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Homology of degenerate real projective quadrics

Mohamad Maassarani

TL;DR

We classify the homology of degenerate real projective quadrics Q_{p,q}^n by relating them to a 2-sheeted cover X_{p,q}^n ≅ (S^{p-1}×S^{q-1})∗S^{n-p-q-1} and applying Mayer–Vietoris, Thom isomorphism, and Kunneth decompositions. The paper first analyzes joins and maps (Section 1), then computes rational and Z/2Z-homology for the covers and quadrics (Section 2–3), and finally reconstructs integer homology from rational and mod-2 data (Section 4). The key results provide explicit homology groups in many cases (notably for q>p>1, and even parameter choices) and give a general method to recover Z-homology from simpler invariants. These findings extend Steenrod and Tucker’s nondegenerate quadrics to degenerate ones, enabling complete integer homology descriptions in several families of signatures.

Abstract

Homology of non degenerate real projective quadrics was studied by Steenrod and Tucker. We Compute the rational and the $\mathbb{Z}/2\mathbb{Z}$ homology of degenerate real projective quadrics. This allows to determine the integer homology of these quadrics.

Homology of degenerate real projective quadrics

TL;DR

We classify the homology of degenerate real projective quadrics Q_{p,q}^n by relating them to a 2-sheeted cover X_{p,q}^n ≅ (S^{p-1}×S^{q-1})∗S^{n-p-q-1} and applying Mayer–Vietoris, Thom isomorphism, and Kunneth decompositions. The paper first analyzes joins and maps (Section 1), then computes rational and Z/2Z-homology for the covers and quadrics (Section 2–3), and finally reconstructs integer homology from rational and mod-2 data (Section 4). The key results provide explicit homology groups in many cases (notably for q>p>1, and even parameter choices) and give a general method to recover Z-homology from simpler invariants. These findings extend Steenrod and Tucker’s nondegenerate quadrics to degenerate ones, enabling complete integer homology descriptions in several families of signatures.

Abstract

Homology of non degenerate real projective quadrics was studied by Steenrod and Tucker. We Compute the rational and the homology of degenerate real projective quadrics. This allows to determine the integer homology of these quadrics.
Paper Structure (17 sections, 55 theorems, 63 equations)

This paper contains 17 sections, 55 theorems, 63 equations.

Table of Contents

  1. On joins
  2. Homology of joins
  3. Join of maps
  4. Double cover and rational homology of degenerate quadrics
  5. The double cover and the antipode
  6. Homology of the double cover and rational homology of degenerate quadrics
  7. The case $p,q,n-p-q>1$
  8. The case $p,q>1$, $n-p-q=1$
  9. The case $p$ or $q$ (not both) equal to $1$ and $n-p-q >1$
  10. The case $p$ or $q$ (not both) equal to $1$ and $n-p-q =1$
  11. The case $p=q=1$ and $n-p-q >1$
  12. The case $p=q=1$ and $n-p-q =1$
  13. Homology of degenerate quadrics over ${\mathbb Z}/2{\mathbb Z}$
  14. The case $p,q>1$
  15. The case $q>p=1$
  16. ...and 2 more sections

Key Result

Proposition 1.1

Denote by $l_k$ the degree $k$ part of $l$. The morphism $l_k$ is surjective for $k\geq 1$.

Theorems & Definitions (96)

  • Proposition 1.1
  • proof
  • Theorem 1.2
  • proof
  • Remark 1.3
  • Proposition 1.4
  • proof
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • ...and 86 more