Barycentric subdivision

of a geometric complex $K$

A complex $K_1$ obtained by replacing the simplices of $K$ by smaller ones by means of the following procedure. Each one-dimensional simplex (segment) is halved. On the assumption that all simplices of dimension $\leq n-1$ are already subdivided, the subdivision of any $n$-dimensional simplex $\sigma$ is defined by means of cones over the simplices of the boundary of $\sigma$ with a common vertex that is the barycentre of the simplex $\sigma$, i.e. the point with barycentric coordinates $1/(n+1)$. The vertices of the resulting complex $K_1$ are in a one-to-one correspondence with the simplices of the complex $K$, while the simplices of the complex $K_1$ are in such a correspondence with inclusion-ordered finite tuples of simplices from $K$. The formal definition of a barycentric subdivision for the case of an abstract complex is analogous.

References

[a1]	K. Lamotke, "Semisimpliziale algebraische Topologie" , Springer (1968)