Bundle tests with a Chern class obstruction

Written by Claude Opus 5.5 (Anthropic), October 2026. Self-checked by the writing AI. Public domain (CC0).

The algebra of the course is built from a sequence of Hermitian vector bundles \(E_0,E_1,\dots\) over a compact manifold and \(24\) maps between successive bundles that satisfy \[ \sum_iX_i^*X_i=I,\qquad\sum_iX_iX_i^*\le\tfrac23\,I . \] This lesson constructs such a sequence for every matrix size \(m\) and every degree bound \(R\), with \(E_0\) trivial of rank \(d_0\), together with an obstruction: the sum of \(m\) copies of \(E_0\oplus\dots\oplus E_R\) contains no trivial subbundle of rank \(2md_0\). In Weighted shifts and the construction a pair of isometries with orthogonal ranges in \(M_m(B)\) is turned into exactly such a subbundle. The obstruction is a Chern class; the maps come from the two lemmas of Expanding maps and a continuous normalization.

The lesson follows [OpenAI-QT, Section 4]. Obstructions of this kind, by characteristic classes of vector bundles, were used by Villadsen [V, Section 3], over products of projective spaces, to construct simple C*-algebras of every stable rank, and by Rørdam [R, Section 3], over products of two-spheres, to construct a simple C*-algebra with a finite and an infinite projection. The Chern classes are taken from the lessons Chern classes and the integral universal ring and The Gysin sequence and projective splitting (course Characteristic classes). The maps use Lemmas 4.1 and 6.1 of Expanding maps and a continuous normalization, and through Lemma 4.1 the hypothesis (SM) on smooth manifolds stated there.

1. Subbundles and complements

The conventions are those of Section 1 of Expanding maps and a continuous normalization: vector bundles are complex, of finite rank, over a compact Hausdorff space, with Hermitian metrics, and a subbundle is a family of subspaces of the fibers whose orthogonal projections form a bundle map. A trivial subbundle of rank \(k\) is a subbundle that is isomorphic to the trivial bundle \(Y\times\mathbb C^k\).

Lemma 1.1. Let \(G\) be a Hermitian vector bundle over a compact Hausdorff space \(Y\), and \(p\) a bundle map of \(G\) such that every \(p(y)\) is an orthogonal projection. Then the images of the \(p(y)\) form a vector bundle \(\operatorname{ran}p\), the images of the \(I-p(y)\) form a vector bundle \(\operatorname{ran}(I-p)\), and \((u,w)\mapsto u+w\) is an isomorphism \(\operatorname{ran}p\oplus\operatorname{ran}(I-p)\to G\).

Proof. The rank of \(p(y)\) is its trace, a continuous function with integer values, hence locally constant. Fix \(y_0\), let \(k\) be the rank of \(p(y_0)\), and let \(\sigma_1,\dots,\sigma_n\) be a local frame of \(G\) near \(y_0\). The vectors \(p(y_0)\sigma_l(y_0)\) span the image of \(p(y_0)\); choose \(k\) of them that form a basis. The corresponding sections \(p\sigma_l\) of \(\operatorname{ran}p\) are continuous, and their Gram determinant is continuous and nonzero at \(y_0\). So they are linearly independent near \(y_0\), and where the rank is \(k\) they span the image of \(p(y)\). The coefficients of a vector of \(\operatorname{ran}p\) in this frame are obtained from its inner products with the frame by the inverse Gram matrix, so they are continuous. Hence \(\operatorname{ran}p\) is a vector bundle. The same applies to \(I-p\). The map \((u,w)\mapsto u+w\) is a continuous bundle map, bijective in each fiber because \(G_y\) is the orthogonal sum of the two images, with the continuous inverse \(v\mapsto(pv,(I-p)v)\). \(\blacksquare\)

2. Chern classes and products of projective spaces

Cohomology is singular cohomology with integer coefficients and the cup product. The lesson Chern classes and the integral universal ring (course Characteristic classes) constructs, for a complex vector bundle \(V\) of rank \(r\) over a Hausdorff space \(B\), Chern classes \(c_k(V)\in H^{2k}(B;\mathbb Z)\) for \(k\ge0\), with \(c_0(V)=1\) and total class \(c(V)=c_0(V)+c_1(V)+\dots\). We use:

