Cohomology - Geometry and Cup products

We can interpret the universal coefficients theorem as a pairing $H^k \times H_k \to \mathbb{Z}$ which is non-degenerate up to torsion.

* Observe that if $A$ is a free abelian group, then $Ext^1(A, M)$ vanishes, as $0\to A \to A$ is a free resolution of $A$ .
* In general, for a finitely generated abelian group, we get $Ext^1(A, \mathbb{Z})$ is the torsion of $A$; as we get a free resolution of the form $$0 \to F_k \to F_n \to A$$ with a diagonal homomorphism.
* In particular, $H^1(X) = Hom(H_1(X), \mathbb{Z})$ .

* Given a space $X$, the singular cohomology is the cohomology of the singular chain complex.
* Given a CW complex $X$, the cellular cohomology is the cohomology of the cellular chain complex; this is naturally isomorphic to singular cohomology.
* These are co-functors.
* We can define relative versions.
* Excision, exactness, dimension and homotopy continue to hold
* Compact support does not hold, i.e., it is not true that given a cohomology class, there is a cocycle $\xi$ representing this so that there is a compact set $K$ such that if $\sigma$ is a singular simplex with image disjoint from $K$, then $\xi(\sigma) = 0$ .

* We assign weights to edges that intersect given (unions of) arcs and curves.
* Co-cycles: transversal arcs/surfaces etc match in simplices one dimension higher.

Torus cohomology

A $\Delta$ -complex is like a CW-complex except

* cells are given by characteristic maps $\sigma: \Delta^n \to X$ where $\Delta^n$ is a simplex.
* the restrictions of $\sigma$ to faces are also characteristic maps.
* We can define simplicial homology in this context.

* Motivation: $H^k(X)\times H^l(X) \to H^{k+l}(X \times X)\overset{\Delta^*}\to H^{k+l}(X)$ .

* Fix $X$ and let $\alpha\in C^k(X)$ and $\beta\in C^l(X)$ . Let $n= k + l$.
* We define for $\sigma:<v_0, v_1,\dots, v_{k+l}> \to X$ , $$\alpha\cup \beta(\sigma) = \alpha(\sigma|_{<v_0, \dots, v_k>})\beta(\sigma|_{<v_k, \dots, v_{k+l}>})$$.
* The same definition works for simplicial homology of $\Delta$ -complexes.

* $\delta (\varphi\cup \psi) = (\delta\varphi)\cup \psi+ (-1)^k\varphi\cup\delta\psi$ ; conclude well-defined on cohomology.
* Thus, cohomology is a ring.

Surface Cup Product

Theorem: We have the relation $$\varphi\cup \psi = (-1)^{kl}\psi\cup\varphi$$ in cohomology.