Singular homology

To start with, for any n 0, we define a singular n-simplex to be any continuous map from the standard n-simplex to om space, r /I — X, and 

Furthermore, let TZ be an arbitrary commutative ring with unit, and set 

At first it does not seem such a good idea to take this extremely large set of singular n-simplices, and then, on top of that, to span a free 7 .-module, taking the elements of this set as a basis. It turns out, however, that most of this module will cancel out, once we pass to homology. 

Similar to what we have done before, we can define the singular boundary operator Sing (X 7 .) — Sing i(A .) only this time the boundary is taken on the standard simplex first, before mapping it to A  

Some properties of singular homology are immediate among these we find the following  

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We state here without proof the central fact pertaining to singular homology, which will allow us to use it instead of simplicial homology, whenever it appears opportune. 

Theorem 3.26. For an arbitrary trisp A, its simplicial and singular homology groups are isomorphic. 

For a locally compact topological space X, let Ftl X) denote the homology group of possibly infinite singular chains with locally finite support (Borel-Moore homology). The... 

Singular characteristics of the ruthenium catalysts are the capability of direct activation of different substrates (alcohols, ethers, formic,orthoformic and other carboxylic acid esters) at a low iodine concentration, and a high flexibility toward carbonylation and/or homologation processes for the substrates used. The catalytic activity of the ruthenium catalysts moreover do not strongly decrease, as occurs with Co or Rh systems, by passing from methyl to higher oxygenated alkyl derivatives. 

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