Let \(\mathbb{CP}^K\) be the space of complex lines in \(\mathbb C^{K+1}\), and \(\gamma_K\subseteq\mathbb{CP}^K\times\mathbb C^{K+1}\) the tautological line bundle, whose fiber at a line \(\ell\) is \(\ell\). The lesson The Gysin sequence and projective splitting (same course) proves the following, with \(x=-e((\gamma_K)_{\mathbb R})=-c_1(\gamma_K)\).

Lemma 2.1. Let \(K_1,\dots,K_R\ge1\), \(Y=\mathbb{CP}^{K_1}\times\dots\times\mathbb{CP}^{K_R}\) with the projections \(\mathrm{pr}_i\colon Y\to\mathbb{CP}^{K_i}\), \(L_i=\mathrm{pr}_i^*\gamma_{K_i}\), and \(x_i=-c_1(L_i)\). Then the monomials \(x_1^{e_1}\cdots x_R^{e_R}\) with \(0\le e_i\le K_i\) form a basis of the abelian group \(H^*(Y;\mathbb Z)\). In particular \(x_1^{K_1}\cdots x_R^{K_R}\ne0\).

Proof. Induction on \(R\), starting with \(R=0\), where \(Y\) is a point and \(H^*(Y;\mathbb Z)=\mathbb Z\) has the basis \(1\). Let \(R\ge1\), let \(B\) be the product of the first \(R-1\) factors, and \(V=B\times\mathbb C^{K_R+1}\). Its bundle of lines is \(B\times\mathbb{CP}^{K_R}=Y\), with \(p\) the projection to \(B\), and its tautological line is \(L_R\). So \(z=-c_1(L_R)=x_R\), and every class on \(Y\) is uniquely \(\sum_{k=0}^{K_R}p^*(b_k)\,x_R^k\). By induction, each \(b_k\) is uniquely an integer combination of the monomials in the classes \(x_1,\dots,x_{R-1}\) formed on \(B\). The line bundles \(L_1,\dots,L_{R-1}\) on \(Y\) are the pullbacks along \(p\) of the corresponding line bundles on \(B\), so by (C2) and because \(p^*\) is a ring homomorphism, \(p^*\) maps these monomials to the monomials in \(x_1,\dots,x_{R-1}\) on \(Y\). Existence and uniqueness of the expansion in the monomials on \(Y\) follow. \(\blacksquare\)

Proposition 2.2. Let \(Y\), \(L_i\) and \(x_i\) be as in Lemma 2.1, \(L_0=Y\times\mathbb C\), \(h_0\ge0\), and let \(G\) be a Hermitian vector bundle over \(Y\) isomorphic to \[ L_0^{\oplus h_0}\oplus L_1^{\oplus K_1}\oplus\dots\oplus L_R^{\oplus K_R} . \] Then \(G\) contains no trivial subbundle of rank greater than \(h_0\).

Proof. Put \(\kappa=K_1+\dots+K_R\). By (C1), (C2) and (C3), \[ c(G)=\prod_{i=1}^R(1-x_i)^{K_i} . \] Expanding the product gives integer multiples of monomials \(x_1^{e_1}\cdots x_R^{e_R}\) with \(0\le e_i\le K_i\), of degree \(2(e_1+\dots+e_R)\). The degree is \(2\kappa\) only if every \(e_i=K_i\), so \[ c_\kappa(G)=(-1)^\kappa\,x_1^{K_1}\cdots x_R^{K_R}\ne0 \] by Lemma 2.1. Suppose that \(T\subseteq G\) is a subbundle isomorphic to \(\varepsilon^k\) with \(k>h_0\). By Lemma 1.1, applied to the orthogonal projection onto \(T\), \(G\) is isomorphic to \(T\oplus T'\), where \(T'\) is the orthogonal complement, of rank \(h_0+\kappa-k<\kappa\). By (C2) and (C3), \(c(G)=c(T)\,c(T')=c(T')\), so \(c_\kappa(G)=c_\kappa(T')=0\) by (C1). This contradicts \(c_\kappa(G)\ne0\). \(\blacksquare\)

The subbundle \(T\) may lie in any position relative to the line summands. The bundle \(G\) has rank \(h_0+\kappa\) and contains the trivial summand \(L_0^{\oplus h_0}\), but no trivial subbundle of larger rank: the top Chern class of \(G\) is the class of the top degree of \(Y\).

