Motivation for local cohomology and local homotopy theories in Algebraic topology.

Question

In general topology, I know about the local topological properties. In algebraic topology homotopy and cohomology theories is also easily understandable. For examples Betti numbers gives information about the numbers of holes in topological spaces. Homotopy groups gives information about simply connectivity and conductibility of spaces. My question is that way we need local theories such as local cohomology and local homotopy. What is the geometric interpretation of these local theories in algebraic topology. Moreover what is difference between local and global theories in algebraic topology?

Answer 1

The local homology groups are defined as $H_k(U,U-x)$ where $U$ is a small neighborhood of $x$ in $X$. This is easily seen to be the same as the reduced homology group of the sphere when $X$ is a manifold. However let us look at real algebraic varieties.

For a space $Y$, we denote by $\overset{\circ}{c}(Y)$ the open cone over $Y$.

Theorem : Let $X$ be a real algebraic variety and $0 \in X$. There is a triangulated space $L_x$, a neighbourhood $U$ of $0$ in $X$, and an homeomorphism $f :U \cong \overset{\circ}{c}(L_x)$, sending $0$ on the vertex of the cone.

Corollary : $H_k(U,U-x)$ is simply $\widetilde{H}_k(L_x)$, the reduced homology groups of $L_x$.

Interesting example are given by the complex varieties $xy = 0, x^3 = y^2, xyz = 0$.

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For the second part of the question, I would say that there is no real difference since cohomology gives global invariants using local data. In a sense, everything is local. More precisely, all reasonable cohomology theory are computed by sheaf theory.

Answer 2

I want to complement Nicolas' answer. The local homology groups of a space can be useful even if we a-priori know that all the local homology groups are isomorphic. For example on a manifold we know that, $H^{n}(U,U\setminus \{x\})\cong H^{n-1}(S^{n-1})\cong\mathbb{Z}$. However, there is a choice made in this isomorphism, namely the choice of the generator of the homology of the sphere.

One can build a space by gluing copies of $H^{n-1}(U,U\setminus \{x\})$ over your manifold. A section of this space exists if and only if the space is orientable. A choice of section which is a generator at every point is a choice of orientation.

This allows one to speak of orientations of topological manifolds, which do not have a tangent bundle, because there is no smooth structure.