1. Basic homological algebra 1 1.1. Chain complexes 2 1.2. Maps and homotopies of maps of chain complexes 2 1.3. Tensor products of chain complexes 3 1.4. Short and long exact sequences 3 1.5. Dual cochain complexes and Hom complexes 4 1.6. Relations between ⊗ and Hom 4 2. The universal coefficient and Künneth theorems 5 2.1. Universal coefficients in homology 5 2.2. The Künneth theorem 6 2.3. ...

In this paper we propose a mathematical framework that can be used for defining cohomology of digital images. We state the Eilenberg-Steenrod axioms and the Universal Coefficient Theorem for this cohomology theory. We show that the Künneth formula doesn’t hold. A cup product is defined and its main properties are proved. 2010 Mathematics Subject Classification. 55N99, 68R99, 68U05, 68U10.

In this article we give characteristic properties of the simplicial homology groups of digital images which are based on the simplicial homology groups of topological spaces in algebraic topology. We then investigate EilenbergSteenrod axioms for the simplicial homology groups of digital images. We state universal coefficient theorem for digital images. We conclude that the Künneth formula for s...

We give a classification theorem for unital separable nuclear simple C∗-algebras with tracial rank no more than one. Let A and B be two unital separable simple nuclear C∗-algebras with TR(A), TR(B) ≤ 1 which satisfy the universal coefficient theorem. We show that A ∼= B if and only if there is an order and unit preserving isomorphism γ = (γ0, γ1, γ2) : (K0(A),K0(A)+, [1A],K1(A), T (A)) ∼= (K0(B...

This is a course in algebraic topology for anyone who has seen the fundamental group and homology. Traditionally (20 years ago), the syllabus included cohomology, the universal coefficient theorem, products, Tor and Ext and duality. We will cover all those topics, while also building a modern framework of derived categories and derived functors using the Cartan–Eilenberg method. We will apply t...

We define united KK-theory for real C*-algebras A and B such that A is separable and B is σ-unital, extending united K-theory in the sense that KKCRT (R, B) = KCRT (B). United KK-theory contains real, complex, and self-conjugate KK-theory; but unlike unaugmented real KK-theory, it admits a universal coefficient theorem. For all separable A and B in which the complexification of B is in the boot...

We describe a duality phenomenon for cohomology theories with the character of Gorenstein rings.