By Andrew H. Wallace
This self-contained textual content is appropriate for complex undergraduate and graduate scholars and will be used both after or simultaneously with classes often topology and algebra. It surveys a number of algebraic invariants: the basic crew, singular and Cech homology teams, and a number of cohomology groups.
Proceeding from the view of topology as a kind of geometry, Wallace emphasizes geometrical motivations and interpretations. as soon as past the singular homology teams, although, the writer advances an realizing of the subject's algebraic styles, leaving geometry apart so one can research those styles as natural algebra. quite a few routines look during the textual content. as well as constructing scholars' considering when it comes to algebraic topology, the routines additionally unify the textual content, on account that lots of them function effects that seem in later expositions. vast appendixes supply important stories of historical past material.
Reprint of the W. A. Benjamin, Inc., ny, 1970 variation.
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Extra resources for Algebraic Topology: Homology and Cohomology
For the space and for its representation. In particular, if different decomposi- tions into simplexes are used for the same space, these will have to be distinguished by the notation. Examples 2-4. If a diagonal is added to each face of a cube the surface of the cube appears as the union of triangles. This is easily seen to be a simplicial On the other hand, the surface of a cube is homeomorphic to the surface of a tetrahedron, and the latter surface is also a simplicial complex. Thus, the two homeomorphic spaces have two entirely different representations as complexes.
DHp a = di1Hp(xo xl ... xp) (Theorem 1-2) = Q1 [(B(x0 x1 ... xp) - (xo xl ... xp) - Hp _ ld(x0 xl ... xp) (Condition (1-8) for dimension p - 1) = Ba - a - HH_ida This verifies Eq. (1-7) on a singular p simplex and, consequently, on any p chain. Next let f: E -+ F be a continuous map and a a singular p simplex on E. f1 Hp a= f1 Q1 HH(xo xl = (fa)1 Hp(xo xl = HH(fa) xp) xp) = HHf1a This verifies condition (1-8). Both (1-7) and (1-8) thus are satisfied for dimension p, on singular p simplexes. H p is now defined on a p chain a = E ai a i by setting Hpa = I aiHPaj The result so obtained can be summed up as follows : 35 1-11.
At the same time, B can be expected to commute with the homomorphisms induced by continuous maps. Geometrically, this would mean that, if a simplex S is embedded in E and if E is continuously mapped into F, then 32 Singular Homology Theory subdividing S and mapping into F gives the same result as carrying S over into F by the continuous map and then subdividing. This will also be proved now. Theorem 1-18. (1) If f: E -+ F is a continuous map and f1 is the induced homomorphism on chain groups then f1 B = Bf1.
Algebraic Topology: Homology and Cohomology by Andrew H. Wallace