. Then for each i there is a y(i) such that supp ρi ⊂ Uy(i) or supp ρi ⊆ S − Y . We consider the smooth function defined on Y ρi gy(i) . F := supp ρi ⊂Uy(i) 5. Consequences of Sard’s Theorem 27 For z ∈ Y we have ρi (z)gy(i) (z) = F (z) = supp ρi ⊂Uy(i) ρi (z)f (z) = f (z).

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