Simplicial Structures in Topology

xii Preface
homology groups, also when the coefficients are in an arbitrary Abelian group G.
One of the first to study this possibility was James W. Alexander [2]; Solomon
Lefschetz [22] wrote a detailed account of this paper. The new homology theory
was soon called cohomology (it seems that it was Hassler Whitney [35] who coined
this new term). As we might expect, the cohomology groups H n (|K|;Z) of a polyhedron
|K| are contravariant functors. The cohomology groups of a polyhedron
are related to its homology groups by the Universal Coefficient Theorem; its proof
(in terms of homologycal algebra) is given in this chapter. The cohomology of a
polyhedron is an invariant stronger than the homology, since the cohomology with
coefficients in a commutative ring with identity element (for instance, the ring of
integers Z) is also a ring. The product in such a ring is called cup product. Inthis
way, we may obtain more precise information on the nature of the polyhedron. This
chapter also introduces the cap product which is a bilinear relation of the type
∩: H p (|K|;Z) × Hp+q(|K|;Z) −→ Hq(|K|;Z) ;
the cap product will be used in Chap. V for proving Poincaré’s Duality Theorem.
Chapter V is divided into three sections: Manifolds, Closed Surfaces, and
Poincaré Duality. In the first one, we introduce n-dimensional manifolds (without
boundary) and triangulable n-manifolds. Then, we study closed surfaces, namely,
path-connected, compact 2-manifolds: by a theorem due to Tibor Radó [29], these
surfaces are triangulable. We prove that these manifolds can be classified into three
types: the sphere S 2 , the connected sums of two-dimensional tori (the torus T 2 and
spheres with g handles), and the connected sums of real projective planes. Subsequently,
by using block homology with coefficients in Z, we prove that these three
kinds of spaces are not homeomorphic. Finally, we prove Poincaré’s Duality Theorem
for connected, triangulable, and orientable n-manifolds V, that is to say, for
triangulable n-manifods V such that Hn(V;Z) ∼ = Z. This very important theorem
states that for every 0 ≤ p ≤ n, H n−p (V;Z) ∼ = Hp(V ;Z) holds true.
In the last chapter (Chap. VI), we introduce another very important functor, from
the category of polyhedra to that of groups (not necessarily Abelian), namely, the
fundamental group, defined by Poincaré (see[27]). Next, we study a family of
functors from the category of polyhedra to that of Abelian groups; we are talking
about the (higher) homotopy groups πn(|K|,x0) with n ≥ 2. Only after Heinz Hopf
wrote his 1931 paper [18], did mathematicians show interest in higher homotopy
groups. In this paper, Hopf proved the existence of infinitely many different homotopy
classes of maps from S 3 to S 2 (Satz 1); indeed, the isomorphism π3(S 2 ,e0) ∼ = Z
(see [26]) is deduced from the exact sequence of homotopy groups associated to the
fibration S 3 → S 2 with fiber S 1 .
It is interesting to note that higher homotopy groups had already been introduced
by Eduard Čech [6] during the Zürich International Mathematics Congress, in 1932;
after that, Witold Hurewicz [19] studied these groups in depth. We approach homotopy
groups by considering the set [S n ,|K|]∗ of all based homotopy classes of all
maps S n →|K| of a polyhedron |K| and providing this set with a group operation,
by means of a natural comultiplication of S n (this is a map from S n to the wedge