3. Ranks and multiplicities

Fix integers \(m,R\ge1\). Put \[ d_0=64m4^R,\qquad d_j=4^jd_0 \] for \(0\le j\le R\), the multiplicities \[ \begin{gathered} a_{0,j}=64m4^{R-j},\\ a_{i,j}=960m4^{R+2i-j-2} \end{gathered} \] for \(0\le j\le R\) and \(1\le i\le j\le R\), and the numbers \[ \begin{aligned} K_i&=m\sum_{j=i}^Ra_{i,j}&&(1\le i\le R),\\ h_0&=m\sum_{j=0}^Ra_{0,j},\\ \eta_j&=K_1+\dots+K_{j+1}&&(0\le j<R). \end{aligned} \] In passing from level \(j\) to level \(j+1\), every multiplicity \(a_{i,j}\) with \(i\le j\) is divided by four, \(a_{i,j+1}=a_{i,j}/4\), and a new multiplicity \(a_{j+1,j+1}=960m4^{R+j-1}=\tfrac{15}4d_j\) appears.

Lemma 3.1.

  1. All \(a_{i,j}\) are positive integers. For \(j<R\) and \(i\le j\), \(a_{i,j}\) is divisible by \(4\) and at least \(256m\).
  2. \(a_{0,j}+a_{1,j}+\dots+a_{j,j}=d_j\) for \(0\le j\le R\).
  3. \(K_i=320m^24^{2i-2}\bigl(4^{R-i+1}-1\bigr)\), a positive integer.
  4. \(h_0=\tfrac43md_0\bigl(1-4^{-(R+1)}\bigr)<2md_0\).
  5. \(\eta_j<\tfrac53md_0\bigl(4^{j+1}-1\bigr)<\tfrac{20}3md_j\) for \(0\le j<R\).

Proof. Part 1. The exponent \(R-j\) is nonnegative, and \(R+2i-j-2\ge R+2-R-2=0\) for \(i\ge1\) and \(j\le R\). For \(j\le R-1\) the exponents are at least \(1\), so \(a_{0,j}\ge256m\) and \(a_{i,j}\ge3840m\); both are divisible by \(4\), since \(64\) and \(960\) are.

Part 2. For \(j=0\) the sum is \(a_{0,0}=64m4^R=d_0\). If the sum is \(d_j\) at level \(j<R\), then at level \(j+1\) it is \(\tfrac14d_j+\tfrac{15}4d_j=4d_j=d_{j+1}\).

Part 3. \[ \begin{aligned} \sum_{j=i}^R4^{R+2i-j-2}&=4^{2i-2}\sum_{j=i}^R4^{R-j}\\ &=4^{2i-2}\bigl(4^{R-i+1}-1\bigr)/3 ; \end{aligned} \] multiply by \(960m^2\).

Part 4. \[ \begin{aligned} \sum_{j=0}^R64m4^{R-j}&=d_0\sum_{j=0}^R4^{-j}\\ &=\tfrac43d_0\bigl(1-4^{-(R+1)}\bigr) ; \end{aligned} \] multiply by \(m\).

