We construct uncountably many simply connected open topological 3-manifolds with genus one ends homeomorphic to the Cantor set. Each constructed 3-manifold has the property that any self homeomorphism of the 3-manifold (which necessarily extends to a homeomorphism of the ends) fixes the ends pointwise. These 3-manifolds are complements of rigid generalized BingWhitehead (BW) Cantor sets. Previous examples of rigid Cantor sets with simply connected complement in R^3 had infinite genus and it was an open question as to whether finite genus examples existed. The examples here exhibit the minimum possible genus, genus one. These rigid generalized BW Cantor sets are constructed using variable numbers of Bing and Whitehead links. Our previous result determining when BW Cantor sets are equivalently embedded in R^3 extends to the generalized construction. This characterization is used to prove rigidity and to distinguish the uncountably many examples. These results were obtained in collaboration with topological group at the Oregon State University, with which we have successfully completed several joint research projects. Results were presented at invited talks presented at several international conferences and some leading universities in the European Union, the United States, the Russian Federation and Japan. Our program group has by now established itself with its successful research of wild embeddings into Euclidean spaces and is considered to be among top groups in this area, which is witnessed also by numerous invitations to conferences and workshops around the world.
COBISS.SI-ID: 16861529
We investigate the concordance properties of "parallel links" P(K), given by the (2,0) cable of a knot K. We focus on the question: if P(K) is concordant to a split link, is K necessarily slice? We show that if P(K) is smoothly concordant to a split link, then many smooth concordance invariants of K must vanish, including the \tau and s-invariants, as well as suitably normalized d-invariants of Dehn surgeries on K. We also investigate the (2,2\ell) cables P_{\ell}(K), and find obstructions to smooth concordance to the sum of the (2,2\ell) torus link and a split link. Paper was published in one of the best mathematical journals and it has received a very positive response from low-dimensional topology experts around the world since it solves some outstanding problems. Strle presented these results in the United States, Canada, Germany and Great Britain. These results were done in collaboration with Scottish topological group at the University of Glasgow (UK), with which we have successfully completed several joint research projects, in cooperation with University of Koeln (Germany), Brandeis University (USA) and McMaster University (Canada).
COBISS.SI-ID: 16946265
We consider open infinite gropes and prove that every continuous map from the minimal grope to another grope is nulhomotopic unless the other grope has a "branch" which is a copy of the minimal grope. Since every grope is the classifying space of its fundamental group, the problem is translated to group theory and a suitable block cancellation of words is used to obtain the result. Cencelj presented these results at international meetings in the United States and Poland. These results were done in collaboration with topological groups at the University of Tennessee (Knoxville) and the University of Florida (Gainesville), with which we have successfully completed several joint research projects.
COBISS.SI-ID: 16655449
Let X be a connected CW complex and let K(G,n) be an Eilenberg-Mac Lane CW complex where G is abelian. As K(G,n) may be taken to be an abelian monoid, the weak homotopy type of the space of continuous functions X \to K(G,n) depends only upon the homology groups of X. The purpose of this paper is to prove that this is true for the actual homotopy type. Precisely, the space \mathrm{map}_\ast \big(X, K(G,n)\big) of pointed continuous maps X \to K(G,n) is shown to be homotopy equivalent to the Cartesian product \prod_{i \leq n} \mathrm{map}_\ast \big(M_i, K(G,n)\big). Here, M_i is a Moore complex of type M\big(H_i(X), i\big). The spaces of functions are equipped with the compact open topology. Smrekar presented these results in Spain and France. Based on his work he was invited to collaborate with distinguished Japanese algebraic topologists. Smrekar also received an award from the European Union for his outstanding research in homotopy theory, namely, the one year Marie Curie Research Fellowship which he spent at the Mathematical Institute in Barcelona. Our research group today ranks among leading groups in the area of topology of CW complexes, which play a very prominent role in homotopy theory and its applications. Our most active members in this area are Pavešič and Smrekar, as well as our doctoral student Franc, who successfully completed dissertation during this program duration.
COBISS.SI-ID: 16643929
The main result of this paper is a characterization of discrete vector fields on noncompact cell complexes that have a proper integral, i.e. they are gradient vector fields of a proper discrete Morse function. Already Robin Forman in his seminal work on discrete Morse theory showed that the characteristic property of integrable discrete vector fields on finite cell complexes is acyclicity. Our doctoral student Jerše extended this result to noncompact cell complexes in another paper. Discrete vector fields with a proper integral should in addition, not contain a so-called forbidden configuration, i.e. a decreasing infinite ray with an increasing ray in the boundary of its descending region. This paper gives a constructive proof of the fact that this property characterizes such discrete vector fields. A specific algorithm for the construction of a proper integral of an acyclic discrete vector field with no forbidden configuration is given. These results were done in collaboration with topological group at the University of Sevilla, with which we have successfully completed several joint research projects. The authors have successfully reported about these results at international meetings in the European Union and the United States.
COBISS.SI-ID: 15865945