Notes on chain complexes

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Topological Methods in Nonlinear Analysis

Do you think my answer is correct? I'm just wondering, because my answer is really short Thanks in advance. Jendrik Stelzner 8, 3 3 gold badges 14 14 silver badges 41 41 bronze badges. Dan Rust Dan Rust Actually, the arrows in my notes are left arrows, but I accidentally wrote them as right arrows here Also, thank you for pointing out the second part, because I actually missed it.

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So if we think about the right arrows as left arrows Is that right? Was there more to the question? There was a question before this that was related to the monomorphism condition. So maybe that's why it says that it is a monomorphism. Sign up or log in Sign up using Google. Sign up using Facebook. Sign up using Email and Password.

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Post as a guest Name. The n -th co homology group H n H n is the group of co cycles modulo co boundaries in degree n , that is,.

An exact sequence or exact complex is a chain complex whose homology groups are all zero. This means all closed elements in the complex are exact. For example, the following chain complex is a short exact sequence. In the middle group, the closed elements are the elements p Z ; these are clearly the exact elements in this group. This is written out in the following commutative diagram. When X and Y are both equal to the n -sphere , the map induced on homology defines the degree of the map f.

Homotopy category of chain complexes - Wikipedia

The concept of chain map reduces to the one of boundary through the construction of the cone of a chain map. A chain homotopy offers a way to relate two chain maps that induce the same map on homology groups, even though the maps may be different. The maps may be written out in a diagram as follows, but this diagram is not commutative. It immediately follows that f and g induce the same map on homology.

  1. Figure 3 from Polytopal complexes: maps, chain complexes and necklaces - Semantic Scholar.
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One says f and g are chain homotopic or simply homotopic , and this property defines an equivalence relation between chain maps. Let X and Y be topological spaces. This shows that two homotopic maps induce the same map on singular homology.

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The name "chain homotopy" is motivated by this example. Let X be a topological space. That is, the boundary of a singular simplex is the alternating sum of restrictions to its faces. Singular homology is a useful invariant of topological spaces up to homotopy equivalence. The degree zero homology group is a free abelian group on the connected components of X.

The cohomology of this complex is called the de Rham cohomology of X. The homology group in dimension zero is isomorphic to the vector space of locally constant functions from M to R. Thus for a compact manifold, this is the real vector space whose dimension is the number of connected components of M. Smooth maps between manifolds induce chain maps, and smooth homotopies between maps induce chain homotopies. Chain complexes of K -modules with chain maps form a category Ch K , where K is a commutative ring. This tensor product makes the category Ch K into a symmetric monoidal category. The identity object with respect to this monoidal product is the base ring K viewed as a chain complex in degree 0. The braiding is given on simple tensors of homogeneous elements by. We have a natural isomorphism.

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  • From Wikipedia, the free encyclopedia. For the type of number, see Bicomplex number. Tool in homological algebra. It has been suggested that Homotopy category of chain complexes be merged into this article.