Part 5. \(a_{i,j}=a_{i,i}4^{-(j-i)}\), so \(K_i<\tfrac43ma_{i,i}=5md_{i-1}\), and \[ \begin{aligned} \eta_j&<5m\bigl(d_0+\dots+d_j\bigr)\\ &=\tfrac53md_0\bigl(4^{j+1}-1\bigr)\\ &<\tfrac53md_04^{j+1}=\tfrac{20}3md_j . \end{aligned} \] \(\blacksquare\)

4. The bundles

Let \(Y=\mathbb{CP}^{K_1}\times\dots\times\mathbb{CP}^{K_R}\), \(L_0=Y\times\mathbb C\), and \(L_i=\mathrm{pr}_i^*\gamma_{K_i}\) with the Hermitian metric of \(\mathbb C^{K_i+1}\). For \(0\le j\le R\) let \[ E_j=\bigoplus_{i=0}^jL_i^{\oplus a_{i,j}} , \] an orthogonal direct sum. By Lemma 3.1, \(E_j\) has rank \(d_j\), and \(E_0=L_0^{\oplus d_0}\) is trivial of rank \(d_0\). The line bundle \(L_i\) occurs in \(E_j\) with multiplicity \(a_{i,j}\) for \(i\le j\), so in \[ G=\bigl(E_0\oplus\dots\oplus E_R\bigr)^{\oplus m} \] it occurs \(K_i\) times for \(i\ge1\) and \(h_0\) times for \(i=0\). By Proposition 2.2 and part 4 of Lemma 3.1, \(G\) contains no trivial subbundle of rank \(2md_0\).

The numbers \(K_i\) are chosen as the total multiplicities of the \(L_i\) in \(G\), so that the top Chern class of \(G\) is the top class of \(Y\). The trivial line \(L_0\) occurs only \(h_0<2md_0\) times in \(G\), because its multiplicity is divided by four at every level, although the \(m\) copies of \(E_0\) alone contain \(md_0\) trivial lines.

Smooth structure. The sets \(U_k\) of the lines \(\mathbb Cz\) with \(z_k\ne0\) are open and cover \(\mathbb{CP}^K\), and \(\mathbb Cz\mapsto(z_l/z_k)_{l\ne k}\) identifies \(U_k\) with \(\mathbb C^K\); the changes of charts are smooth. Over \(U_k\), \(\mathbb Cz\mapsto z/z_k\) is a nowhere-zero section of \(\gamma_K\) whose coordinates are the chart coordinates and \(1\). So \(\mathbb{CP}^K\) is a compact smooth manifold of real dimension \(2K\), \(\gamma_K\) is a smooth line subbundle of \(\mathbb{CP}^K\times\mathbb C^{K+1}\), and the metric of \(\mathbb C^{K+1}\) is smooth on it. Products, pullbacks along the smooth projections and orthogonal direct sums preserve smoothness. So \(Y\) is a compact smooth manifold and the \(E_j\) are smooth Hermitian bundles.

Smaller bases. For \(0\le j<R\) let \(Y_{j+1}=\mathbb{CP}^{K_1}\times\dots\times\mathbb{CP}^{K_{j+1}}\), of complex dimension \(\eta_j\). The bundles \(E_j\) and \(E_{j+1}\) involve only \(L_0,\dots,L_{j+1}\), so they are the pullbacks along the projection \(Y\to Y_{j+1}\) of the bundles defined by the same formulas on \(Y_{j+1}\); these are denoted by the same letters.

5. Maps that double dimensions

Lemma 5.1. Let \(0\le j<R\). There are continuous bundle maps \(X_1,\dots,X_{24}\colon E_j\to E_{j+1}\) over \(Y\) with \[ \begin{gathered} \sum_{i=1}^{24}X_i^*X_i=I_{E_j},\\ \sum_{i=1}^{24}X_iX_i^*\le\tfrac23\,I_{E_{j+1}} . \end{gathered} \]

Proof. We work over \(Y_{j+1}\) and write \(d=d_j\). Decompose \(E_{j+1}=F_{\mathrm{old}}\oplus F_{\mathrm{new}}\) with \[ \begin{gathered} F_{\mathrm{old}}=\bigoplus_{i=0}^jL_i^{\oplus a_{i,j}/4},\\ F_{\mathrm{new}}=L_{j+1}^{\oplus a_{j+1,j+1}} , \end{gathered} \] of ranks \(d/4\) and \(\tfrac{15}4d\).

