Foam cobordism and the Sah-Arnoux-Fathi invariant

Abstract

This is the first in a series of papers where scissor congruence and K-theoretical invariants are related to cobordism groups of foams in various dimensions. A model example is provided where the cobordism group of weighted one-foams is identified, via the Sah-Arnoux-Fathi invariant, with the first homology of the group of interval exchange automorphisms and with the Zakharevich first K-group of the corresponding assembler. Several variations on this cobordism group are computed as well.

Figures (48)

• Figure 1: Three types of points on a 2-foam. Left to right: a regular point, seam points on a seam interval, a vertex.
• Figure 2: One of the four possible choices of facet orientations and order of thin facets near a seam of an oriented 2-foam. Facet orientation as indicated by the three "cap" semicircular arrows. The two thin facets are shown as tangent to each other along the seam, which is a convenient convention for tracking thin facets.
• Figure 3: Left: ordering of seams near a vertex, with facets labelled $f_1$, $f_2$, $f_3$, $f_{12}$, $f_{23}$, $f_{123}$. The orderings are from smaller to larger indices: $(f_1,f_2)$, $(f_2,f_3)$, $(f_1,f_{23})$, $(f_{12},f_3)$. Right: one out of two possible facet orientations near a vertex is shown. Orientations and orderings must be compatible along each seam, as explained earlier and in Figure \ref{['fig3_005']}.
• Figure 4: Three parallel cross-sections near a vertex of a 2-foam, with the middle cross-section going through the vertex. Small arrows show the order of facets along the seams.
• Figure 5: Vertex of a (weighted) two-foam is analogous to that of a branched surface, c.f. Oer88.

Theorems & Definitions (41)

• Remark 2.1
• Proposition 2.2
• proof
• Remark 2.3
• Remark 2.4
• Proposition 2.5
• Theorem 2.6
• proof
• Remark 2.7
• Remark 2.8