Abstract
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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Adams, J.F.: Algebraic Topology: A student’s Guide. Cambridge University Press, Cambridge (1972)
Adams, J.F.: Stable Homotopy and Generalized Homology. University of Chicago, Chicago (1974)
Adhikari, M.R., Adhikari, A.: Basic Modern Algebra with Applications. Springer, New Delhi (2014)
Brown, E.H.: Cohomology theories. Ann. Math. 75, 467–484 (1962)
Bott, R.: The stable homotopy of the classical groups. Ann. Math. Second Series 70, 313–337 (1959)
Croom, F.H.: Basic Concepts of Algebraic Topology. Springer, New York (1978)
Dieudonné, J.: A History of Algebraic and Differential Topology, 1900-1960. Modern Birkhäuser, Boston (1989)
Dold, A.: Relations between ordinary and extraordinary homology. Algebraic Topology Colloquium, Aarhus, pp. 2–9 (1962)
Dold, A.: Lectures on Algebraic Topology. Springer, New York (1972)
Dold, A., Thom, R.: Quasifaserungen und unendliche symmetrische Produkte. Ann. Math. 67(2), 239–281 (1958)
Eilenberg, S., Steenrod, N.: Foundations of Algebraic Topology. Princeton University Press, Princeton (1952)
Fulton, W.: Algebraic Topology, A First Course. Springer, New York (1975)
Gray, B.: Homotopy Theory, An Introduction to Algebraic Topology. Acamedic Press, New York (1975)
Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)
Huber, P.J.: Homotopy theory in general categories. Math. Ann. 144, 361–385 (1961)
Hilton, P.J.: An Introduction to Homotopy Theory. Cambridge University Press, Cambridge (1983)
Hilton, P.J., Wylie, S.: Homology Theory. Cambridge University Press, Cambridge (1960)
James, I.M. (ed.): Handbook of Algebraic Topology. North Holland, Amsterdam (1995)
Luke, G., Mishchenko, A.S.: Vector Bundles and Their Applications. Kluwer Academic Publishers, Boston (1998)
Mayer, J.: Algebraic Topology. Prentice-Hall, New Jersey (1972)
Massey, W.S.: A Basic Course in Algebraic Topology. Springer, New York (1991)
Maunder, C.R.F.: Algebraic Topology. Cambridge University Press, Cambridge (1980)
Mitra, S.: A study of some notions of algebraic topology through homotopy theory. Ph.D. thesis, University of Calcutta (2007)
Mosher, R., Tangora, M.C.: Cohomology Operations and Applications in Homotopy Theory. Harper and Row, New York (1968)
Spanier, E.: Algebraic Topology. McGraw-Hill, New York (1966)
Steenrod, N.E., Epstein, D.B.A.: Cohomology Operations. Princeton University Press, Princeton (1962)
Switzer, R.M.: Algebraic Topology-Homotopy and Homology. Springer, Berlin (1975)
Vick, J.W.: Homology Theory: Introduction to Algebraic Topology. Springer, New York (1994)
Whitehead, G.W.: Generalized homology theories. Trans. Am. Math. Soc. 102, 227–283 (1962)
Whitehead, G.W.: Elements of Homotopy Theory. Springer, New York (1978)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2016 Springer India
About this chapter
Cite this chapter
Adhikari, M.R. (2016). Spectral Homology and Cohomology Theories. In: Basic Algebraic Topology and its Applications. Springer, New Delhi. https://doi.org/10.1007/978-81-322-2843-1_15
Download citation
DOI: https://doi.org/10.1007/978-81-322-2843-1_15
Published:
Publisher Name: Springer, New Delhi
Print ISBN: 978-81-322-2841-7
Online ISBN: 978-81-322-2843-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)