A topological product Tverberg theorem

We prove a generalization of the topological Tverberg theorem. One special instance of our general theorem is the following: Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Delta"> <mml:semantics> <mml:mi mathvariant="normal"> Δ </mml:mi> <mml:annotation encoding="application/x-tex">\Delta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> denote the 8-dimensional simplex viewed as an abstract simplicial complex, and suppose that its vertices are arranged in a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="3 times 3"> <mml:semantics> <mml:mrow> <mml:mn>3</mml:mn> <mml:mo> × </mml:mo> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">3\times 3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> array. Then for any continuous map <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f colon normal upper Delta right-arrow double-struck upper R cubed"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>:</mml:mo> <mml:mi mathvariant="normal"> Δ </mml:mi> <mml:mo stretchy="false"> → </mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">f:\Delta \to \mathbb {R}^3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> it is possible to partition the rows or the columns of the vertex array into two parts, such that the disjoint faces <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma"> <mml:semantics> <mml:mi> σ </mml:mi> <mml:annotation encoding="application/x-tex">\sigma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="tau"> <mml:semantics> <mml:mi> τ </mml:mi> <mml:annotation encoding="application/x-tex">\tau</mml:annotation> </mml:semantics> </mml:math> </inline-formula> induced by the two parts satisfy <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f left-parenthesis sigma right-parenthesis intersection f left-parenthesis tau right-parenthesis not-equals normal empty-set"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi> σ </mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo> ∩ </mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi> τ </mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo> ≠ </mml:mo> <mml:mi mathvariant="normal"> ∅ </mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">f(\sigma )\cap f(\tau ) \neq \emptyset</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Our result also has consequences for geometric transversals and topological Helly theorems.