In this research, we explore the properties and applications of the mapping cone and its variant, the pinched mapping cone. The mapping cone is a construction that arises naturally in algebraic topology and is used to study the homotopy type of spaces. It has several key properties, including its homotopy equivalence to the cofiber of a continuous map, and its ability to compute homotopy groups using the long exact sequence associated with the cofiber. We also provide an overview of the properties and applications of the mapping cone and the pinched mapping cone in algebraic topology. This work highlights the importance of these constructions in the study of homotopy theory and the calculation of homotopy groups. The study also points to the potential for further research in this area which includes the study of higher homotopy groups and the applications of these constructions to other areas of mathematics.
References
[1]
Lotfi, V. (2021) A Cone-Dominance Approach for Discrete Alternative Multiple Criteria Problems with Indifference Regions. Open Access Library Journal, 8, 1-20. https://doi.org/10.4236/oalib.1108093
[2]
Baziw, E. and Verbeek, G. (2021) Cone Bearing Estimation Utilizing a Hybrid HMM and IFM Smoother Filter Formulation. International Journal of Geosciences, 12, 1040-1054. https://doi.org/10.4236/ijg.2021.1211055
[3]
Tom Dieck, T. (1988) Algebraic Topology and Transformation Groups. Proceedings of a Conference Held in Göttingen, FRG, Vol. 1361, 23-29 August 1987. https://doi.org/10.1007/BFb0083029
[4]
Tom Dieck, T. (2008) Algebraic Topology. Vol. 8, European Mathematical Society. University of Göttingen, Göttingen. https://doi.org/10.4171/048
[5]
Fiorenza, D. and Manetti, M. (2007) Structures on Mapping Cones. Algebra & Number Theory, 1, 301-330. https://doi.org/10.2140/ant.2007.1.301
[6]
Elsner, A.E., Burns, S.A. and Webb, R.H. (1993) Mapping Cone Photopigment Optical Density. JOSA A, 10, 52-58. https://doi.org/10.1364/JOSAA.10.000052
[7]
Tsunoda, K., et al. (2004) Mapping Cone- and Rod-Induced Retinal Responsiveness in Macaque Retina by Optical Imaging. Investigative Ophthalmology & Visual Science, 45, 3820-3826. https://doi.org/10.1167/iovs.04-0394
[8]
Alkhezi, Y. (2019) General Properties of a Morphism on the Pinched Tensor Product over Rings. Far East Journal of Mathematical Sciences, 119, 183-194. https://doi.org/10.17654/MS119020183
[9]
Alkhezi, Y. (2020) Shift and Morphism of the Pinched Tensor Product over Special Rings. Far East Journal of Mathematical Sciences, 122, 47-57. https://doi.org/10.17654/MS122010047
[10]
Alkhezi, Y. (2020) Tensors and the Clifford Algebra Special Case Pinched Tensor Product. Far East Journal of Mathematical Sciences, 124, 127-140. https://doi.org/10.17654/MS124020127
[11]
Christensen, L.W. and Jorgensen, D.A. (2014) Tate (Co)Homology via Pinched Complexes. Transactions of the American Mathematical Society, 366, 667-689. https://doi.org/10.1090/S0002-9947-2013-05746-7
[12]
Alkhezi, Y. (2021) On the Geometric Algebra and Homotopy. Journal of Mathematics Research, 13, 1-52. https://doi.org/10.5539/jmr.v13n2p52
[13]
Alkhezi, Y. and Jorgensen, D. (2019) On Tensor Products of Complete Resolutions. Communications in Algebra, 47, 2172-2184.
[14]
Christensen, L.W., Foxby, H.B. and Holm, H. (2012) Derived Category Methods in Commutative Algebra. https://www.math.ttu.edu/~lchriste/download/dcmca.pdf
