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I am reading The Geometric Realization of a Semi-Simplicial Complex by J. Milnor and here are the definitions:enter image description here

I find it difficult to visualize without specific examples. Can anyone help to provide some typical examples of geometric realization of semi-simplicial complexes?

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  • $\begingroup$ Maybe this is helpful: arxiv.org/pdf/0809.4221 $\endgroup$ Nov 6, 2014 at 11:58
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    $\begingroup$ @sanjab, exactly what I am looking for! Thank you!! $\endgroup$
    – Zuriel
    Nov 6, 2014 at 12:02
  • $\begingroup$ Yes @DanielRust, the notations seem somehow different but still understandable. Historically it is possible that the "normal" notations are later than Milnor's, if this is true, people swapped his notation for some reason and it may not be Milnor's fault at all. For $s_i$, notice that in his definition of $\Delta_n$, $t_0$ is always $0$ and thus there is no need to delete it. $\endgroup$
    – Zuriel
    Nov 6, 2014 at 12:44

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Let $K_0=\{a,b\}$ and let $K_1=\{X,Y\}$ with $\partial_0(X)=\partial_1(Y)=a$ and $\partial_1(X)=\partial_0(Y)=b$. This is a $\Delta$-complex.

The space $\bar{K}$ is a disjoint union of two discrete points given by $a$ and $b$, and two disjoint intervals given by $X$ and $Y$. The equivalence relation just says that we join the beginning of the $X$ interval and the end of the $Y$ interval to the point $a$, and the end of the $X$ interval and the beginning of the $Y$ interval to the point $b$. Draw a picture of this and you will see that the geometric realisation $|K|=\bar{K}/{\sim}$ is homeomorphic to a circle.

If we instead want to represent the circle by a simplicial complex $K$, then we need at least three $1$-simplices. Let $K_0=\{a,b,c\}$ and let $K_1 = \{\{a,b\},\{b,c\},\{c,a\}\}$ with $$\partial_0(\{a,b\})=\partial_1(\{c,a\})=a\\ \partial_0(\{b,c\}) = \partial_1(\{a,b\})=b\\ \partial_0(\{c,a\}) = \partial_1(\{b,c\})=c$$ and this time we have three edges which are glued together in the geometric realisation according to the ordering of their boundary faces given by the image of the $\partial_i$.

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  • $\begingroup$ Thank you for the example! Just to clarify one thing: In the definition of semi-simplicial complex, apart form the face map, there should also be degeneracy maps, right? How do you define those maps from $K_0$ to $K_1$ in your example? $\endgroup$
    – Zuriel
    Nov 6, 2014 at 12:16
  • $\begingroup$ Apparently I described a $\Delta$-set (Delta set) instead of a semi-simplicial complex. It's hard to know what definition is being used in the extract, especially as some people now equate semi-simplicial complexes with delta sets in terminology, but I don't think this has always been the case. $\endgroup$
    – Dan Rust
    Nov 6, 2014 at 12:42
  • $\begingroup$ For instance in this paper of Eilenberg and Zilber, they essentially give the modern $\Delta$-set definition for what they call semi-simplicial complexes. $\endgroup$
    – Dan Rust
    Nov 6, 2014 at 12:44
  • $\begingroup$ I think likely Milnor's semi-simplicial complex is equivalent to our simplicial set today, since both face maps and degeneracy maps are present. $\endgroup$
    – Zuriel
    Nov 6, 2014 at 12:49
  • $\begingroup$ In that case my example doesn't fit this definition, as you need at least three $1$-dimensional simplicies to define a simplicial complex which is homeomorphic to a circle. I'll correct my example later (although I'm sure you could now do it yourself!) but I'm afraid I have a class to teach now. Good luck! $\endgroup$
    – Dan Rust
    Nov 6, 2014 at 12:51

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