Questions asked in lecture: Is \( \Id_X \in \Mor(X, X) \) unique? Do we really need the axiom (for functors) that \( F(\Id_X) = \Id_{F(X)} \)?

These questions were answered in lecture.

A remark: suppose that \(\FF\) is a field; suppose that we are working in the category \(\Vec_\FF\). Then there is a natural transformation from the identity functor to the double dual functor. This is furthermore a natural isomorphism if we restrict to the subcategory of finite dimensional vector spaces.

Question asked in lecture: Does \(\Ext_R(\cdot, Q)\) vanish if \(Q\) is an injective \(R\)-module?

Answer: Yes. This follows quickly if you use the "injective resolution" version of the definition. So the challenge here is to prove that the two definitions are equivalent.

Question asked in lecture: Is there a version of the UCT when \(R\) is not a principal ideal domain?

Answer: I don't know. But several things go wrong in the proof.

Questions asked in lecture: What is the relationship between singular cochains and differential forms? Is there a relationship between singular cohomology and algebraic geometry?

One answer to the first is de Rham's theorem. For example, we can use the Whitney approximation theorem to replace singular chains by smooth chains (in fact, this is a chain homotopy equivalence between the singular and smooth chain groups, proving that the homology theories are isomorphic). We next define the "integration homomorphism" from forms to smooth cochains. This descends to cohomology by Stokes' theorem. We now use open covers, Mayer-Vietoris, and the five lemma to prove that the integration homomorphism is an isomorphism.

Given the above, one answer to the second question consists of "Go read about the Hodge conjecture." Or ask an algebraic geometer!

Question: What is an application of relative MV for homology?

Answer: It is needed for the proof of the "patching theorem" (3.26) which constructs the fundamental class for an \(R\)-orientable closed manifold.

Question asked in lecture: What is the relation between the Chow ring (algebraic geometry) and the other versions of cohomology?

Almost this exact question is asked on mathoverflow; one answer again leads us fairly neatly to the Hodge conjecture.

Question: Suppose we are given non-trivial cohomology classes \(\phi\) and \(\psi\) in dimensions \(k\) and \(\ell\). We are asked to decide if \(\phi \cup \psi\) is non-trivial. However, we compute that \(H_{k + \ell} = 0\). How to proceed? (For example, this happens in the assignment about the Klein bottle.)

Answer in the form of a hint: Use the definitions, not the theorems.

Question: What is the CW structure on \(P^\infty\)? Which norm are we using to define the infinite-dimensional sphere? What is \(P^\infty\) "for"? What is the sphere in other infinite-dimensional spaces?

Answer: \(P^\infty\) is the ascending union of finite dimensional projective spaces. So we can obtain a cell structure by requiring that the $k$-skeleton is exactly the $k$-dimensional projective space. As for the norm - all reasonable choices give the same topology - to be concrete, let's use the $L^2$ norm. \(P^\infty\) is one of the first "classifying spaces".

Question: Does de Rham cohomology give a cohomology theory?

Answer: The simplest answer is "no", as \(H^*_{\mathrm{deR}}\) is defined on smooth manifolds, and "most" CW complexes are not manifolds (let alone smooth).

However, it is reasonable to restrict to manifold pairs \((M, N)\) (where \(N\) is a closed submanifold of \(M\)) and ask which of the axioms are satisfied. A bit of work (using "forms that vanish" on \(N\)) should give us the long exact sequence axiom, for pairs. De Rham's theorem gives a cochain map (?) from this long exact sequences to the one for \(H^*_{\mathrm{sing}}\); also, most of the arrows are isomorphsisms. The five lemma promotes the remaining arrows to isomorphisms from \(H^*_{\mathrm{deR}}(M, N)\) to \(H^*_{\mathrm{sing}}(M, N)\). From this we deduce the disjointness, excision, and homotopy axioms. (Unless I've made a mistake!)

The cap product on homology and cohomology. Its absolute and relative versions. Naturality of the cap product. Poincare duality for compact and non-compact manifolds (statements). Cohomology with compact supports. Direct sets, systems, and limits; examples. Cohomology with compact supports as a direct limit. Local duality maps as direct limits. Mayer-Vietoris for Poincare duality (3.36). Direct limits of diagrams and the proof of Poincare duality via "patching". Applications of Poincare duality: orientability of codimension-one submanifolds of spheres, Euler characteristic in odd dimensions, intersection pairings.

Question: Is there a manifold without a "nice" cell structure? (That is, is there a manifold that actually needs the fancy proof of Poincare duality?)

Answer: Yes, the E8 manifold.