--- title: Rudyak's Conjecture and LS-Category url: https://www.emergentmind.com/topics/rudyak-s-conjecture type: topic --- # Rudyak's Conjecture and LS-Category Searching arXiv for recent papers on Rudyak's conjecture and related LS-category results. Rudyak's conjecture is the assertion that a degree one map between closed manifolds cannot decrease Lusternik–Schnirelmann category. In the reduced convention used throughout the cited work, if \(f:M\to N\) is a degree one map between closed manifolds, then the conjectured inequality is \[ \operatorname{cat}(M)\ge \operatorname{cat}(N). \] Here \(\operatorname{cat}(X)\) is the least integer \(k\) such that \(X\) admits an open cover by \(k+1\) subsets, each nullhomotopic in \(X\). The conjecture is motivated by the algebraic consequences of degree one maps: for closed oriented manifolds they preserve the fundamental class, induce split epimorphisms in homology, and induce split monomorphisms in cohomology for every coefficient ring, suggesting that the source should not be simpler than the target in the LS-categorical sense [2508.18534]. ## 1. Formulation and basic significance Rudyak’s conjecture is usually stated for closed orientable manifolds in the form \[ \deg f = 1 \quad \Longrightarrow \quad \operatorname{cat}(M)\ge \operatorname{cat}(N). \] A degree one map \(f:M^n\to N^n\) satisfies \[ f_*([M])=[N], \] so it is a topological form of domination. The expectation that such maps should not raise geometric complexity is reinforced by the standard fact that \[ f_*:H_*(M;R)\to H_*(N;R) \] is a split epimorphism and \[ f^*:H^*(N;R)\to H^*(M;R) \] is a split monomorphism for every coefficient ring. Any cohomological lower bound for \(\operatorname{cat}(N)\) detected by a nonzero class can therefore be pulled back to \(M\) [2508.18534]. The conjecture is also tied to classical variational interpretations of LS-category. For a closed manifold \(M\), \(\operatorname{cat}(M)+1\) is a lower bound for the number of critical points of any smooth function on \(M\). This places the conjecture at the intersection of homotopy theory, manifold topology, and critical point theory [2508.18534]. The reduced convention is standard in the literature discussed here: \(\operatorname{cat}(X)=0\) for contractible \(X\). In this normalization, the conjecture is a monotonicity statement for a homotopy invariant that is sensitive both to cohomology and to unstable homotopy structure [2109.08011]. ## 2. Early partial results and structural reductions The conjecture predates the recent surgery-theoretic work. Rudyak introduced it and proved it under restrictions on dimension, connectivity, and category of the domain; he later proved it for manifolds of dimension \(\le 4\) [2508.18534]. Another early positive case is the collapsing map \[ M\# N \to N, \] which fits the connected-sum formula \[ \operatorname{cat}(M\# N)=\max\{\operatorname{cat}(M),\operatorname{cat}(N)\}. \] This shows that the conjecture holds for one of the most basic families of degree one maps [2008.06002]. A major structural reduction was obtained by Dranishnikov through products of lens spaces. For odd \(n\), relatively prime \(p\) and \(q\), there are integers \(k,l\) with \(lp+kq=1\) such that \[ M=k(L_p^m\times S^n)\#\,l(S^m\times L_q^n) \] admits a degree one map to \[ L_p^m\times L_q^n. \] When \(mn+1, \] then Rudyak’s conjecture would fail [1409.8316]. The same paper computed \[ \operatorname{cat}(L_p^n\times L_q^n)=n+1 \] for every \(n=2k-1\) and distinct primes \(p,q\ge k\), and established the upper bound \[ \operatorname{cat}(L_p^n\times L_q^n)\le 2n-2 \] for all odd \(n\) and odd relatively prime \(p,q\). These results do not prove the conjecture, but they eliminate a broad class of possible counterexamples and show that natural test cases often realize the smallest value allowed by cup-length [1409.8316]. ## 3. Surgery-theoretic verification in the LS-category setting A decisive advance was the surgery approach of Dranishnikov and Scott. Their theorem applies to a normal map of degree one \[ f:M\to N \] between closed orientable smooth manifolds, with \(N\) \((r-1)\)-connected for some \(r\ge 1\). If \[ \dim N \le 2r\,\operatorname{cat}(N)-3, \] then \[ \operatorname{cat}(M)\ge \operatorname{cat}(N). \] This weakens Rudyak’s earlier geometric assumptions from stably parallelizable manifolds to normal maps and improves the dimension inequality from \[ \dim N \le 2r\,\operatorname{cat}(N)-4 \] to \[ \dim N \le 2r\,\operatorname{cat}(N)-3 \] [2008.06002]. The mechanism is Wall’s surgery obstruction theory. For a normal degree one map in dimension \(n\ge 5\), there is an obstruction \[ \theta(f)\in L_n(\pi_1(N)), \] and \(\theta(f)=0\) if and only if \(f\) is normally bordant, through surgeries in dimensions \(\le \dim M/2\), to a homotopy equivalence. LS-category is encoded by the Ganea fibration \[ p_q:G_q(N)\to N, \] whose fiber is the \((q+1)\)-fold join \(*^{q+1}\Omega N\), with \[ \operatorname{cat}(X)\le q \iff p_q \text{ admits a section.