Sixteen maps for small subspaces. Fix \(i\le j\) and put \(D=a_{i,j}\), which is divisible by \(4\) and at least \(256m\) by Lemma 3.1. Apply Lemma 4.1 of Expanding maps and a continuous normalization over a point, with \(q=16\), source \(\mathbb C^D\), target \(\mathbb C^{D/4}\) and the pairs \((s,2s)\) for \(1\le s\le m\). Here \(2s\le2m<D/4\), and, using \(D\ge64m\) and \(s\le m\), \[ \begin{aligned} &(16s-2s)\bigl(\tfrac D4-2s\bigr)-s(D-s)\\ &\quad=\tfrac52sD-27s^2\\ &\quad\ge160ms-27ms=133ms>0 . \end{aligned} \] So there are linear maps \(A^{(i)}_1,\dots,A^{(i)}_{16}\colon\mathbb C^D\to\mathbb C^{D/4}\) such that for every subspace \(S\) with \(1\le\dim S\le m\) the images \(A^{(i)}_kS\) span a space of dimension greater than \(2\dim S\). Since \(L_i^{\oplus D}=L_i\otimes\mathbb C^D\), the maps \(\mathrm{id}\otimes A^{(i)}_k\colon L_i^{\oplus D}\to L_i^{\oplus D/4}\) are smooth bundle maps; in a fiber, a nonzero vector of the line identifies them with the \(A^{(i)}_k\). For \(1\le k\le16\) let \(T_k\colon E_j\to E_{j+1}\) be the block diagonal map with the blocks \(\mathrm{id}\otimes A^{(i)}_k\), \(0\le i\le j\), into \(F_{\mathrm{old}}\), and with zero component in \(F_{\mathrm{new}}\).

The flag argument. We show that \(\dim\sum_{k=1}^{16}T_kS\ge2\dim S\) for every subspace \(S\) of a fiber of \(E_j\) with \(\dim S\le m\). Write the fiber of \(E_j\) as \(V_0\oplus\dots\oplus V_j\) and the fiber of \(F_{\mathrm{old}}\) as \(F_0\oplus\dots\oplus F_j\), following the line bundles, and \(T_k=\bigoplus_iT_{k,i}\) with \(T_{k,i}\colon V_i\to F_i\). Put \(V_{\le i}=V_0\oplus\dots\oplus V_i\) and \(F_{\le i}=F_0\oplus\dots\oplus F_i\), both zero for \(i=-1\), and \[ \begin{gathered} W=\sum_{k=1}^{16}T_kS,\\ S_i=S\cap V_{\le i},\qquad W_i=W\cap F_{\le i} . \end{gathered} \] Let \(\pi_i\colon V_{\le i}\to V_i\) and \(\pi_i'\colon F_{\le i}\to F_i\) be the projections onto the last summand. The kernel of \(\pi_i\) on \(S_i\) is \(S_{i-1}\), so \(\dim\pi_i(S_i)=\dim S_i-\dim S_{i-1}\); likewise \(\dim\pi_i'(W_i)=\dim W_i-\dim W_{i-1}\). Since \(T_k\) is block diagonal, \(T_kS_i\subseteq W\cap F_{\le i}=W_i\) and \(\pi_i'T_k=T_{k,i}\pi_i\) on \(V_{\le i}\). Hence \[ \begin{aligned} \sum_{k=1}^{16}T_{k,i}\bigl(\pi_iS_i\bigr)&=\pi_i'\Bigl(\sum_{k=1}^{16}T_kS_i\Bigr)\\ &\subseteq\pi_i'(W_i) . \end{aligned} \] The subspace \(\pi_i(S_i)\) of \(V_i\) has dimension at most \(\dim S\le m\), so the left side has dimension at least \(2\dim\pi_i(S_i)\). As \(W\subseteq F_{\le j}\), summing over \(i\) gives \[ \begin{aligned} \dim W&=\sum_{i=0}^j\bigl(\dim W_i-\dim W_{i-1}\bigr)\\ &\ge2\sum_{i=0}^j\bigl(\dim S_i-\dim S_{i-1}\bigr)\\ &=2\dim S . \end{aligned} \]

