Supervisors info:
Ιάκωβος Ανδρουλιδάκης, Αναπληρωτής Καθηγητής, Τμήμα Μαθηματικών, ΕΚΠΑ
Διονύσιος Λάππας, Αναπληρωτή Καθηγητής, Τμήμα Μαθηματικών, ΕΚΠΑ
Αντώνιος Μελάς, Καθηγητής, Τμήμα Μαθηματικών, ΕΚΠΑ
Summary:
In this thesis we present a proof of the Hurewicz theorem. The Hurewicz theorem (relative case) provides an isomorphism between homology groups and quotients of homotopy groups of (n-1)-connected pairs of spaces. It was stated by Witold Hurewicz in 1935, who also introduced the notion of homotopy equivalence between spaces, defined absolute and relative homotopy groups of dimension n ≥ 2 and formed the long sequence of these groups. Although we devote a chapter to CW approximation of spaces and the excision theorem in recognition of their significance in algebraic topology, our proof of the Hurewicz theorem does not make use of them, like many others do. Instead, it is based on a more homological approach, which follows the proofs found in Whitehead's book "Elements of homotopy theory" (1978), Spanier's book "Algebraic topology" (1966) and Dieck's book "Algebraic topology" (2008).
The thesis is structured as described below:
In Chapter 1 the necessary background about topological spaces, topologies and topological properties is introduced. Basic operations on spaces, such as cylinders, cones, suspensions and mapping cylinders, are defined. Also, particular spaces that are significant in algebraic topology are presented. More specifically, we introduce loop spaces, H-spaces, simplices, simplicial complexes and CW-complexes. There is also a special reference to the real projective n-space.
Chapter 2 contains information regarding homotopy theory. The notions of homotopy between maps, deformation retraction and homotopy equivalent spaces are presented. Their presentation is followed by the introduction of the fundamental group and higher homotopy groups of a space. The group construction for n>1 is given by the fundamental group of loop spaces. After that we generalise to relative homotopy groups, or in other words homotopy groups that refer to pairs of spaces, and we form, with the help of loop spaces, of mapping fibres and with the application of functors, long sequences of these groups that turn to be exact.
In Chapter 3 homology theory is defined axiomatically. Singular homology is chosen in order to show the existence of a homology theory. Its uniqueness up to isomorphism is ensured as well. At the end simplicial and cellular homology are presented briefly and some computations of homology groups are executed.
Chapter 4 includes some major theorems that are considered basic in algebraic topology. More specifically, we shortly discuss simplicial approximation, we state and prove the cellular approximation and the CW approximation theorems in detail, while we mention without proof the excision theorem of homotopy.
In Chapter 5, finally, the Hurewicz relative theorem is shown, using a homological approach. The absolute Hurewicz theorem is just stated, since its proof follows from the relative case.
Keywords:
Hurewicz theorem, algebraic topology, homotopy theory, homology theory, homotopy groups of higher dimensions.
