Long Exact Sequence in Homotopy

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Group Cohomology

Covers the concept of group cohomology, focusing on chain complexes, cochain complexes, cup products, and group rings.

Relative Homotopy Groups

Covers relative homotopy groups, establishing long exact sequences and defining boundary homomorphisms.

Induced Homomorphisms on Relative Homology Groups

Covers induced homomorphisms on relative homology groups and their properties.

Model Categories: Properties and Structures

Covers the properties and structures of model categories, focusing on factorizations, model structures, and homotopy of continuous maps.

Simplicial and Singular Homology Equivalence

Demonstrates the equivalence between simplicial and singular homology, proving isomorphisms for finite s-complexes and discussing long exact sequences.

Homotopic Extension Problem

Explores solving the homotopic extension problem, constructing relative CW complexes, and ensuring uniqueness in CW approximations.

Cellular Homology: Applications

Delves into applying cellular homology to compute homology groups and Euler characteristic, showcasing its practical implications.

Homotopy Theory: Cylinders and Path Objects

Covers cylinders, path objects, and homotopy in model categories.

Cellular Approximation Theorem

Focuses on proving the Cellular Approximation Theorem, introducing polyhedra and piecewise linear maps.

Homotopy Equivalence in Chain Complexes

Explores homotopy equivalence in chain complexes, emphasizing path object construction and left/right homotopy characterization.