Eight maps for large subspaces. Apply Lemma 4.1 of Expanding maps and a continuous normalization over \(Y_{j+1}\), of real dimension \(2\eta_j\), to the bundles \(E_j\) and \(F_{\mathrm{new}}\), of ranks \(d\) and \(\tfrac{15}4d\), with \(q=8\) and the pairs \((s,2s)\) for \(m<s\le d\). Here \(2s\le2d<\tfrac{15}4d\), and, using \(s\le d\), \(s>m\) and part 5 of Lemma 3.1, \[ \begin{aligned} &(8s-2s)\bigl(\tfrac{15}4d-2s\bigr)-s(d-s)\\ &\quad=\tfrac{43}2sd-11s^2\\ &\quad\ge\tfrac{21}2sd>\tfrac{20}3md>\eta_j . \end{aligned} \] This gives smooth bundle maps \(T_{17},\dots,T_{24}\colon E_j\to F_{\mathrm{new}}\subseteq E_{j+1}\) such that \(\dim\sum_{k=17}^{24}T_kS>2\dim S\) for every subspace \(S\) of a fiber with \(m<\dim S\le d\).

Normalization. Adding maps does not decrease the span of the images. So \(\dim\sum_{k=1}^{24}T_kS\ge2\dim S\) for every subspace \(S\) of every fiber of \(E_j\): the first sixteen maps suffice when \(\dim S\le m\), the last eight when \(\dim S>m\). Lemma 6.1 of Expanding maps and a continuous normalization, over \(Y_{j+1}\), gives bundle maps \(X_1,\dots,X_{24}\colon E_j\to E_{j+1}\) with the two relations. Their pullbacks to \(Y\) have the same relations. \(\blacksquare\)

The two lists have different tasks. Large subspaces cannot be doubled inside \(F_{\mathrm{old}}\), whose rank \(d/4\) is less than \(2\dim S\) as soon as \(\dim S>d/8\). Small subspaces cannot be treated by the incidence lemma over the large base \(Y_{j+1}\): for a line, \(s=1\), and \(q\) maps into \(F_{\mathrm{new}}\) its expression is \(\bigl(\tfrac{15}4(q-2)-1\bigr)d-(2q-5)\), while for \(j=0\) the base has complex dimension \(\eta_0=K_1=5md_0\bigl(1-4^{-R}\bigr)\), which grows with \(m\). The number of maps has to be the same for all \(m\), because the algebra of Weighted shifts and the construction has a fixed number of generators. So small subspaces are handled by maps that are constant on each block, for which the dimension count takes place over a point.

6. The bundle tests

Theorem 6.1 (bundle tests; [OpenAI-QT, Section 4]). Let \(m,R\ge1\) and \(d_0=64m4^R\). There are a compact smooth manifold \(Y\), Hermitian vector bundles \(E_j\) over \(Y\) for \(j\ge0\), and continuous bundle maps \(X_i^{(j)}\colon E_j\to E_{j+1}\) for \(1\le i\le24\) and \(j\ge0\), with the following three properties. (1) \(E_0\) is trivial of rank \(d_0\). (2) For every \(j\ge0\), \[ \begin{gathered} \sum_{i=1}^{24}X_i^{(j)*}X_i^{(j)}=I_{E_j},\\ \sum_{i=1}^{24}X_i^{(j)}X_i^{(j)*}\le\tfrac23\,I_{E_{j+1}} . \end{gathered} \] (3) \(\bigl(E_0\oplus\dots\oplus E_R\bigr)^{\oplus m}\) contains no trivial subbundle of rank \(2md_0\).

