Sumários

Other forms of duality

30 maio 2019, 10:30 • Gustavo Granja

Remarks on the H_{n-1} and H^n of a manifold following from the computation of the top homology. Compactly supported cohomology of a locally compact space. Functoriality with respect to proper maps and open embeddings. Examples. The system of fundamental classes for an arbitrary oriented manifold and statement of Poincaré duality for such. Sketch proof. Manifolds with boundary. Existence of collarings. Orientations. Statement of Lefschetz duality. Cech cohomology and Alexander duality for the homology of a complement of a compact set in a sphere. Example: RP^2 does not embed in R^3.

Poincaré duality

29 maio 2019, 10:00 • Gustavo Granja

Finished the proof of the Lemma from last time. Theorem: If M is a connected n-manifold and R is a ring then 1. H_i(M;R)=0 for i>n; H_n(M;R) = 0 if M is not compact; the 2-torsion in R is M is compact and not R-orientable; R if M is compact and R-orientable. The idea of Poincaré duality. Definition and properties of the cap product. Relation with the cup product. Statement of Poincaré duality for a compact R-oriented manifold: cap product with the fundamental class is an isomorphism. Corollary: Any odd dimensional compact manifold has 0 Euler characteristic. Corollary: If M is compact and R-orientable and H^i(M;R) \to \Hom(H_i(M),R) is an iso for each i then H^k(M;R) \otimes H^{n-k}(M;R) \to R given by \alpha \otimes \beta --> <\alpha \cup \beta , [M]> is a perfect pairing. Example: The cohomology ring of projective spaces again. Definition of cohomology with compact support and the statement of Poincaré duality for arbitrary R-oriented manifolds. 

The top homology of a manifold

28 maio 2019, 10:00 • Gustavo Granja

The cohomology ring of projective spaces using the Kunneth formula (following Hatcher). The orientation double cover \tilde M of a manifold. Example: RP^2. Prop: Let M be a connected manifold. Then M is orientable iff \tilde M is disconnected. If \pi_1(M) does not contain a subgroup of index 2 then M is orientable. Definition of the cover M_R \to M whose fiber over x is H_n(M,M\x;R) for R a ring. Relation between M_R and \tilde M. Lemma: Let M be a manifold and A a compact subset. Then 

H_i(M,M\A;R)=0 for all i>n=\dim(M). H_n(M,M\A;R) is naturally identified with the germs of sections of M_R \to M over A.

The cross and cup products

22 maio 2019, 10:00 • Gustavo Granja

Properties of the Eilenberg-Zilber maps: unitarity, associativity, commutativity. The cross product in homology and its properties. Naturality of the EZ maps gives relative EZ and hence relative cross products for good pairs. Cross product with ring coefficients. The Kunneth formula. Example: The cross product gives a generator for the top class in the product of two spheres. Using cartesian products of characteristic maps for the characteristic maps in a product of cell complexes one obtains an isomorphism of chain complexes C_*^{CW}(X\times Y) \cong C_*^{CW}(X) \otimes C_*^{CW}(Y).


The cohomology cross product. Properties including evaluation on the homology cross product. Example: Applying UCT we see that a generator for H^{k+l}(S^k \times S^l) is given by the cross product of the top generators of each factor. The cup product and its properties (cohomology is functorial graded commutative ring). Relation between cross and cup products. The cohomology Kunneth formula. Example: The cohomology ring of S^k \times S^l. 

The Eilenberg-Zilber Theorem

21 maio 2019, 10:00 • Gustavo Granja

Remarks and examples concerning the splitting of the universal coefficient formula. Prop: For G,H abelian groups we have Tor(G,H) \cong Tor(H,G); 2. Tor(G,H)=0 for all H if and only if G is torsion free. In particular \otimes Q is exact and hence H_*(X;\Q)=H_*(X) \otimes Q. The Euler characteristic of a space with finitely generated homology. Prop: If a chain complex is finitely generated the alternating sum of the ranks of the chain groups computes the Euler characteristic. The trace of an endomorphism of a finitely generated abelian group and the Lefschetz number of a self chain map of a finitely generated chain complex. Additivity of the trace on short exact sequences. Statement of the Lefschetz fixed point Theorem (see Hatcher 2.C or Dold for the proof). The tensor product of chain complexes. For the following a reference is Vick or Dold in the bibliography:

Free functors with models. The acyclic model theorem. The Eilenberg-Zilber Theorem: There are natural chain homotopy equivalences C_*(X) \otimes C_*(Y) \cong C_*(X\times Y). Explicit formulas for the chain equivalences: the Alexander-Whitney and the Eilenberg-MacLane shuffle map.