#### Glasnik Matematicki, Vol. 45, No.1 (2010), 219-290.

### SOME CELLULAR SUBDIVISIONS OF SIMPLICIAL COMPLEXES

### Sibe Mardešić

Department of Mathematics, University of Zagreb, Bijenička cesta 30, 10002 Zagreb, P.O. Box 335, Croatia

*e-mail:* `smardes@math.hr`

**Abstract.** In a previous paper the author has associated with
every inverse system of compact CW-complexes **X** with limit *X* and every
simplicial complex *K* with geometric realization |*K*| a resolution of
*X* × |*K*|, which
consists of spaces having the homotopy type of polyhedra. In a subsequent paper it
is shown that this construction is functorial. The proof depends essentially on
particular cellular subdivisions of *K*. The purpose of this paper is to
describe in detail these subdivisions and establish their relevant
properties. In particular, one defines two subdivisions
*L*(*K*) and *N*(*K*) of *K*.
Each cell from *L*(*K*), respectively from *N*(*K*), is contained
in a simplex σ *K* and it is
the direct sum *a* *b*,
respectively *c* *d*,
of certain simplices contained in σ.
One defines new subdivisions *L*'(*K*) and *N*'(*K*) of *K*
by taking for their cells the direct sums
*L*(*a*) *b*,
respectively *c* *N*(*d*).
The main result asserts that there is an
isomorphism of cellular complexes
: *L*'(*K*) → *N*'(*K*),
which induces a
selfhomeomorphism
θ : |*K*| → |*K*|.

**2000 Mathematics Subject Classification.**
55U10, 52B11, 54C56.

**Key words and phrases.** Convex polytope, simplicial complex,
cellular complex, subdivision of a complex, isomorphism of cellular complexes, resolution of a space, shape.

**Full text (PDF)** (free access)
DOI: 10.3336/gm.45.1.14

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