Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Generalized Grassmann invariant-redrawn

Kiyoshi Igusa

TL;DR

The work develops a picture-based framework for third group homology, encoding $H_3G$ via Peiffer diagrams (pictures) and linking Dehn-type relations to a kernel $\ker\partial_2$ to realize a combinatorial model. It defines the generalized Grassmann invariant $\chi: K_3({\mathbb Z}[\pi])\to H_0(\pi;{\mathbb Z}_2[\pi])$, proves naturality and surjectivity, and situates $K_3$ within a broader Whitehead/pseudoisotopy context through exact sequences and higher-categorical invariants. The paper then connects pictures to stability diagrams, introducing ghost handle slides and augmented picture monoids to reconcile Morse pictures with representation-theoretic walls and Cartan-subalgebra geometry, and sketches dual invariants for torsion-free classes. Overall, it interlaces combinatorial picture theory, Steinberg/K-theory, and stability concepts to illuminate transgression phenomena and potential higher-degree generalizations in $A$-theory and Waldhausen frameworks. These constructions yield concrete tools for calculating $K_3$ and understanding pseudoisotopy obstructions via geometric/combinatorial data.

Abstract

This is my old unpublished paper called "The generalized Grassmann invariant". It shows how "pictures" also known as "Peiffer diagrams" represent elements of $H_3G$ for any group $G$ and shows that $K_3(\mathbb Z [G])$ is isomorphic to a group of deformation classes of pictures for the Steinberg group of $\mathbb Z[G]$. A picture representing an element of order $16$ in $K_3(\mathbb Z)\cong \mathbb Z_{48}$ is also constructed. In this updated version of the paper, we modify only the pictures and leave the text more or less unchanged. We also added an Appendix to explain the new pictures using representations of quivers and root systems of type $A_n$. Often, some roots are missing in the Morse pictures. We give two ideas to replace these roots. One uses "ghost handle slides" to obtain a standard picture. The second idea uses the (real) Cartan subalgebra $H$ to obtain a "relative" picture for a torsion class and adds "ghost modules" which are directly related to the generalized Grassmann invariant. Additions and changes are in blue except the pictures are black with colored ghosts.

Generalized Grassmann invariant-redrawn

TL;DR

The work develops a picture-based framework for third group homology, encoding via Peiffer diagrams (pictures) and linking Dehn-type relations to a kernel to realize a combinatorial model. It defines the generalized Grassmann invariant , proves naturality and surjectivity, and situates within a broader Whitehead/pseudoisotopy context through exact sequences and higher-categorical invariants. The paper then connects pictures to stability diagrams, introducing ghost handle slides and augmented picture monoids to reconcile Morse pictures with representation-theoretic walls and Cartan-subalgebra geometry, and sketches dual invariants for torsion-free classes. Overall, it interlaces combinatorial picture theory, Steinberg/K-theory, and stability concepts to illuminate transgression phenomena and potential higher-degree generalizations in -theory and Waldhausen frameworks. These constructions yield concrete tools for calculating and understanding pseudoisotopy obstructions via geometric/combinatorial data.

Abstract

This is my old unpublished paper called "The generalized Grassmann invariant". It shows how "pictures" also known as "Peiffer diagrams" represent elements of for any group and shows that is isomorphic to a group of deformation classes of pictures for the Steinberg group of . A picture representing an element of order in is also constructed. In this updated version of the paper, we modify only the pictures and leave the text more or less unchanged. We also added an Appendix to explain the new pictures using representations of quivers and root systems of type . Often, some roots are missing in the Morse pictures. We give two ideas to replace these roots. One uses "ghost handle slides" to obtain a standard picture. The second idea uses the (real) Cartan subalgebra to obtain a "relative" picture for a torsion class and adds "ghost modules" which are directly related to the generalized Grassmann invariant. Additions and changes are in blue except the pictures are black with colored ghosts.

Paper Structure

This paper contains 14 sections, 31 theorems, 111 equations, 9 figures.

