#### Glasnik Matematicki, Vol. 42, No.1 (2007), 195-211.

### THE *S*_{n}-EQUIVALENCE OF COMPACTA

### Nikica Uglešić and Branko Červar

University of Zadar, Studentski dom, F. Tuđmana 24 D, 23000 Zadar,
Croatia

*e-mail:* `nuglesic@unizd.hr`
Department of Mathematics, University of Split, Teslina ulica 12/III,
21000 Split, Croatia

*e-mail:* `brankoch@pmfst.hr`

**Abstract.** By reducing the Mardešić *S*-equivalence to a
finite case, i.e., to each
*n*
{0} **N**
separately, we have derived the notions of
*S*_{n}-equivalence
and *S*_{n+1}-domination of compacta.
The *S*_{n}-equivalence for all *n*
coincides with the *S*-equivalence. Further, the
*S*_{n+1}-equivalence implies
*S*_{n+1}-domination, and the
*S*_{n+1}-domination implies
*S*_{n}-equivalence. The
*S*_{0}-equivalence is a trivial equivalence relation, i.e., all non
empty compacta are mutually *S*_{0}-equivalent. It is proved that
the *S*_{1}-equivalence is strictly finer than the
*S*_{0}-equivalence, and that the
*S*_{2}-equivalence is strictly
finer than the *S*_{1}-equivalence.
Thus, the *S*-equivalence is
strictly finer than the *S*_{1}-equivalence. Further, the
*S*_{1}-equivalence classifies compacta which are
homotopy (shape) equivalent to ANR's up to the homotopy (shape) types. The
*S*_{2}-equivalence class of an FANR coincides with its
*S*-equivalence class as well as with its shape type class.
Finally, it is conjectured that, for every *n*, there exists
*n'* > *n* such that the
*S*_{n'}-equivalence is strictly
finer than the *S*_{n}-equivalence.

**2000 Mathematics Subject Classification.**
55P55.

**Key words and phrases.** Compactum, ANR, shape, *S*-equivalence.

**Full text (PDF)** (free access)
DOI: 10.3336/gm.42.1.14

**References:**

- K. Borsuk,
*Some quantitative properties of shapes*,
Fund. Math. **93** (1976), 197-212.

MathSciNet

- D. A. Edwards and R. Geoghegan,
*Shapes of complexes,
ends of manifolds, homotopy limits and the Wall obstruction*,
Ann. of Math. (2) **101** (1975), 521-535, Correction. Ibid.
**104** (1976), 389.

MathSciNet
CrossRef

- K. R. Goodearl and T. B. Rushing,
*Direct limit groups
and the Keesling-Mardesic shape fibration*,
Pacific J. Math. **86** (1980), 471-476.

MathSciNet

- H. M. Hastings and A. Heller,
*Splitting homotopy
idempotents*, in: Shape theory and geometric topology (Dubrovnik,
1981), Lecture Notes in Math. **870**, Springer, Berlin-New
York, 1981, 23-36.

MathSciNet

- J. Keesling and S. Mardesic,
*A shape fibration with fibers of different shape*, Pacific J. Math.
**84** (1979), 319-331.

MathSciNet

- S. Mardesic,
*Comparing fibres in a shape
fibration*, Glasnik Mat. Ser. III **13(33)** (1978),
317-333.

MathSciNet

- S. Mardesic and J. Segal, Shape Theory, North-Holland
Publishing Co., Amsterdam-New York-Oxford 1982.

MathSciNet

- S. Mardesic and N. Uglesic,
*A category whose isomorphisms induce an equivalence relation coarser
than shape*, Topology Appl. **153** (2005), 448-463.

MathSciNet
CrossRef

- K. Morita,
*The Hurewicz and the Whitehead theorems in
shape theory*, Sci. Rep. Tokyo Kyoiku Daigaku Sect. A **12**
(1974), 246-258.

MathSciNet

- N. Uglesic,
*A note on the Borsuk quasi-equivalence*, submitted.

- N. Uglesic and B. Cervar,
*A subshape spectrum for compacta*,
Glasnik Mat. Ser. III **40(60)** (2005), 351-388.

MathSciNet

- C. T. C. Wall,
*Finiteness conditions for CW-complexes*, Ann.
of Math. (2) **81** (1965), 59-69.

MathSciNet
CrossRef

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