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I am taking the following as the definition of a $\Delta$-complex.

(i) one starts with an indexing set $I_n$ for each $n \in \mathbb{Z}_{\ge 0}$.

(ii) for each $\alpha \in I_n$, one takes a copy $\sigma_\alpha^n$ of the standard $n$-simplex.

(iii) one forms the disjoint union of all of these simplices, for all $n\geq 0$.

(iv) now require that for each $(n-1)$-dimensional face of $\sigma_\alpha^n$, there is an associated $\sigma_\beta^{n-1}$ for some $\beta \in I_{n-1}$.

(v) now form the quotient space by identifying each $(n-1)$-dimensional face of each $\sigma_\alpha^n$ with $\sigma_\beta^{n-1}$ using the canonical homeomorphism. In particular, these homeomorphisms preserve the ordering of vertices.

The sources I've consulted don't make it clear what a subcomplex is. Can someone give me a rigorous definition?

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up vote 1 down vote accepted

With that (slightly strange way of phrasing the) definition, a subcomplex should be a sequence $(J_n)_{n\geq0}$ such that

  • for each $n\geq0$ we have $J_n\subseteq I_n$, and

  • if $\alpha\in J_n$ and $\beta\in I_{n-1}$ are such that $\sigma_b^{n-1}$ is associated to one of the faces of $\sigma_\alpha^n$, then $\beta\in J_{n-1}$.

N.B.: you did not tell us where you got that definition from, nor what other sources you consulted. There are lots of nice expositions of this subject... With time, I've learned to appreciate C. R. F. Maunder's Algebraic Topology, for example.

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The definition is from lectures I took on algebraic topology that don't have typeset notes, so I cannot link you to an electronic version. However, I did reproduce above the lecturer's definition word-for-word. It appears not to be too different from the one in Hatcher's Algebraic Topology. What definition would you normally use for $\Delta$-complexes? Thank you for the recommendation; I'll try to find it in my library. – Sputnik May 18 '11 at 2:27
Incidentally, I've since learned that Eilenberg and Zilber first invented the notion of $\Delta$-complexes under the name "semi-simplicial complexes" in this paper. – Sputnik May 18 '11 at 15:24

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