} \] Assuming \(\operatorname{cat}(M)\le q<\operatorname{cat}(N)=q+1\), one obtains a lift over \(M\), extends it across the surgery trace \(W\), and then transfers it to the homotopy-equivalent boundary component \(M'\), producing a section of \(p_q\) over \(N\), a contradiction [2008.06002]. The crucial numerical input is connectivity of the Ganea fiber. If \(N\) is \((r-1)\)-connected, then \(\Omega N\) is \((r-2)\)-connected, and the fiber \(*^{q+1}\Omega N\) is \((r(q+1)-2)\)-connected. The inequality \[ \dim N \le 2r\,\operatorname{cat}(N)-3 \] is exactly what guarantees that the lift extends over the low-dimensional surgery cells [2008.06002]. ## 4. Generalization to sectional category and higher topological complexity The surgery framework was later extended from LS-category to sectional category. For a fibration \(p:E\to B\), the sectional category \(\operatorname{secat}(p)\) is the least integer \(k\) such that \(B\) is covered by \(k+1\) open sets, each admitting a local section of \(p\). Schwarz’s theorem gives \[ \operatorname{secat}(p)\le n \quad\Longleftrightarrow\quad \ast_B^{\,n+1}p \text{ admits a section}, \] where \(\ast_B^{\,n+1}p\) is the \((n+1)\)-fold fiberwise join. LS-category appears as the special case \[ \operatorname{cat}(X)=\operatorname{secat}(ev_1:PX\to X). \] This places Rudyak’s conjecture in a broader comparison problem for fibrations over manifolds [2109.08011]. The main theorem of this generalized setting considers a commuting square \[ \begin{tikzcd} E^M \arrow[r, "\bar f"] \arrow[d, swap, "p^M"] & E^N \arrow[d, "p^N"] \\ M \arrow[r, "f"] & N \end{tikzcd} \qquad\text{with}\qquad p^N\circ \bar f = f\circ p^M, \] where \(f:M\to N\) is a normal degree one map of closed smooth manifolds, the fiber \(F^N\) of \(p^N\) is \((r-2)\)-connected, \(f\) has zero surgery obstruction, and \[ 5\le \dim N \le 2r\,\operatorname{secat}(p^N)-3. \] Under these hypotheses, \[ \operatorname{secat}(p^M)\ge \operatorname{secat}(p^N). \] The proof follows the Dranishnikov–Scott strategy, replacing Ganea fibrations by fiberwise joins and using the connectivity formula \[ \operatorname{conn}(\ast^{q+1}F^N)=r(q+1)-2 \] when \(F^N\) is \((r-2)\)-connected [2109.08011]. This encompasses higher topological complexity. For \(k\ge 2\), \[ \Delta_k^X:X^I\to X^k, \qquad \mathrm{TC}_k(X)=\operatorname{secat}(\Delta_k^X), \] with \(\mathrm{TC}(X)=\mathrm{TC}_2(X)\). The generalized theorem yields, under the corresponding obstruction and range hypotheses, \[ \mathrm{TC}_k(M)\ge \mathrm{TC}_k(N). \] In the simply connected case, explicit Wall \(L\)-groups and Browder’s product formula permit a sharper result: if \(N\) is \((r-1)\)-connected with \(r\ge 2\), \(5\le k\dim N \le 2r\,\mathrm{TC}_k(N)-3\), and \[ \dim N \not\equiv 0 \pmod 4, \] then \[ \mathrm{TC}_k(M)\ge \mathrm{TC}_k(N). \] This shows that the monotonicity principle underlying Rudyak’s conjecture extends beyond LS-category to a wider obstruction-theoretic context [2109.08011]. ## 5. Low-dimensional simply connected manifolds A 2025 result establishes Rudyak’s conjecture for all simply connected closed smooth manifolds of dimension at most \(8\). The main theorem states that a degree one map \[ f:M\to N \] between simply connected \(n\)-manifolds with \(n\le 8\) satisfies \[ \operatorname{cat}(M)\ge \operatorname{cat}(N). \] This settles the conjecture throughout the \(1\)-connected low-dimensional range up to dimension \(8\), while leaving open non-simply connected manifolds in dimensions \(5\)–\(8\), simply connected manifolds in dimensions \(>8\), and the unrestricted conjecture [2508.18534]. The proof combines several ingredients. Whitehead’s estimate gives \[ \operatorname{cat}(X)\le \frac{\dim X}{k+1} \] for a \(k\)-connected CW complex, so for simply connected spaces \[ \operatorname{cat}(X)\le \frac{\dim X}{2}. \] This sharply restricts the possible categories in low dimensions. A second key input is Schwarz’s description of LS-category via Ganea fibrations: \[ \operatorname{cat}(X)\le n \iff p_n^X:G_n(X)\to X \text{ admits a section.