Topology group

– dr.sc. Zdravko Čuka, FGAG, Split
– Ivančica Mirošević, FESB, Split

Research areas:

• General topology
• Algebraic topology
• Shape and coarse shape theory
• Continuum theory
• Generalized inverse sequences

Research description:

Research area is the general and algebraic topology, particularly a research where techniques of approximating spaces by limits of ANR and polyhedral inverse systems are applied.
Another research area is the shape theory, particularly the coarse shape theory, which may be considered as its full generalization. Recently, some topological and algebraic coarse shape invariants has been studied which enable applications to topology and homotopy theory, as well. Among others, the most interesting invariants are the coarse shape groups and coarse shape path connectedness.
Concerning continuum theory the category CU in which u.s.c. functions are morphisms and compact metric spaces are objects is introduced. Inverse sequences in CU are also considered and it is shown that they form category, denoted with ICU. Results are applied to prove that inverse limits of inverse sequences with inverse limits as terms are homeomorphic. Further, the definition of topological entropy due to Adler, Konheim, and McAndrew is generalized to set-valued functions from a closed subset of the interval to closed subsets of the interval. These set-valued functions are viewed, via their graphs, as closed subsets of [0,1]2. Motivation for observing graphs and their Mahavier products (notion introduced by Greenwood and Kennedy) is that when set-valued functions are iterated in the usual sense, information is lost – and often lost very fast. Mahavier products are a convenient way to study subsets of a generalized inverse limit space. They also make it easier to study “finite” generalized inverse limits – which are not interesting at all in inverse limits, but are interesting in their own right in generalized inverse limits.

Contacts with academic and other institutions:

– University of Maribor, Slovenia
– Waseda University, Tokyo, Japan
– University of Skopje, Institute of Mathematics, Macedonia
– Warsaw University of Technology, Varšava, Poland
– University of Richmond, Richmond VA, USA
– Lamar University, Beaumont TX, USA
– University of Perugia, Department of Mathematics and computer science, Italy

Description of the current cooperation:

During 2007. and 2011. V. Matijević has spent 10 days on Warsaw University of Technology, Varšava, Poljska, as a invited speaker of topology seminar
– During 2009.V. Matijević has spent 5 days on University of Maribor, Slovenia, as a invited speaker of topology seminar
– During 2010. N. Koceić has spent 5 days on University of Skopje, Institute of Mathematics, Macedonia, as a invited speaker of topology seminar
– During 2010. and 2013. G. Erceg has spent 5 days on University of Maribor, Slovenia
in scientific collaboration with colleagues from topology seminar in Maribor
– During 2012. V. Matijević has spent 5 days on University of Skopje, Institute of Mathematics, Macedonia, as a invited speaker of topology seminar
– During 2013. N. Koceić and V. Matijević have spent 15 days on Waseda University, Tokyo, Japan, as invited speakers of two topology conferences
– During 2014. N. Koceić, V. Matijević and G. Erceg have spent 7 days on University of Richmond, Richmond VA, USA, as invited speakers of topology conference. G. Erceg, also spend an extra time working on its Ph. D. thesis with his mentor
– During 2014. N. Koceić has spent 3 days on University of Maribor, Slovenia
– as a invited speaker of topology seminar
– During 2014. G. Erceg has spent 15 days on Lamar University, Beaumont TX, SAD in scientific collaboration with his mentor
– During 2016. N. Koceić has spent. 7 days on University of Skopje, Institute of Mathematic , Macedonia, as a invited speaker of topology conference
– During 2017. N. Koceić and V. Matijević have spent 7 days on University of Perugia, Department of mathematics and computer science, Italija, within Teaching Staff Mobility (Erasmus).

Plans for future research:

Lately, covering and overlay spaces over (locally) compact topological groups are studied. Overlay spaces over some different classes of connected topological groups will be explored.
In the research area of coarse shape theory introducing of topology on the coarse shape group which would turn them into topological groups will be explored. Studying of some other approaches (intrinsic at the first place) to the coarse shape is included in the plan for the future. Further, possible applications of the coarse shape theory to dynamical systems will be explored.
Concerning researching in area of continuum theory, the plan is to generalize new notion of topological entropy on a wider class of spaces (compact metric spaces), then eventually to find conditions to get trivial topological entropy, further calculating entropy for new interesting examples and potentially applying obtained results in theory of generalized inverse limits and more generalized in continuum theory and for topological dynamical systems. Also, the plan for the future involve studying generalized inverse limits defined by Ingram and Mahavier in some new category as an inverse limits in categorical sense. In results done so far, they are not.

List of selected research papers:

 K. Eda, J. Mandić and V. Matijević, Torus-like continua which are not self-covering spaces, Topology Appl. 153 (2005), 359-369.
 K. Eda and V. Matijević, Finite-sheeted covering maps over 2-dimensional connected, compact abelian groups, Topology Appl. 153 (2006), 1033–1045.
 V. Matijević, Finite-sheeted covering maps over Klein bottle weak solenoidal spaces, Glasnik Mat. 42 (62) (2007), 19-41.
 K. Eda and V. Matijević, Finite-index supergroups and subgroups of torsionfree abelian groups of rank two, Journal Alg.319 (2008), 3567-3587.
 L.J. Hernandez Paricio and V. Matijević, Fundamental groups and finite sheeted coverings, Journal Pure Appl. Alg. 214 (2009), 281-296.
 N. Uglešić and V. Matijević, On expansions and pro-pro-categories, Glasnik Mat.45 (2010), 173-217.
 V. Matijević, A note on finite-sheeted covering maps from 2-dimensional compact abelian groups, Topology Appl. 153 (2010) 2746-2756.
 K. Eda and V. Matijević, Covering maps over solenoids which are not covering homomorphisms, Funadmenta Math. 221 (2013).
 K. Eda and V. Matijević, Existence and uniqueness of topological group structures on covering spaces over groups, Fundamenta Math. (to appear) submitted.
 N. Koceić Bilan, N. Uglešić, The coarse shape, Glasnik Matematički 42 (62) (2007), 145-187.
 A. Kadlof, N. Koceić Bilan, N. Uglešić, Borsuk.s quasi equivalence is not transitive, Fundamenta Mathematicae 197 (2007), 215-227.
 N. Koceić Bilan, The coarse shape groups, Topology and its Application 157 (2010), 894-901.
 N. Koceić Bilan, On some coarse shape invariants, Topology and its Appl. 157 (2010) 2679-2685.
 N. Koceić Bilan, N. Uglešić, The coarse shape path connectedness, Glasnik Mat. 46 (66) (2011), 489-503
 N. Koceić Bilan, Towards the algebraic characterization of (coarse) shape path connectedness, Topology and its Applications 160 (2013) 538-545
 N. Koceić Bilan, Computing coarse shape groups of solenoids, Mathematical communications 14 (2014), 243-251
 N. Koceić Bilan, I. Jelić, On intersections of the exponential and logarithmic curves, Annales Mathematicae et Informaticae 43 (2014) 159-170
 N. Koceić Bilan, Continuity of coarse shape groups, Homology, Homotopy and Applications, vol.18(2), (2016), 209-215
 I. Banič, M. Črepnjak, G. Erceg, M. Merhar and U. Milutinović, Inducing functions between inverse limits with upper semicontinuous bonding functions, Houston J. Math. Vol. 41, No. 3 (2015), 1021-1037