I took these notes in fall 2016 for Math 215A, taught by Professor Gunnar Carlsson.
A homotopy between and is such that and . A homotopy rel is constant on . A homotopy of points is a path , with . A homotopy of paths is , with . A homotopy of loops is , with . Homotopy equivalence is an isomorphism in the homotopy category : and are homotopy equivalent if and are homotopic to the identity. The fundamental group of is the set of loops () at a base point modulo equivalence under homotopy, with the group operation of concatenation. If is path-connected, then . If is simply-connected, then , by definition. If is convex, then . . for and path-connected van Kampen theorem: assuming , , , and are all open and path-connected and the base point is in . The free product is taken with amalgamation with respect to . if and are locally contractible.
Induced homomorphism: A map induces a homomorphism , since maps loops to loops. Concretely: is a functor. If retracts into , then is injective for the inclusion . A retraction is a map such that . If has a deformation retract into , then is an isomorphism. A deformation retract is a homotopy from to such that and is a retraction. We can also think of it as a homotopy rel from to a retraction into . For a covering space with projection , every point in has a neighborhood for which breaks up into disjoint open sets, each homeomorphic to by . (Stack of pancakes!) The subgroups of correspond to path-connected covering spaces (assuming is path-connected, locally path-connected, and semilocally simply-connected):
A chain complex is a sequence of abelian groups (or objects in any abelian category) and homomorphisms where . represents the -simplices of a space, called -chains. is the boundary map, which takes a simplex to its boundary. are -dimensional cycles. are -dimensional boundaries, indicating -dimensional cycles that become homotopic after adding -simplices. , and is a short exact sequence. A chain map is a morphism of chain complexes, i.e. a sequence of homomorphisms such that . A chain map induces homomorphisms , taking . is a chain map. For any sequence of homomorphisms , the map is a chain map, but it induces the zero homomorphism. Thus, if the difference between two chain maps for some map , called a chain homotopy, then , and and are chain-homotopic. The homology of a chain complex are defined as . is a functor. represents “holes” enclosed by -dimensional sheets. represents connected components. An exact sequence is a chain complex with trivial homology, i.e. . Note that this condition already implies that . is exact iff is injective. is exact iff is surjective. is exact iff is an isomorphism. A short exact sequence is an exact sequence of the form . Zig-zag lemma: if is a short exact sequence of chain complexes, then the following sequence of homology groups is exact: with the connecting homomorphism .
Homology of spaces
In simplicial homology, we set to be the free abelian group generated by the -simplices of a -complex and boundary map generated by In singular homology, we set to be the free abelian group generated by all maps and boundary map generated by These define functors and induce . A map induces a chain map by mapping to the composition (which in turn induces homomorphisms ). If and are homotopic, then and are chain-homotopic (and ). If and are homotopy equivalent, then . This motivates the category . Additionally, reduced homology sets , with defined as extended linearly. The homology groups are denoted and satisfy except . . and for . The relative chain complex for is defined as , with boundary maps given by (well-defined by a diagram chase). The relative homology groups are the homology groups of . Chains in become trivial. , and . A map , i.e. with , induces a chain map . Good pair: if is closed and a deformation retract of some neighborhood of , with the isomorphism induced by . Excision theorem: If , then with the isomorphism induced by . Equivalently, if , then . Long exact sequence of a pair: Applying the zig-zag lemma to the short exact sequence the following sequence is exact: with connecting homomorphism for (and thus is a relative cycle). Long exact sequence of a triple: If , we can easily generalize using the short exact sequence Mayer-Vietoris sequence: Suppose . Applying the zig-zag lemma to the short exact sequence the following sequence is exact: with connecting homomorphism for and .
\todo Example of calculating simplicial homology
Brouwer fixed point theorem
Relate fundamental group and homology
The cochain complex is obtained by applying the contravariant functor to a chain complex : with coboundary map , so . The cohomology groups are . . . . . . Universal coefficient theorem: The homology and cohomology groups are related by the split exact sequence A free resolution of an abelian group is an exact sequence of the form where is free. Homomorphisms can be extended to a chain map , unique up to chain-homotopy. For any two free resolutions and of the same abelian group, . Every abelian group has a free resolution of the form with freely generated by generators of , and the kernel of the surjection. Ext functor: is the first cohomology group of a free resolution of excluding , i.e. the first homology group of . if is free. . If where is the torsion part, then .