thesis published Isaac Mabillard author 32BF9DAA-F248-11E8-B48F-1D18A9856A87 Uli Wagner supervisor UlWa department GradSch department Motivated by topological Tverberg-type problems in topological combinatorics and by classical results about embeddings (maps without double points), we study the question whether a finite simplicial complex K can be mapped into Rd without triple, quadruple, or, more generally, r-fold points (image points with at least r distinct preimages), for a given multiplicity r ≤ 2. In particular, we are interested in maps f : K → Rd that have no global r -fold intersection points, i.e., no r -fold points with preimages in r pairwise disjoint simplices of K , and we seek necessary and sufficient conditions for the existence of such maps. We present higher-multiplicity analogues of several classical results for embeddings, in particular of the completeness of the Van Kampen obstruction for embeddability of k -dimensional complexes into R2k , k ≥ 3. Speciffically, we show that under suitable restrictions on the dimensions(viz., if dimK = (r ≥ 1)k and d = rk \ for some k ≥ 3), a well-known deleted product criterion (DPC ) is not only necessary but also sufficient for the existence of maps without global r -fold points. Our main technical tool is a higher-multiplicity version of the classical Whitney trick , by which pairs of isolated r -fold points of opposite sign can be eliminated by local modiffications of the map, assuming codimension d – dimK ≥ 3. An important guiding idea for our work was that suffciency of the DPC, together with an old result of Özaydin's on the existence of equivariant maps, might yield an approach to disproving the remaining open cases of the the long-standing topological Tverberg conjecture , i.e., to construct maps from the N -simplex σN to Rd without r-Tverberg points when r not a prime power and N = (d + 1)(r – 1). Unfortunately, our proof of the sufficiency of the DPC requires codimension d – dimK ≥ 3, which is not satisfied for K = σN . In 2015, Frick [16] found a very elegant way to overcome this \codimension 3 obstacle&quot; and to construct the first counterexamples to the topological Tverberg conjecture for all parameters(d; r ) with d ≥ 3r + 1 and r not a prime power, by a reduction1 to a suitable lower-dimensional skeleton, for which the codimension 3 restriction is satisfied and maps without r -Tverberg points exist by Özaydin's result and sufficiency of the DPC. In this thesis, we present a different construction (which does not use the constraint method) that yields counterexamples for d ≥ 3r , r not a prime power. https://research-explorer.ista.ac.at/download/1123/6809/Thesis_final version_Mabillard_w_signature_page.pdf application/pdf https://research-explorer.ista.ac.at/download/1123/9178/2016_Mabillard_Thesis.pdf application/pdfno Institute of Science and Technology Austria2016 eng 2663-337X 55 https://research-explorer.ista.ac.at/record/2159 Mabillard, Isaac. <i>Eliminating Higher-Multiplicity Intersections: An r-Fold Whitney Trick for the Topological Tverberg Conjecture</i>. Institute of Science and Technology Austria, 2016. I. Mabillard, Eliminating Higher-Multiplicity Intersections: An r-Fold Whitney Trick for the Topological Tverberg Conjecture, Institute of Science and Technology Austria, 2016. Mabillard, I. (2016). <i>Eliminating higher-multiplicity intersections: an r-fold Whitney trick for the topological Tverberg conjecture</i>. Institute of Science and Technology Austria. Mabillard, Isaac. “Eliminating Higher-Multiplicity Intersections: An r-Fold Whitney Trick for the Topological Tverberg Conjecture.” Institute of Science and Technology Austria, 2016. Mabillard I. Eliminating higher-multiplicity intersections: an r-fold Whitney trick for the topological Tverberg conjecture. 2016. I. Mabillard, “Eliminating higher-multiplicity intersections: an r-fold Whitney trick for the topological Tverberg conjecture,” Institute of Science and Technology Austria, 2016. Mabillard I. 2016. Eliminating higher-multiplicity intersections: an r-fold Whitney trick for the topological Tverberg conjecture. Institute of Science and Technology Austria. 11232018-12-11T11:50:16Z2026-07-29T11:26:07Z