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In this paper, we investigate a Cartan calculus on the homology of free loop spaces which is introduced by Kuribayashi, Wakatsuki, Yamaguchi and the author [ Cartan calculi on the free loop spaces , Preprint, arXiv:2207.05941, 2022]. In particular, it is proved that the Cartan calculus can be described by the loop product and the loop bracket in string topology. Moreover, by using the descriptions, we show that the loop product behaves well with respect to the Hodge decomposition of the homology of free loop spaces. [ABSTRACT FROM AUTHOR]

The B-operators (abbreviation for Brownian-type operators) are upper triangular 2\times 2 block matrix operators that satisfy certain algebraic constraints. The purpose of this paper is to characterize the weak, the strong and the uniform stability of B-operators, respectively. This is achieved by giving equivalent conditions for the convergence of powers of a B-operator in each of the corresponding topologies. A more subtle characterization is obtained for B-operators with subnormal (2,2) entry. The issue of the strong stability of the adjoint of a B-operator is also discussed. [ABSTRACT FROM AUTHOR]

A subset A of a vector space X is called \alpha-lineable whenever A contains, except for the null vector, a subspace of dimension \alpha. If X has a topology, then A is \alpha-spaceable if such subspace can be chosen to be closed. The vast existing literature on these topics has shown that positive results for lineability and spaceability are quite common. Recently, the stricter notions of (\alpha,\beta)-lineability/spaceability were introduced as an attempt to shed light on more subtle issues. In this paper, among other results, we prove some general criteria for the notion of (\alpha,\beta)-lineability/spaceability and, as applications, we extend recent results of different authors. [ABSTRACT FROM AUTHOR]

The main result of this paper is that, under PFA, for every regular space X with F(X) = \omega we have |X| \le w(X)^\omega; in particular, w(X) \le \mathfrak {c} implies |X| \le \mathfrak {c}. This complements numerous prior results that yield consistent examples of even compact Hausdorff spaces X with F(X) = \omega such that w(X) = \mathfrak {c} and |X| = 2^\mathfrak {c}. We also show that regularity cannot be weakened to the Hausdorff property in this result because we can find in ZFC a Hausdorff space X with F(X) = \omega such that w(X) = \mathfrak {c} and |X| = 2^\mathfrak {c}. In fact, this space X has the strongly anti-Urysohn (SAU) property that any two infinite closed sets in X intersect, which is much stronger than F(X) = \omega. Moreover, any non-empty open set in X also has size 2^\mathfrak {c}, and thus our example answers one of the main problems of both Juhász, Soukup, and Szentmiklóssy [Topology Appl. 213 (2016), pp. 8–23] and Juhász, Shelah, Soukup, and Szentmiklóssy [Topology Appl. 323 (2023), Paper No. 108288, 15 pp.] by providing in ZFC a SAU space with no isolated points. [ABSTRACT FROM AUTHOR]

We explain the existence of open sets of complex quasi-modes in terms of Hörmander's commutator condition. [ABSTRACT FROM AUTHOR]

This paper deals with the existence of periodic solutions and attractiveness for some partial functional differential equations in Banach spaces. We assume that the first linear part generates a strongly continuous semigroup, while the delayed part is periodic with respect to the first argument. We prove that the existence of a bounded solution implies the existence of a periodic solution. Several results regarding uniqueness and global attractiveness of periodic solutions are also established. The analysis relies on a fixed point theorem of Chow and Hale's type and uses some arguments of weak topology. Our theorems extend in a broad sense some new and classical related results. An application to a transport equation with delay is also presented. [ABSTRACT FROM AUTHOR]

In this paper, Lie group representations on Hilbert spaces are studied in relation with operator ranges. Let R be an operator range of a Hilbert space H. Given the set Λ of R-invariant operators, and given a Lie group representation ρ : G → GL(H), we discuss the induced semigroup homomorphism ρ : ρ−1(Λ) → B(R) for the operator range topology on R. In one direction, we work under the assumption ρ−1 (Λ) = G, so ρ : G →B(R) is in fact a group representation. In this setting, we prove that ρ is continuous (and smooth) if and only if the tangent map dρ is R-invariant. In another direction, we prove that for the tautological representations of unitary or invertible operators of an arbitrary infinite-dimensional Hilbert space H, the set ρ−1(Λ) is neither a group for a large set of nonclosed operator ranges R nor closed for all nonclosed operator ranges R. Both results are proved by means of explicit counterexamples. [ABSTRACT FROM AUTHOR]

In this paper, we generalize topological results known for noncompact manifolds with nonnegative Ricci curvature to spaces with nonnegative N-Bakry Émery Ricci curvature. We study the Splitting Theorem and a property called the geodesic loops to infinity property in relation to spaces with nonnegative N-Bakry Émery Ricci curvature. In addition, we show that if Mn is a complete, noncompact Riemannian manifold with nonnegative N-Bakry Émery Ricci curvature where N > n, then Hn−1(M,Z) is 0. [ABSTRACT FROM AUTHOR]

