Talk:Simplicial homology

Early revisions

Added bulk to the stub. The aesthetics might need tampering with (the size of equations seems random)

I agree -- I did some edits, hopefully it looks better now. The dsplayed formulas (using math mode, and tex code) look more-or-less OK to me. But using math mode for inline formulas has rather awful effects. Best would be to void that altogether, whenever possible, I think. Turgidson 16:27, 6 January 2007 (UTC)

There were some errors in this article. The definition of cycles and Z_k made no sense. This can only be defined after defining the boundary operator. (see second reference, p. 106) Also, the sign of the example was not consistent with the defintion. I've changed the example, but one could also change the sign in the definition. I'm not sure which sign is more common, but it does not really matter.134.96.51.208 13:20, 2 November 2007 (UTC)

numerical implementation section

IMO this section isn't directly relevant to simplicial homology. It is not actually discussing implementations of simplicial homology itself as that's fairly straightforeward and has been done well-before the 1999 reference given. This section is about computing homology of objects that you have partial information on, where one makes estimations on the topology. In that regard it probably deserves its own article with links to various homology and topology articles. I massaged the language to make it more on topic. Rybu (talk) 21:03, 27 May 2009 (UTC)

"a simplex is a tuple"

really? are there really n! simplexes with vertexes x_1, .., x_n? or maybe just 2, as I think!! http://planetmath.org/node/32828 tuples different by an even permutation should be identified!--2.236.204.53 (talk) 17:59, 6 October 2013 (UTC)

Bad calculation of homology group

Ker∂≅ℤ⊕ℤ⊕ℤ, Im∂ ≅ ℤ⊕ℤ is not sufficient to show Ker∂/Im∂ ≅ ℤ - consider ℤ/2ℤ, 2ℤ ≅ ℤ. An isomorphism must be constructed carrying the boundary image to ℤ⊕ℤ⊕0 ≤ ℤ⊕ℤ⊕ℤ. Finishing the calculation properly requires some integer gaussian elimination, which I'm not sure Wikipedia even covers clearly right now. ᛭ LokiClock (talk) 01:55, 8 September 2014 (UTC)

Or perhaps after changing the article to use a specific, constructed isomorphism to ℤ⊕ℤ⊕ℤ, the phrase "x is an integer linear combination of y, z and can therefore be eliminated" can be used. ᛭ LokiClock (talk) 02:00, 8 September 2014 (UTC)

Definitions should be made more precise.

In order to define some notion of homology, simplices must be oriented in some way. One way how to do it is to choose a global ordering of all vertices and then only consider ordered tuples of vertices with increasing order. Or we may choose an orientation of each simplex independently and identify $[v_{0},...,v_{n}]$ with $sgn(\pi )[v_{\pi (0)},...,v_{\pi (n)}]$, such as in Prasolov. Another notion of simplicial homology is in Hatcher, where it is based on the more general Delta-complexes, but even there orientation is crucial.

Also, I miss in this article some kind of notion of induced maps. A simplicial map $f:K\to L$ induces a map on the simplicial homology but one has to be careful to define the induced chain map properly: each simplex $[v_{0},...,v_{n}]$ that is mapped to a "degenerate simplex" -- that is, $f(v_{i})=f(v_{j})$ for some $i\neq j$ -- should be mapped to zero on the chain level.

It seems to me that the current version of this important article is very poor in general :( Franp9am (talk) 20:27, 15 May 2015 (UTC)