In the future please link to abstract pages rather than pdf files. Another tool, the homotopy axiom, will come in the next chapter. From a formal point of view, the mayervietoris sequence can be derived from the eilenbergsteenrod axioms for homology theories using the long exact sequence in homology. R, the mayer vietoris exact sequence, and the kunneth formula see below. Similar constructions are available in a wide variety of other contexts, such as abstract algebra, groups, lie algebras, galois theory, and algebraic. Mathematics gr6402 fall 2017 tuesday and thursday 10. We use the mayervietoris sequence to determine the betti numbers. Show that a mayer vietoris sequence exists in this framework. By duality see universal coefficient theorem an analogous statement holds for the homology of x x, u u and v v. The mayervietoris sequence is an important computational tool in cohomology, as in homology. Thanks for contributing an answer to mathematics stack exchange. Degree, linking numbers and index of vector fields 12. We also prove some graphtheoretical analogues of standard results in di erential geometry, in particular, a graph version of stokes theorem and the mayervietoris sequence in cohomology.
Third, there is the mayervietoris sequence, which allows us to compute. This long exact sequence of cohomology groups is called the meyervietoris. Chapter 1 manifolds and varieties via sheaves as a. I found tus book an introduction manifolds, where a computation is presented via mayer vietoris sequences. Mayervietoris sequence for differentiablediffeological. A gentle introduction to homology, cohomology, and sheaf cohomology jean gallier and jocelyn quaintance department of computer and information science. I eventually got my partitions of unity sorted out, and managed to do this the fake mayer vietoris way i was looking for.
The mayer vietoris sequence is a technique for computing the cohomology. Extensive use of figures, taken from page 150 hatcher. Explaining application of mayer vietoris to klein bottle and torus. This book offers a selfcontained exposition to this subject and to the theory of characteristic classes from the curvature point of view. In mathematics, homology is a general way of associating a sequence of algebraic objects such as abelian groups or modules to other mathematical objects such as topological spaces. You can use that to show the two maps in those sequences are isos, but i realized its a bit easier to just pull the sequence out a bit further. From a formal point of view, the mayer vietoris sequence can be derived from the eilenbergsteenrod axioms for homology theories using the long exact sequence in homology. The mayervietoris sequence is a powerful tool, which will let us understand a number of facts about hqm.
Differentiable manifoldsthe mayervietoris sequences in. Asking for help, clarification, or responding to other answers. We also prove some graphtheoretical analogues of standard results in differential geometry, in particular, a graph version of stokes theorem and the mayer vietoris. Statement suppose that a manifold xcan be written as the union of two open submanifolds, uand v. A gentle introduction to homology, cohomology, and sheaf. Homology groups were originally defined in algebraic topology. We also prove some graphtheoretical analogues of standard results in di erential geometry, in particular, a graph version of stokes theorem and the mayer vietoris sequence in cohomology. The corresponding long exact sequence in cohomology as discussed above is what is traditionally called the mayer vietoris sequence of the cover of x x by u u and v v in a a cohomology. Excision property and mayer vietoris sequence conversely, let us assume that we have an element c0 n 1 such that i 1c 0 n 1 0 f 0 n 1 c 0 n 1. We also prove some graphtheoretical analogues of standard results in differential geometry, in particular, a graph version of stokes theorem and the mayervietoris. A similar proof is used in chapter 10, where i proved poincar. We already have a mayervietoris sequence for cohomology based on differen tiable singular cubes.
This is a nontrivial fact that can be shown for example by combining the computation of h0x. In the future please link to abstract pages rather than pdf files, e. The uniqueness of the cohomology of cw complexes 149 chapter 20. The mayervietoris sequence for hde rham cohomology.
Two homotopic maps from x to y induce the same homomorphism on cohomology just as on homology. The mayer vietoris sequence is a powerful tool, which will let us understand a number of facts about hqm. Two homotopic maps from x to y induce the same homomorphism on cohomology just as on homology the mayer vietoris sequence is an important computational tool in cohomology, as in homology. We defined it as follows i translated from french, so sorry if i use some wrong terminology. The mayervietoris property in differential cohomology. Another spectral sequence arises when u fu ig i2i is an open covering of x.
We then develop the mayer vietoris sequence, perform a few computations, including a. In this form, we obtain a tool for computing the cohomology of a manifold covered by sets with known cohomology. The result as stated in 1931 is very di erent from the. Note that the boundary homomorphism increases rather than. They are easily shown to be diffeomorphism invariants, but surprisingly they turn out also to be topological invariants. Note that this also makes sense if u and v are disjoint, if we. This makes cohomology into a contravariant functor from topological spaces to abelian groups or rmodules. It requires no prior knowledge of the concepts of algebraic topology or cohomology.