Compositions of invertibility preserving maps for some monoids and their application to Clifford algebras

For some monoids, we give a method of composing invertibility preserving maps associated to "partial involutions." Also, we define the notion of "determinants for finite dimensional algebras over a field." As examples, we give invertibility preserving maps for Clifford algebras into a field and determinants for Clifford algebras into a field, where we assume that the algebras are generated by less than or equal to 5 generators over the field. On the other hand, "determinant formulas for Clifford algebras" are known. We understand these formulas as an expression that connects invertibility preserving maps for Clifford algebras and determinants for Clifford algebras. As a result, we have a better sense of determinant formulas. In addition, we show that there is not such a determinant formula for Clifford algebras generated by greater than 5 generators

Examples of rational toral rank complex

In "A Hosse diagram for rational toral tanks," we see a CW complex ${\mathcal T}(X)$, which gives a rational homotopical classification of almost free toral actions on spaces in the rational homotopy type of $X$ associated with rational toral ranks and also presents certain relations in them. We call it the {\it rational toral rank complex} of $X$. It represents a variety of toral actions. In this note, we will give effective 2-dimensional examples of it when $X$ is a finite product of odd spheres. This is a combinatorial approach in rational homotopy theory.Comment: 8 page

Proof of some properties of transfer using noncommutative determinants

A transfer is a group homomorphism from a finite group to an abelian quotient group of a subgroup of the group. In this paper, we explain some of the properties of transfers by using noncommutative determinants. These properties enable us to understand transfers more naturally