By Massey

ISBN-10: 038797430X

ISBN-13: 9780387974309

This publication is meant to function a textbook for a path in algebraic topology in the beginning graduate point. the most subject matters lined are the type of compact 2-manifolds, the basic staff, overlaying areas, singular homology conception, and singular cohomology thought. those subject matters are constructed systematically, heading off all unecessary definitions, terminology, and technical equipment. anyplace attainable, the geometric motivation at the back of many of the strategies is emphasised. The textual content contains fabric from the 1st 5 chapters of the author's prior ebook, ALGEBRAIC TOPOLOGY: AN advent (GTM 56), including just about all of the now out-of- print SINGULAR HOMOLOGY thought (GTM 70). the fabric from the sooner books has been rigorously revised, corrected, and taken brand new.

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**Extra resources for A basic course in algebraic topology**

**Example text**

5) Representing elements of Z[i] as pairs of integers, implement two CoCoA functions GaussGCD(. ) and GaussLCM(. ) which compute the greatest common divisor and least common multiple of two Gaußian integers, respectively. 6) Using CoCoA, program a factorization algorithm GaussFactor(. ) for elements z ∈ Z[i]. Hint: Proceed as for integers, searching for divisors in the set of all elements a + bi such that a2 + b2 divides ϕ(z). Tutorial 5: Squarefree Parts of Polynomials In this tutorial we shall explore how one can eﬀectively compute the squarefree part of a univariate polynomial over certain fields.

Hint: Let (a) ∩ (b) = (c) , let ab = rc , and let c = sa . Show that s cannot be a unit. Then deduce that r has to be a unit. 2 Unique Factorization b) Use a) to prove that two factorizations of any element are the same up to order and units. c) Conclude that R is a factorial domain. √ Exercise√7. Consider the ring R = Z[ −5] . a. √ Hint: Show that both 2 and 1 + −5 are common divisors. Exercise 8. Let R be a factorial domain and a, b ∈ R \ {0} two coprime elements. Prove that the polynomial ax+b is an irreducible element of R[x] .

A) Show that a P -submodule M ⊆ P r is a monomial module if and only if for every m ∈ M and every t ∈ Supp(m) we have t ∈ M . b) Write a CoCoA function Is Monomial(. ) which, for a list of vectors generating a P -submodule M ⊆ P r , checks if M is monomial and which returns TRUE or FALSE. ) c) Implement a CoCoA function MonComps(. ) which, for a list of terms generating a monomial P -submodule M ⊆ P r , computes the list of monomial ideals I1 , . . b. d) Write a CoCoA program MinMonomials(. ) which takes a list of terms generating a monomial P -submodule of P r and computes the minimal monomial system of generators of that module.

### A basic course in algebraic topology by Massey

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