Proof. Take \(Y\) and \(E_0,\dots,E_R\) from Section 4; parts 1 and 3 were shown there. For \(j<R\) take the maps of Lemma 5.1. For \(j\ge R\) put \(E_{j+1}=E_j\oplus E_j\), with the two inclusions \(\iota_1,\iota_2\), and \[ \begin{gathered} X_1^{(j)}=2^{-1/2}\iota_1,\qquad X_2^{(j)}=2^{-1/2}\iota_2,\\ X_i^{(j)}=0\quad(3\le i\le24) . \end{gathered} \] Then \(\sum_iX_i^{(j)*}X_i^{(j)}=\tfrac12\bigl(\iota_1^*\iota_1+\iota_2^*\iota_2\bigr)=I_{E_j}\) and \(\sum_iX_i^{(j)}X_i^{(j)*}=\tfrac12\bigl(\iota_1\iota_1^*+\iota_2\iota_2^*\bigr)=\tfrac12I_{E_{j+1}}\). The bundles beyond level \(R\) do not enter part 3. \(\blacksquare\)

In Weighted shifts and the construction the test with parameters \((m,R)\) is used as follows: words of length at most \(R\) in the maps and their adjoints, applied to \(E_0\), stay in \(E_0\oplus\dots\oplus E_R\). So a pair of isometries with orthogonal ranges in \(M_m(B)\), approximated by matrices of polynomials of degree at most \(R\), would produce a trivial subbundle of rank \(2md_0\) in \(\bigl(E_0\oplus\dots\oplus E_R\bigr)^{\oplus m}\).

Remark 6.2 (seven maps). Theorem 6.1 holds with seven maps in place of twenty-four. First, seven maps suffice in each list of Lemma 5.1. Over a point, with \(D\ge64m\) and \(1\le s\le m\), \[ \begin{aligned} &(7s-2s)\bigl(\tfrac D4-2s\bigr)-s(D-s)\\ &\quad=\tfrac14sD-9s^2\\ &\quad\ge16ms-9ms=7ms>0 , \end{aligned} \] and over \(Y_{j+1}\), with \(m<s\le d\), \[ \begin{aligned} &(7s-2s)\bigl(\tfrac{15}4d-2s\bigr)-s(d-s)\\ &\quad=\tfrac{71}4sd-9s^2\\ &\quad\ge\tfrac{35}4sd>\tfrac{20}3md>\eta_j . \end{aligned} \] Second, the two lists can be merged. Let \(A_1,\dots,A_7\colon E_j\to F_{\mathrm{old}}\) be block diagonal maps as in the first list and \(B_1,\dots,B_7\colon E_j\to F_{\mathrm{new}}\) maps as in the second, and put \(T_k=A_k+B_k\colon E_j\to F_{\mathrm{old}}\oplus F_{\mathrm{new}}\). For a subspace \(S\) of a fiber, the projection of \(\sum_kT_kS\) to \(F_{\mathrm{old}}\) is \(\sum_kA_kS\), and its projection to \(F_{\mathrm{new}}\) is \(\sum_kB_kS\). A subspace is at least as large as each of its projections. So \(\dim\sum_kT_kS\ge2\dim S\) for every \(S\), by the flag argument when \(\dim S\le m\) and by the second list when \(\dim S>m\), and Lemma 6.1 of Expanding maps and a continuous normalization applies to \(T_1,\dots,T_7\). The continuation beyond level \(R\) uses two maps. The consequence for the additivity defect is drawn at the end of Weighted shifts and the construction.

Six maps would not be enough for this dimension count: for \(q=6\) the expression over a point is \(4s\bigl(\tfrac D4-2s\bigr)-s(D-s)=-7s^2<0\) (Exercise 7.4).

7. Exercises

7.1. For \(m=R=1\), compute \(d_0\), \(d_1\), the multiplicities, \(K_1\), \(h_0\) and \(\eta_0\), and describe \(Y\), \(E_0\), \(E_1\) and \(G\).

7.2. Let \(\gamma_K^\perp\subseteq\mathbb{CP}^K\times\mathbb C^{K+1}\), \(K\ge1\), be the orthogonal complement of the tautological line bundle. Show that \(c(\gamma_K^\perp)=1+x+\dots+x^K\), and conclude that \(\gamma_K^\perp\) has no nowhere-zero continuous section.

