By Massey

This ebook is meant to function a textbook for a path in algebraic topology first and foremost graduate point. the most subject matters coated are the type of compact 2-manifolds, the basic team, masking areas, singular homology idea, and singular cohomology concept. those issues are built systematically, warding off all unecessary definitions, terminology, and technical equipment. anywhere attainable, the geometric motivation at the back of some of the recommendations is emphasised. The textual content contains fabric from the 1st 5 chapters of the author's past publication, ALGEBRAIC TOPOLOGY: AN creation (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 conscientiously revised, corrected, and taken modern.

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To give an example, consider the polynomial ring K X1 , X2 , . . in inﬁnitely many variables over a ﬁeld K. As a module over itself, it is generated by the unit element 1. Hence, it is ﬁnitely generated. However, the submodule (X1 , X2 , . ) ⊂ K X1 , X2 , . . , which is given by the ideal generated by all variables, is not ﬁnitely generated. We want to carry assertion (ii) of Proposition 5 over to modules of ﬁnite presentation. Proposition 6. Let ✲ 0 M f ✲ M g ✲ ✲ M 0 be an exact sequence of R-modules.

As an example, consider a family (Mi )i∈I of submodules in M . Then the inclusion maps ιi : Mi ⊂ ✲ M determine an R-module homomorphism Φ : HomR (M, N ) ✲ HomR (Mi , N ), ϕ ✲ (ϕ ◦ ιi )i∈I . i∈I It is easily seen that Φ is an isomorphism for all R-modules N if and only if M is the direct sum of the submodules Mi ⊂ M . Also note that for this assertion we can replace the inclusion maps ιi by arbitrary R-module homomorphisms; they will automatically be injective if Φ is an isomorphism for all N . This gives us the opportunity to characterize direct sums in terms of a universal mapping property.

Is ∈ I. Sometimes the notation x = xi1 ⊕ . . ⊕ xis is used in this context in order to stress the fact that the terms xiσ are unique. (3) Let (Mi )i∈I be a family of R-modules. Then the cartesian product Mi i∈I is an R-module under componentwise addition and scalar multiplication; it is called the direct product of the Mi . 4 Modules Ni ⊂ i∈I Mi , where Ni = i∈I we may view each Mj as a submodule of is direct and we have Mi = (xi )i∈I ∈ i∈I 35 for i = j , for i = j Mj 0 Mi . The sum of these submodules Mi ; xi = 0 for almost all i ∈ I .