For digraphs D and H, a mapping f : V (D)→V (H) is a homomorphism of D to H if uv ∈ A(D) implies f(u)f(v) ∈ A(H). Let H be a fixed directed or undirected graph. The homomorphism problem for H asks whether a directed or undirected graph input digraph D admits a homomorphism to H. The list homomorphism problem for H is a generalization of the homomorphism problem for H, where every vertex x ∈ V (...

In the early 1980's, Johnson defined a homomorphism $\mathcal{I}_{g}^1\to\bigwedge^3 H_1(S_{g},\mathbb{Z})$, where $\mathcal{I}_{g}^1$ is Torelli group of closed, connected and oriented surface genus $g$ with boundary component $S_g$ corresponding without component. This known as homomorphism. We study map induced by on rational homology groups apply it to abelian cycles determined disjoint bou...

Notes taken by Stuart White during the lectures and not proof read. Errors will abound. 1. Nuclearity: Approximation and perturbation Recall: Definition 1.1. A C∗-algebra. Definition 1.2. A cp map Theorem 1.3. Stinespring’s theorem. Every cp map is a compression of a ∗-homomorphism. Definition 1.4. φ : A → B cp has order zero if a ⊥ b =⇒ φ(a) ⊥ φ(b). 1 Theorem 1.5. (i) φ : A→ B has order zero i...

For a given pair of maps f, g : X → M from an arbitrary topological space to an n-manifold, the Lefschetz homomorphism is a certain graded homomorphism Λfg : H(X) → H(M) of degree (−n). We prove a Lefschetztype coincidence theorem: if the Lefschetz homomorphism is nontrivial then there is an x ∈ X such that f(x) = g(x).

We survey known results about the complexity of surjective homomorphism problems, studied in the context of related problems in the literature such as list homomorphism, retraction and compaction. In comparison with these problems, surjective homomorphism problems seem to be harder to classify and we examine especially three concrete problems that have arisen from the literature, two of which r...