By Paul Kirk James F. Davis

The quantity of algebraic topology a graduate scholar focusing on topology needs to study might be intimidating. in addition, by way of their moment 12 months of graduate stories, scholars needs to make the transition from realizing uncomplicated proofs line-by-line to realizing the general constitution of proofs of adverse theorems. to aid scholars make this transition, the cloth during this publication is gifted in an more and more refined demeanour. it's meant to bridge the space among algebraic and geometric topology, either via supplying the algebraic instruments geometric topologist wishes and by means of targeting these parts of algebraic topology which are geometrically encouraged. necessities for utilizing this e-book comprise easy set-theoretic topology, the definition of CW-complexes, a few wisdom of the elemental group/covering area conception, and the development of singular homology. so much of this fabric is in short reviewed at the start of the publication. the subjects mentioned by means of the authors contain ordinary fabric for first- and second-year graduate classes. The center of the exposition comprises chapters on homotopy teams and on spectral sequences. there's additionally fabric that may curiosity scholars of geometric topology (homology with neighborhood coefficients and obstruction conception) and algebraic topology (spectra and generalized homology), in addition to guidance for extra complicated subject matters corresponding to algebraic $K$-theory and the s-cobordism theorem. a special function of the e-book is the inclusion, on the finish of every bankruptcy, of numerous tasks that require scholars to provide proofs of considerable theorems and to put in writing notes accompanying their reasons. engaged on those tasks permits scholars to grapple with the "big picture", teaches them tips on how to supply mathematical lectures, and prepares them for partaking in learn seminars. The ebook is designed as a textbook for graduate scholars learning algebraic and geometric topology and homotopy conception. it's going to even be invaluable for college kids from different fields equivalent to differential geometry, algebraic geometry, and homological algebra. The exposition within the textual content is obvious; precise circumstances are provided over advanced common statements.

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We then splice this diagram with the previous one to obtain the inductive step. It is important to notice that the short exact sequence 0 → PA → PB → PC → 0 is not (necessarily) a split short exact sequence of chain complexes, even though each chain module is projective. 27. Let 0 → A → B → C → 0 be a short exact sequence of R-modules and let 0 → PA → PB → PC → 0 be a short exact sequence of deleted projective resolutions provided by the horseshoe lemma. Let N be an R-module. Then there are long exact sequences: 42 2.

Earlier in this chapter you were asked in an exercise to compute Tor(A, B) and Ext(A, B) for ﬁnitely generated abelian groups. Lest you learn all the theory without any examples, we give a way of stating the result. Let torsion(A) denote the subgroup of A consisting of elements of ﬁnite order. 6. Universal coeﬃcient theorems and 43 Ext(A, B) ∼ = torsion(A) ⊗Z B. Exercise 28. For any commutative ring R show that ∼ Extq (A, M ) ⊕ Extq (B, M ) Extq (A ⊕ B, M ) = R and R R R R ∼ TorR q (A ⊕ B, M ) = Torq (A, M ) ⊕ Torq (B, M ).

34 2. Homological Algebra If at any stage in the above construction the kernel Kn is projective, then one may stop there since · · · → 0 → 0 → Kn → Pn−1 → · · · → P0 → M → 0 is a projective resolution. We omit typing the 0’s. We also record the following lemma which we used in constructing resolutions. 18 (Splicing lemma). If the sequences A → B − β β◦α → D → E are exact, then A → B −−→ D → E is exact. 0→C− Exercise 24. Prove the splicing lemma. 19. 1. If R is a ﬁeld, and M is any R-module, then Id 0 → M −→ M → 0 is a projective resolution.