In the 80’s, Gunnar Carlsson introduced a conjecture pertaining to the singular homology of certain finite CW complexes. In 2018, Iyengar and Walker demonstrated that a related conjecture was false using tools from homological algebra and a result on maximal rank of multiplication maps between exterior algebras over vector spaces in a paper by Conca, Herbig, and Iyengar. This presentation will showcase an investigation of maximal rank of a generalized form of multiplication map not discussed in the prior papers. In particular, such a generalized form of multiplication map between exterior algebras is induced by a certain partition of a finite dimensional vector space’s basis. Although more counterexamples can be produced in this way, this research indicates why vector spaces of dimension 2n induce certain multiplication maps that are well-behaved, and why more research is needed for the case of vector spaces of dimension dn.