} \] The hard cases are reduced to analyzing the possibility that \(\operatorname{cat}(M)\le 2<\operatorname{cat}(N)\), especially in dimensions \(7\) and \(8\) [2508.18534]. Surgery enters in a different way from the earlier dimension-connectivity theorem. One shows that if \(M'\) is obtained from a simply connected manifold \(M\) with \(\operatorname{cat}(M)\le 2\) by surgery in dimensions \(k\le 4\), then \[ \operatorname{cat}(M')\le 2. \] This permits modification of the source while preserving low category. In dimensions \(7\) and \(8\), if \(M\) is spin, the degree one map is bordant by \(2\)- and \(3\)-surgeries to a map inducing isomorphisms on \(\pi_i\) for \(i\le 3\). A Moore–Postnikov analysis of the Ganea fibration over \(N\) then forces \(\operatorname{cat}(N)\le 2\), excluding counterexamples. If \(M\) is non-spin and \(n\ge 7\), the paper proves \[ \operatorname{cat}(M)\ge 3, \] again ruling out the only possible low-category obstruction [2508.18534]. The six-dimensional case is treated by James’s theorem. For a \(3k\)-dimensional \((k-1)\)-connected manifold \(N\), if \(f:M\to N\) has degree one, then \[ \operatorname{cat}(M)\ge \operatorname{cat}(N). \] For \(k=2\), this yields the simply connected six-dimensional case directly [2508.18534]. ## 6. Candidate counterexamples and anomalous product phenomena Despite the positive results, the conjecture remains open in full generality, and several papers identify plausible routes to counterexamples. One such route arises from Iwase’s manifolds \(M_2\) and \(M_3\), which are \(16\)-dimensional manifolds with \[ {\rm cat}_{LS}(M_2)={\rm cat}_{LS}(M_3)=3 \] and anomalous product behavior. In particular, \(M_3\) satisfies \[ {\rm cat}_{LS}(M_3\times M_3)\le 5<2\,{\rm cat}_{LS}(M_3), \] so the expected square-product equality fails [2011.04819]. The same work constructs degree \(2\) and degree \(3\) maps \[ g:S^{14}\times S^2\to M_2, \qquad h:S^{14}\times S^2\to M_3, \] and uses a connected-sum construction for coprime degrees to obtain a degree one map \[ f:\; 2(M_3\times S^2\times S^{14})\#-\,(M_2\times S^2\times S^{14})\longrightarrow M_2\times M_3. \] The source has \[ {\rm cat}_{LS}\le 4. \] Therefore, if \[ {\rm cat}_{LS}(M_2\times M_3)\ge 5, \] then Rudyak’s conjecture would fail. The paper proves only that \[ {\rm cat}_{LS}(M_2\times M_3)>4, \] so the decisive product-category computation remains unresolved [2011.04819]. This picture complements the earlier lens-space reduction. In both settings, the conjecture is transformed into a concrete LS-category problem for a product manifold: \(L_p^n\times L_q^n\) in one case, \(M_2\times M_3\) in another. The available computations support the conjecture in many cases, but they also indicate that any eventual counterexample, if it exists, is likely to involve delicate product behavior and unstable homotopy phenomena [1409.8316]. ## 7. Present status and mathematical outlook The current state of the subject is mixed. On the positive side, Rudyak’s conjecture is known for manifolds of dimension \(\le 4\), for all simply connected closed smooth manifolds of dimension at most \(8\), for connected-sum collapse maps, and for broad surgery-theoretic ranges governed by connectivity and dimension inequalities [2508.18534]. The surgery program also shows that the conjectural monotonicity principle extends naturally from LS-category to sectional category and to higher topological complexity under explicit normality, obstruction, connectivity, and dimension hypotheses [2109.08011]. At the same time, the conjecture remains open in general. The available proofs depend on additional structure: normality and vanishing surgery obstruction in the surgery-theoretic framework, low-dimensional obstruction theory in the \(1\)-connected theorem, or special computations for products of lens spaces and Iwase-type manifolds. These hypotheses are essential for the methods currently available, even though they are not known to be intrinsic to the conjecture itself [2008.06002]. Two limitations stand out in the recent literature. First, the connected-sum maneuver that removes surgery obstructions for LS-category does not extend transparently to general sectional category, because there is no clear general notion of connected sum of fibrations and no analogue of \[ \operatorname{cat}(M_1\# M_2)=\max\{\operatorname{cat}(M_1),\operatorname{cat}(M_2)\} \] for arbitrary sectional-category invariants [2109.08011]. Second, the unresolved candidate counterexamples are all product-sensitive: lens-space products for small arithmetic parameters and the mixed Iwase product \(M_2\times M_3\) [2011.04819]. In this sense, Rudyak’s conjecture now functions less as an isolated statement and more as a unifying problem linking degree one maps, Ganea–Schwarz theory, surgery obstruction, classifying maps, spin and non-spin phenomena, and higher motion-planning invariants. The known results strongly constrain possible counterexamples, but they do not yet determine whether the conjecture is universally true. Source: https://www.emergentmind.com/topics/rudyak-s-conjecture