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grounding in the more elementary parts of

**algebraic topology**, although these are treated wherever possible in an up-to-date way. The reader interested in pursuing the subject further will find ions for further reading in the notes at the end of each chapter. Chapter 1 is a survey of results in algebra and analytic**topology**thatThis course. This course correspondingly has two parts. Part I is point{ set

**topology**, which is concerned with the more analytical and aspects of the theory. Part II is an introduction to**algebraic****topology**, which associates**algebraic**structures such as groups to topological spaces.This is where

**algebraic****topology**comes in. The idea is to associate**algebraic**invariants of a topological space. Here \invariants" means that two homeomorphic spaces should have the same invariants. Thus to show two spaces are not homeomorphic, it su ces to show they have di erent invariants. So, to summarise the entire course:**Topology**is hard.consists of three three-quarter courses, in analysis, algebra, and

**topology**. The ﬁrst two quarters of the**topology**sequence focus on manifold theory and diﬀerential geometry, including diﬀerential forms and, usually, a glimpse of de Rham cohomol-ogy. The third quarter focuses on**algebraic****topology**. I have been teaching theAn Introduction to

**Algebraic****Topology**or: why are we learning this stuff, anyway? Reuben Stern This version: November 22, 2017 Abstract These are notes outlining the basics of**Algebraic****Topology**, written for students in the Fall 2017 iteration of Math 101 at Harvard.This course provides an introduction to

**Algebraic****Topology**, more precisely, to basic reasoning and constructions in**Algebraic****Topology**and some classical invariants such as the fundamental group, singular homology, and cellular homology. The basic idea of**Algebraic****Topology**is to translate topological problemsBasic questions of

**Algebraic****Topology**: 1.Given spaces Xand Y, is X∼Y? 2.What is [X,Y]? Definition.A pair of spaces (X,A) is a space Xand a subset A⊆X. A map of pairs is f: (X,A) →(Y,B) is a continuous map f: X→Y such that f(A) ⊆B. Maps of pairs f 0,f 1: (X,A) →(Y,B) are homotopic, written f 0 ∼f 1, if f 0,f 1: X→Y