### Notes

Notes will be posted occasionally to supplement the material in lectures and references.

• #### Spaces and Questions

###### Sunday, Mar 22, 2020.

Note: The title alludes to a paper of Gromov, but needless to say this note is much more prosaic. The goal of Algebraic Topology is to answer questions about spaces - obviously including topological questions but sometimes indirectly other questions as well. We have already seen many results that can help us answer these - invariance of dimension, the Jordan-Brouwer separation theorem, the fixed point theorems of Brouwer and Lefschetz etc.

• #### Free resolutions

###### Thursday, Mar 26, 2020.

Homology and cohomomolgy of modules, which are $Ext$ ant $Tor$ in the universal coefficients theorem, are defined in terms of free resolutions. Here we define these and sketch the proofs of their existence and uniqueness. Fix henceforth a ring $R$ and modules $M$ and $N$ over $R$. Note that $R$ may not be commutative, and we take $M$ and $N$ to be bi-modules. Free resolutions A free resolution of $M$ is an exact sequence $\dots \to F_n \to \dots \to F_1 \to F_0 \to M \to 0$ such that each module $F_k$ is free over $R$.

• #### Cohomology - Geometry and Cup products

###### Saturday, Mar 28, 2020.

Pairing and Universal coefficients 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.

Cup products: relative, manifolds with boundary * Relative cochains $C^*(X, A)$ are cochains that vanish on $A$. * For manifolds with boundary, there is no co-cycle condition at the boundary, so co-chains are similar to properly embedded submanifolds with boundary. * On the other hand, relative co-cycles correspond to closed manifolds. Geometry of Poincaré duality * Given a cell decomposition of an oriented $n$-dimensional manifold $M$, we can construct a dual cell decomposition.