Table of Contents

  1. The space of pictures for $H_3G$
  2. $K_3({\mathbb{Z}}[\pi])$ and $\overline P(St({\mathbb{Z}}[\pi])$
  3. The Generalized Grassmann Invariant
  4. Naturality of $\chi$
  5. $\chi(\pi_3^s(B\pi\cup pt))=0$.
  6. $\chi:K_3({\mathbb{Z}})\to {\mathbb{Z}}_2$ is surjective
  7. Application to Pseudoisotopy
  8. Appendix: Pictures as stability diagrams
  9. Picture group
  10. Stability conditions
  11. Walls $D(M)$
  12. Ghost handle slides
  13. Cartan subalgebra and generalized Grassmann invariant
  14. Torsion-free classes and duality

Key Result

Theorem 1.4

If ${\mathcal{Y}}$ is a reduced set of relations for $G$ then $\overline P(G)\cong \ker\partial_2$ as $G$-modules.

Figures (9)

  • Figure 1: Picture for $1\to 2$ has 3 indecomposable representations $S_1,S_2,P_1$. The walls $D(S_1)$, $D(S_2)$ are the $y$ and $x$ axes respectively since they are $(1,0)^\perp$ and $(0,1)^\perp$. The wall $D(P_1)$ is the set of all $x$ in $(1,1)^\perp$ so that $x\cdot \mathop{\mathrm{\underline{dim}}}\nolimits S_2\le0$ since $S_2\subset P_1$. The picture group has generators $x(M)$ corresponding to each indecomposable module $M$. On the right is the geometrically correct picture in the Cartan subalgebra $H$. The open region where $h_1<h_2<h_3$ is the subset of $H$ on which $\alpha_1,\alpha_2$ and $\beta$ are negative.
  • Figure 2: Picture for $1\to 2\to 3$ with labels $D(M)$, written simply as $M$, on the left and with corresponding subsets of $H$ on the right. Coordinates $h_i$ are decreasing in the central region and $h_i<h_{i+1}$ at points outside the $h_i=h_{i+1}$ circle. Compare this with Figure \ref{['lem: 2.2']}(b) where $e_{ij}$ labels are on the sets where $h_i=h_j$ using $i,j,k,\ell=1,2,3,4$.
  • Figure 3: Picture for $1\to 2\quad 3$ which has only 4 indecomposable modules: $S_1,S_2,S_3$ and $P_1$ with $\mathop{\mathrm{\underline{dim}}}\nolimits P_1=(1,1,0)=\alpha_{13}$. The walls $D(S_1),D(S_2),D(S_3)$, shown on the left, are given by the equations $h_1=h_2$, $h_2=h_3$, $h_4=h_5$ and $D(P_1)$ is given, as before, by $h_1=h_3\ge h_2$ which means $D(P_1)$ is outside the $S_2$ circle and inside the $S_1$ circle. Coordinates in the Cartan subalgebra are shown on the right. This should be compared with Figure \ref{['lem: 2.2']}(a) with indices $i,j,k,\ell,m$ replaced with $1,2,3,4,5$.
  • Figure 4: Picture for $1\leftarrow 2\to 3$ which has 6 walls, but the wall $D(P_2)$ vanishes in the Morse theory diagram since $x(P_2)$ is in the kernel of the representation $\rho_{uvw}:G(A_3^+)\to T_{34}$. We will fix this by introducing the "ghost handle slide" $z=z_{34}^{\overline{uvw}}$.
  • Figure 5: Picture for $1\to 2\leftarrow 3$ has 6 walls, but the wall $D(I_2)$ vanishes in the Morse theory diagram since $x(I_2)$ is in the kernel of the corresponding representation $\rho_{uvw}:G(A_3^-)\to T_{12}$ where $T_{12}\subset T_4({\mathbb{Z}}[\pi])$. We will insert a "ghost" element $z=z_{12}^{\overline{uvw}}\in \widetilde{T}_{12}$ to fill in this gap.
  • ...and 4 more figures

Theorems & Definitions (83)

  • Definition 1.1
  • Example 1.2
  • Definition 1.3
  • Theorem 1.4
  • Definition 1.5
  • Lemma 1.6
  • proof
  • Lemma 1.7
  • proof
  • Corollary 1.8
  • ...and 73 more