@phdthesis{1123,
  abstract     = {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" 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.     },
  author       = {Mabillard, Isaac},
  issn         = {2663-337X},
  pages        = {55},
  publisher    = {Institute of Science and Technology Austria},
  title        = {{Eliminating higher-multiplicity intersections: an r-fold Whitney trick for the topological Tverberg conjecture}},
  year         = {2016},
}