By Anant R. Shastri

Building on rudimentary wisdom of genuine research, point-set topology, and uncomplicated algebra, **Basic Algebraic Topology** offers lots of fabric for a two-semester direction in algebraic topology.

The ebook first introduces the mandatory primary techniques, similar to relative homotopy, fibrations and cofibrations, class conception, phone complexes, and simplicial complexes. It then specializes in the elemental staff, overlaying areas and basic features of homology concept. It provides the vital gadgets of research in topology visualization: manifolds. After constructing the homology conception with coefficients, homology of the goods, and cohomology algebra, the publication returns to the learn of manifolds, discussing Poincaré duality and the De Rham theorem. a short creation to cohomology of sheaves and Čech cohomology follows. The center of the textual content covers larger homotopy teams, Hurewicz’s isomorphism theorem, obstruction concept, Eilenberg-Mac Lane areas, and Moore-Postnikov decomposition. the writer then relates the homology of the complete house of a fibration to that of the bottom and the fiber, with functions to attribute sessions and vector bundles. The e-book concludes with the fundamental concept of spectral sequences and several other purposes, together with Serre’s seminal paintings on greater homotopy groups.

Thoroughly classroom-tested, this self-contained textual content takes scholars all of the technique to changing into algebraic topologists. historic feedback during the textual content make the topic extra significant to scholars. additionally appropriate for researchers, the publication presents references for extra interpreting, offers complete proofs of all effects, and comprises various workouts of various levels.

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**Example text**

The following elementary result answers all the above objections satisfactorily and therefore we shall stick to our definition of a path. 8 (Invariance under subdivision) Let γ : [0, 1] → X be a path, and 0 < t1 < · · · < tn < 1. Then γ is path homotopic to γ|[0,t1 ] ∗ · · · ∗ γ[tn ,1] . Proof: It is enough to prove this for n = 1 with t1 = a. By repeated application of this we get the general case. So, for 0 < a < 1, consider the following parameterisation of the unit interval: α : [0, 1] → [0, 1] given by α(t) = 2at, 0 ≤ t ≤ 1/2, 2t − 1 + (2 − 2t)a, 1/2 ≤ t ≤ 1.

By induction hypothesis, it follows that γ is path homotopic the constant loop. 32 π1 (Sn ) = (1), n ≥ 2. Proof: Write Sn = U ∪ V, where U = Sn \ {(0, . . , 0, 1)}, V = Sn \ {(0, . . , 0, −1}, Then by stereographic projection we know that both U and V are homeomorphic to Rn and hence contractible. Also, it is clear that U, V are both open and U ∩ V is connected. (This is where you need the hypothesis that n ≥ 2. It follows that all the hypotheses in the above theorem are satisfied and hence π1 (Sn ) = (1).

Let G : I × I −→ R be a function such that (i) exp ◦G = H and (ii) for all t ∈ I the function s → G(t, s) is continuous and G(t, 0) = 0. 22. We claim that G is actually a continuous function on the whole of I × I. For this it is enough to prove that it is so restricted to each l l+1 sub-square Sk,l := [ nk , k+1 n ] × [ n , n ]. This we do by induction on l. For l = 0, consider 1 Sk,0 . Clearly H(Sk,0 ) ⊂ S \ {−1}. Therefore G(Sk,0 ) is contained in the disjoint union n 1 1 n− ,n + 2 2 Since G(t, 0) = 0 for all t, it follows that G(t × [0, n1 ]) ⊂ (− 12 , 12 ) by continuity of G|t×I .