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This chapter continues to study homotopy theory through different products defined between homotopy groups such as the Whitehead product introduced by J.H.C. Whitehead in 1941, the Samelson product introduced by H.

This chapter assembles together some basic concepts and results of set theory, algebra, analysis, set topology, Euclidean spaces, manifolds with standard notations for smooth reading of the book. It is assumed that the readers are familiar with these basic concepts. However, for their detailed study, the books Adhikari and Adhikari (Basic Modern Algebra with Applications, 2014), Dugundji (Topology, 1966), Herstein (Topics in Algebra, 1964), Maunder (Algebraic Topology, 1970), Spanier (Algebraic Topology, 1966), and some other books are referred in Bibliography.

This chapter focuses the history on the emergence of the ideas leading to new areas of study in algebraic topology and conveys the contributions of some mathematicians who introduced new concepts or proved theorems of fundamental importance or inaugurated new theories in algebraic topology starting from the creation of homotopy, fundamental group, and homology group by H. Poincare (1854–1912) in 1895, which are the first most profound and far reaching inventions in algebraic topology.

This chapter opens with a study of homotopy theory by introducing its elementary basic concepts such as homotopy of continuous maps, homotopy equivalence, H-group, H-cogroup, contractible space, retraction, deformation with illustrative geometrical examples and applications.

This chapter conveys a study of a special class of topological spaces, called CW-complexes introduced by J.H.C. Whitehead (1904–1960) in 1949 with their homotopy properties to meet the need for development of algebraic topology. This class of spaces is broader and has some better categorical properties than simplicial complexes, but still retains a combinatorial nature that allows for computation (often with a much smaller complex). Algebraic topologists now feel that the category of CW-complexes is a good category for homotopy and homology theories.

This chapter conveys the geometry of finite simplicial complexes which provides a convenient way to study manifolds and builds up interesting topological spaces called polyhedra from these complexes followed by a study of their homotopy properties with computing their fundamental groups and develops certain analytical geometric tools for subsequent chapters.

This chapter continues to study homology and cohomology theories through the concept of a spectrum and constructs its associated homology and cohomology theories, called spectral homology and cohomology theories. It also introduces the concept of generalized (or extraordinary) homology and cohomology theories. Moreover, this chapter conveys the concept of an \(\Omega \)-spectrum and constructs a new \(\Omega \)-spectrum \(\underline{A}\), generalizing the Eilenberg–MacLane spectrum K(G, n). It constructs a new generalized cohomology theory \(h^*(~ ~ ; \underline{A})\) associated with this spectrum \(\underline{A}\), which generalizes the ordinary cohomology theory of Eilenberg and Steenrod. This chapter works in the category \( {\mathcal {C}}\) whose objects are pairs of spaces having the homotopy type of finite CW-complex pairs and morphisms are continuous maps of such pairs. This is a full subcategory of the category of pairs of topological spaces and maps of pairs, and this admits the construction of mapping cones. Let \({\mathcal {C}_0}\) be the category whose objects are pointed topological spaces having the homotopy type of pointed finite CW-complexes and morphisms are continuous maps of such spaces. There exist the (reduced) suspension functor \(\Sigma : {\mathcal {C}_0} \rightarrow {\mathcal {C}_0}\) and its adjoint functor \(\Omega : {\mathcal {C}_0} \rightarrow {\mathcal {C}_0}\) which is the loop functor.

This chapter continues the study of homotopy theory though the concept of fundamental groups invented by H.Poincare (1854–1912) in 1895, which conveys the first transition from topology to algebra by assigning a group structure on the set of relative homotopy classes of loops in a functorial way.

This chapter continues the study of homotopy theory through fiber bundles, vector bundles, and K-theory.

This chapter opens with homology and cohomology theories which play a key role in algebraic topology. Homology and cohomology groups are also topological invariants like homotopy groups and Euler characteristic. Homology (cohomology) theory is a sequence of covariant (contravariant) functors from the category of chain (cochain) complexes to the category of abelian groups (modules). A key feature of these functors is their homotopy invariance in the sense that homotopic maps induce the same homomorphism in homology (cohomology). In particular, topological spaces of the same homotopy type have isomorphic homology (cohomolgy) groups.

This chapter conveys homotopy theory through an important class of CW-complexes called Eilenberg–MacLane spaces introduced by S. Eilenberg (1915–1998) and S. MacLane (1909–2005) in 1945.

This chapter continues to study homotopy theory displaying construction of a sequence of covariant functors \(\pi _n\) given by W. Hurewicz (1904–1956) in 1935 from topology to algebra by extending the concept of fundamental group, which is the first influential functor of homotopy theory invented by H. Poincare (1854–1912) in 1895.

This chapter displays some similarities and further interesting relations between homology and homotopy groups of topological spaces in addition to some relations between these theories discussed earlier.

This chapter presents an approach formulating axiomatization of homology and cohomology theories which makes the subject algebraic topology elegant and provides a quick access to further study. These axioms, now called Eilenberg and Steenrod axioms for homology and cohomology theories, were announced by S. Eilenberg (1915–1998) and N. Steenrod (1910–1971) in 1945 but first appeared in their celebrated book “The Foundations of Algebraic Topology” in 1952.

The aim of this paper is to find a generalization of topological groups. The concept arises out of the investigation to obtain a group structure on the set [X,Y], of homotopy classes of maps from a space X to a given space Y for all X which is natural with respect to X. We also study the generalized topological groups. Finally, associated with each generalized topological group we construct a contra variant functor from the homotopy category of pointed topological spaces and base point preserving continuous maps to the category of groups and homomorphism.

A k-regular spanning subgraph of G is called a k-factor of G. In this paper we give a necessary and sufficient condition for the inserted graph I(G) of a non-trivial graph G to have a 1-factor. If denotes the degree of the vertex v in G and if for all vertices v in G, where is an even integer then we prove that the inserted graph I(G) of G has a 2m-factor. Also we prove that if m is a non-negative integer and G is a connected graph with even number of edges such that for all vertices v in G, then the inserted graph I(G) of G has a -factor.

Semirings of formal power series over semirings, in particular, over k-semifields, are studied. It is also shown that the semiring of formal power series S[[x]] over a k-semifield S becomes a local semiring. Moreover, the Jacobson radical of S[x]] over a k-semifield S is described.