Notes:Fundamentals of Algebraic Topology - Weintraub

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TODO: Make up for the rush this was created in

Chapter 1

Section 1.1: Background

A pair [ilmath](X,A)[/ilmath] consists of a space [ilmath]X[/ilmath] and a subspace [ilmath]A\in\mathcal{P}(X)[/ilmath]. We let [ilmath]f:X\rightarrow Y[/ilmath] denote a map from [ilmath]X[/ilmath] to [ilmath]Y[/ilmath] as usual.

Similarly we let [ilmath]f:(X,A)\rightarrow(Y,B)[/ilmath] denote a map from the pair [ilmath](X,A)[/ilmath] to [ilmath](Y,B)[/ilmath], i.e. a map [ilmath]f:X\rightarrow Y[/ilmath] whose restriction is [ilmath]f\big\vert_A:A\rightarrow B[/ilmath]; or more simply, a map [ilmath]f:X\rightarrow Y[/ilmath] with [ilmath]f(A)\subseteq B[/ilmath]

For most purposes we can identify the pair [ilmath](X,\emptyset)[/ilmath] with [ilmath]X[/ilmath]. Then [ilmath]f:X\rightarrow Y[/ilmath] is a special case of [ilmath]f:(X,\emptyset)\rightarrow (Y,\emptyset)[/ilmath]. The author notes this undermines clarity. Ha.

A homeomorphism, [ilmath]f:X\rightarrow Y[/ilmath] is a continuous map with a continuous inverse map, [ilmath]g:Y\rightarrow X[/ilmath]. A homeomorphism [ilmath]f:(X,A)\righarrow(Y,B)[/ilmath] is defined similarly.

  • I think this means [ilmath]f:(X,A)\rightarrow (Y,B)[/ilmath] with inverse [ilmath]g:(Y,B)\rightarrow (X,A)[/ilmath], this ends up with [ilmath]f(A)=B[/ilmath]

The spaces or pairs are said to be homeomorphic.

This author uses [ilmath]\approx[/ilmath] for homeomorphic spaces (or "pairs", [ilmath](X,A)\approx(Y,B)[/ilmath].

For a pair [ilmath](X,Y)[/ilmath] we let [ilmath]X/Y[/ilmath] denote the quotient space.

We also have the homeomorphism:

  • [ilmath]f:\frac{[0,1]}{(\{0\}\cup\{1\})}\rightarrow\mathbb{S}^1[/ilmath] by [ilmath]f:t\mapsto e^{2\pi jt}[/ilmath] which he does often use in the next sections.

Given two spaces [ilmath]X[/ilmath] and [ilmath]Y[/ilmath] we write [ilmath]X\times Y[/ilmath] for their product, as usual. We define the product of pairs by:

  • [ilmath](X,A)\times(Y,B):=\big(X\times Y, (X\times B)\cup(A\times Y)\big)[/ilmath]

Section 1.2: Homotopy