Results on projectors, regular values, odd maps, support complexes and Borsuk numbers, with their assumptions

The course proves the statements below; each link leads to the statement with its assumptions and its proof. The lessons are self-checked by the writing AI; independent mathematical review is not complete.

• Projector geometry — Lemma 1.1. The rank-one projections of R^k form a compact copy of projective space of diameter sqrt 2; distance sqrt 2 means orthogonal lines.
• From a cover to an admissible map — Lemma 2.3. A cover of X_k by m sets of diameter less than sqrt 2 gives a smooth map to the simplex whose supports separate orthogonal lines.
• Sard's theorem — Theorem 2.1. The critical values of a smooth map between Euclidean spaces form a null set.
• Regular level sets — Proposition 3.1. On the smooth open region, a regular fibre is a submanifold with tangent kernel. In compact M it is compact exactly when closed in M. For a whole fibre, global continuity is sufficient; Example 3.3 shows local smoothness alone is not.
• Local counting — Lemma 4.2. For a globally continuous map from a closed manifold, an invertible derivative at each point of the chosen fibre yields a finite fibre and the local modulo-two count; the manifold-target degree conclusion has the stated target hypotheses.
• Homology of projective spaces modulo two — Proposition 3.1. H_i(RP^n; F_2) = F_2 for 0 <= i <= n, with connecting isomorphisms in the transfer sequence of S^n -> RP^n.
• Odd self-maps of spheres — Theorem 4.1. An odd continuous map S^n -> S^n has degree one modulo two.
• Borsuk-Ulam theorem — Corollary 4.2. There is no odd continuous map S^n -> S^d with d < n.
• Degree zero over a product (OpenAI) — Proposition 5.1. A map L x RP^(h-1) -> RP^(a-1) induced by a map odd in the sphere variable has degree zero modulo two when dim L >= 1.
• Square roots — Lemma 1.1. Positive semidefinite square roots exist, are unique, continuous, and smooth on positive definite operators.
• Positive extension (OpenAI) — Lemma 2.1. The Gaussian average of an admissible map is continuous, has trace sum, the exact support of the range, and restricts to subspaces.
• Odd extension (OpenAI) — Proposition 3.1. The odd extension to symmetric operators has l1-norm the trace norm, exact sign supports, and is smooth at nonsingular operators.
• Number of labels (OpenAI) — Proposition 1.1. An admissible map on the lines of R^k needs at least k(k+1)/2 labels; at equality its extension has degree one and every pair is a face.
• Supports at equality (OpenAI) — Theorem 2.1. With k(k+1)/2 labels every facet has k labels and is hit an odd number of times over each regular value.
• Simplicial and singular homology modulo two — Lemma 3.1. For a finite simplicial complex the simplicial and singular homologies with F_2 coefficients agree.
• Facet cycle (OpenAI) — Theorem 3.3. At equality the sum of the facets is a cycle modulo two.
• Six-label triangle systems — Lemma 1.3. Links are five-cycles, no transposition preserves the system, and five labels determine it.
• No ten labels on R^4 (OpenAI) — Theorem 2.5. There is no admissible map from the lines of R^4 to the nine-dimensional simplex.
• Borsuk's conjecture fails in dimension nine (OpenAI) — Theorem 3.1. X_4 has diameter sqrt 2 and cannot be covered by ten subsets of smaller diameter.
• All dimensions from nine — Corollary 3.2. For every d >= 9 a compact subset of R^d needs more than d+1 pieces of smaller diameter.

Reading order

Read the lessons in order; each uses the notation and results of the previous ones.