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 JordanBrouwer separation theorem, the fixed point theorems of Brouwer and Lefschetz etc.
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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 bimodules. 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 $R$.
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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 nondegenerate 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.
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Poincare duality
Tuesday, Mar 31, 2020.
Cup products: relative, manifolds with boundary * Relative cochains $C^*(X, A)$ are cochains that vanish on $A$. * For manifolds with boundary, there is no cocycle condition at the boundary, so cochains are similar to properly embedded submanifolds with boundary. * On the other hand, relative cocycles correspond to closed manifolds. Geometry of PoincarĂ© duality * Given a cell decomposition of an oriented $n$dimensional manifold $M$, we can construct a dual cell decomposition.
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Algebraic Topology in Lowdimensional topology
Thursday, Apr 2, 2020.
Notes are in pdf (if you see this in the all notes page you will have to click on the specific note).