7.3. Over a point, let \(V=V_1\oplus V_2\) and \(F=F_1\oplus F_2\), and let \(A_k=A_k^{(1)}\oplus A_k^{(2)}\), \(1\le k\le q\), be block diagonal maps \(V\to F\) such that for \(b=1,2\) the maps \(A_k^{(b)}\) send every nonzero vector of \(V_b\) to vectors spanning a space of dimension at least \(2\). Show directly that the maps \(A_k\) send every nonzero vector of \(V\) to vectors spanning a space of dimension at least \(2\).

7.4. Verify the identities \[ \begin{aligned} &(16s-2s)\bigl(\tfrac D4-2s\bigr)-s(D-s)\\ &\qquad=\tfrac52sD-27s^2,\\ &(8s-2s)\bigl(\tfrac{15}4d-2s\bigr)-s(d-s)\\ &\qquad=\tfrac{43}2sd-11s^2, \end{aligned} \] and show that the first expression with \(16\) replaced by \(6\) is negative for every \(s\ge1\).

7.5. Suppose that the multiplicity of \(L_0\) were not divided by four, that is, \(a_{0,j}=d_0\) for all \(j\). Show that then \(G\) would contain a trivial subbundle of rank \(2md_0\).

8. Solutions

7.1. \(d_0=256\), \(d_1=1024\); \(a_{0,0}=256\), \(a_{0,1}=64\), \(a_{1,1}=960\); \(K_1=960\), \(h_0=320\) and \(\eta_0=960\), which is less than \(\tfrac{20}3\cdot256\). So \(Y=\mathbb{CP}^{960}\), \(E_0=L_0^{\oplus256}\), \(E_1=L_0^{\oplus64}\oplus L_1^{\oplus960}\), and \(G=E_0\oplus E_1\cong L_0^{\oplus320}\oplus L_1^{\oplus960}\), of rank \(1280\). By Proposition 2.2 it contains no trivial subbundle of rank greater than \(320\), in particular none of rank \(2md_0=512\).

7.2. By Lemma 1.1, \(\gamma_K\oplus\gamma_K^\perp\) is isomorphic to the trivial bundle of rank \(K+1\). By (C1), (C2) and (C3), \((1-x)\,c(\gamma_K^\perp)=1\), and in \(\mathbb Z[x]/(x^{K+1})\) the inverse of \(1-x\) is \(1+x+\dots+x^K\). So \(c_K(\gamma_K^\perp)=x^K\ne0\). A nowhere-zero section \(\sigma\) spans a trivial line subbundle: its projections \(\sigma\sigma^*/\|\sigma\|^2\) are continuous, and \((y,\lambda)\mapsto\lambda\sigma(y)\) is an isomorphism from the trivial line bundle. Its orthogonal complement in \(\gamma_K^\perp\) has rank \(K-1\), and as in the proof of Proposition 2.2 this would force \(c_K(\gamma_K^\perp)=0\).

7.3. Let \(v=v_1+v_2\ne0\) and \(W\) the span of the \(A_kv\). If \(v_2\ne0\), the projection of \(W\) to \(F_2\) is the span of the \(A_k^{(2)}v_2\), of dimension at least \(2\), and \(W\) is at least as large as its projection. If \(v_2=0\), \(W\) is the span of the \(A_k^{(1)}v_1\). This is the flag argument of Lemma 5.1 for lines.

7.4. Expanding the products, \[ \begin{aligned} &14s\cdot\tfrac D4-28s^2-sD+s^2\\ &\qquad=\tfrac52sD-27s^2,\\ &6s\cdot\tfrac{15}4d-12s^2-sd+s^2\\ &\qquad=\tfrac{43}2sd-11s^2 . \end{aligned} \] With \(6\) in place of \(16\): \(4s\cdot\tfrac D4-8s^2-sD+s^2=-7s^2\).

7.5. Then \(L_0\) occurs \(m(R+1)d_0\ge2md_0\) times in \(G\), and \(L_0^{\oplus2md_0}\) is itself a trivial subbundle of rank \(2md_0\).

References