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I have the following definitions:

Definition 1: A simplicial complex $K$ is a family of finite nonempty subsets of a set $V_k$ (the elements of $V_k$ are called vertices) such that:

1) if $vin V_k$, then ${v}in K$.

2) if $Sin K$ and $emptyset neq Tsubseteq S$, then $Tin K$.

The elements of $K$ are called simplexes.

Definition 2: The geometric realization of a simplicial complex $K$ is defined as $|K|= {alpha: V_K to [0,1] ; | ; alpha^{-1}((0,1]) in K , ;sum_{vin V_k} alpha(v)=1 }$.

Definition 3: The barycentric subdivision of a simplicial complex $K$ is defined as the simplicial complex $sd(K)$ such that $V_{sd(K)}=K$ and a simplex in $sd(K)$ has the form ${s_0, s_1, ...,s_q}$, where $s_i in K $ and $s_{i-1} subset s_i $, ($s_{i-1}$ is a proper subset of $s_i$ ), for all $i$.

I'm asked to show that $|sd(K)|$ is homeomorphic to $|K|$.

I already know that for $S={v_0,v_1,...,v_q} in K$, $|S|={alpha in |K| ; | ; alpha^{-1} ((0,1]) subseteq S}$ is homeomorphic to $Delta^q = {(t_0,t_1,...,t_q)in mathbb{R}^{n+1} ; | ; t_i geq 0, sum_{i=0}^{n} t_i =1}$ (actually, the homeomorphism is an isometry).

Here, the topology we are using is the one induced by the metric $d:|K|$x$|K|to mathbb{R}$ such that $d(alpha, beta) = sqrt{sum_{vin V_K} (alpha (v) - beta (v))^2}$.

Now, I'm trying to construct an homeomorphism from $|K|$ to $|sd(K)|$, but everything I can think of, just doesn't work. I would be very thankful if you could give me a hint on how to construct this homeomorphism, or if you could tell me another way to proceed on this. Thank you so much!

Let us think of a function $alpha:V_Kto [0,1]$ as a formal sum $sum_{vin V_K}alpha(v)cdot v$. The homeomorphism $|sd(K)|to |K|$ is then given by mapping $Sin V_{sd(K)}=K$ to the formal sum $sum_{vin S}frac{1}{|S|}cdot v$, and "extending linearly". In other words, a point $alphain|sd(K)|$ can be written as a formal sum $sum_{Sin K} alpha(S)cdot S$ and then we map this formal sum to $|K|$ by replacing each $S$ with $sum_{vin S}frac{1}{|S|}cdot v$ to get the formal sum $sum_{Sin K}sum_{vin S}frac{1}{|S|}cdot v$ which we can think of as a formal sum of elements of $V_K$. I'll leave it to you to show that this is a homeomorphism. (As a hint for showing it is a bijection, given a point of $|K|$, you can figure out what simplex of $|sd(K)|$ it is in the image of by ordering its nonzero coefficients from smallest to biggest. For instance, a sum like $0.2a+0.3b+0.5c$ for three vertices $a,b,c$ can be split up as $0.6(frac{1}{3}a+frac{1}{3}b+frac{1}{3}c)+0.2(frac{1}{2}b+frac{1}{2}c)+0.2c$, to write it as a convex combination of expressions of the form $sum_{vin S}frac{1}{|S|}cdot v$.)

The intuition here is that for each geometric simplex $|S|$ in $|K|$, we add a new vertex at the barycenter of $S$, the point $sum_{vin S}frac{1}{|S|}cdot v$ that is the average of all its vertices. We can then take convex combinations of the barycenters of $|S|$ and its faces to subdivide $|S|$ into many smaller simplices. For a very simple example, suppose $K$ is just a line segment, with two vertices and an edge connecting them. Then $sd(K)$ has three vertices: two are just the vertices that $K$ had, but one of them comes from the edge. We think of this new vertex as being the midpoint of the edge of $K$, which we can connect to the two vertices of $K$ to split the edge into two smaller edges. So, $|sd(K)|$ is just two line segments joined together, which is homeomorphic to a single line segment $|K|$.