This paper presents a simple proof of Wloc2,2 regularity of Lipschitz uniform local minimizers of vectorial variational problems. The method is based on the idea that inner variations provide constraints on the structure of singularities of local minimizers. [ABSTRACT FROM AUTHOR]

In this paper we show that certain generalizations of the Cr-Whitney topology, which include the Hölder-Whitney and Sobolev-Whitney topologies on smooth manifolds, satisfy the Baire property, to wit, the countable intersection of open and dense sets is dense. [ABSTRACT FROM AUTHOR]

We consider the iteration of quasiregular maps of transcendental type from Rd to Rd. In particular we study quasi-Fatou components, which are defined as the connected components of the complement of the Julia set. Many authors have studied the components of the Fatou set of a transcendental entire function, and our goal in this paper is to generalise some of these results to quasi-Fatou components. First, we study the number of complementary components of quasi-Fatou components, generalising, and slightly strengthening, a result of Kisaka and Shishikura. Second, we study the size of quasi-Fatou components that are bounded and have a bounded complementary component. We obtain results analogous to those of Zheng, and of Bergweiler, Rippon and Stallard. These are obtained using techniques which may be of interest even in the case of transcendental entire functions. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a topology on the set of isomorphism classes of finitely generated modules over an associative algebra. Then we focus on the relative topology on the set of isomorphism classes of maximal Cohen-Macaulay modules over a Cohen-Macaulay local ring. We discuss the irreducible components over certain hypersurfaces. [ABSTRACT FROM AUTHOR]

A well-known result in [Hsu-Smith-Waltman, Trans. Amer. Math. Soc. (1996)] states that in a competitive semiflow defined on X+ = X+1 × X+2, the product of two cones in respective Banach spaces, if (u*, 0) and (0,v*) are the global attractors in X+1 × {0} and {0} × X+2 respectively, then one of the following three outcomes is possible for the two competitors: either there is at least one coexistence steady state, or one of (u*, 0), (0,v*) attracts all trajectories initiating in the order interval I = [0, u*] × [0, v*]. However, it was demonstrated by an example that in some cases neither (u*,0) nor (0,v*) is globally asymptotically stable if we broaden our scope to all of X+. In this paper, we give two sufficient conditions that guarantee, in the absence of coexistence steady states, the global asymptotic stability of one of (u*, 0) or (0, v*) among all trajectories in X+. Namely, one of (u*,0) or (0,v*) is (i) linearly unstable, or (ii) linearly neutrally stable but zero is a simple eigenvalue. Our results complement the counterexample mentioned in the above paper as well as applications that frequently arise in practice. [ABSTRACT FROM AUTHOR]

We prove that order convergence on a Boolean algebra turns it into a compact convergence space if and only if this Boolean algebra is complete and atomic. We also show that on an Archimedean vector lattice, order intervals are compact with respect to order convergence if and only the vector lattice is complete and atomic. Additionally we provide a direct proof of the fact that uo convergence on an Archimedean vector lattice is induced by a topology if and only if the vector lattice is atomic. [ABSTRACT FROM AUTHOR]

Let X be a completely regular Hausdorff space, let E and F denote Banach spaces. Let C_b(X,E) denote the space of E-valued bounded continuous functions on X and let \beta be the strict topology on this space. We establish the relationship between nuclear operators T:C_b(X,E)\rightarrow F between the locally convex space (C_b(X,E),\beta) and the Banach space F and their representing operator-valued Borel measures. [ABSTRACT FROM AUTHOR]

Dana P. Williams raised in [Proc. Amer. Math. Soc., Ser. B 3 (2016), pp. 1-8] the following question: Must a second countable, locally compact, transitive groupoid have open range map? This paper gives a negative answer to that question. Although a second countable, locally compact transitive groupoid G may fail to have open range map, we prove that we can replace its topology with a topology which is also second countable, locally compact, and with respect to which G is a topological groupoid whose range map is open. Moreover, the two topologies generate the same Borel structure and coincide on the fibres of G. [ABSTRACT FROM AUTHOR]

Measure differential equations can model non-classical problems like the quantum effects. In this paper we will solve extremal problems for eigenvalues of measure differential equations by exploiting the approximation of general measures by smooth measures and the continuity results of eigenvalues in weak* topology of measures. [ABSTRACT FROM AUTHOR]

There is a long-standing problem, posed by A.T.-M. Lau [ Fixed point theory and its applications , Academic Press, New York-London, 1976, pp. 121–129], whether left amenability is sufficient to ensure the existence of a common fixed point for every jointly weak^{\ast } continuous nonexpansive semigroup action on a nonempty weak^{\ast } compact convex set in a dual Banach space. In this note we discuss the current status of this problem and give a partial solution in the case of weak^{\ast } compact convex sets with the Radon–Nikodým property. [ABSTRACT FROM AUTHOR]

As a generalization of almost everywhere convergence to vector lattices, unbounded order convergence has garnered much attention. The concept of boundedly uo-complete Banach lattices was introduced by N. Gao and F. Xanthos, and has been studied in recent papers by D. Leung, V. G. Troitsky, and the aforementioned authors. We will prove that a Banach lattice is boundedly uo-complete iff it is monotonically complete. Afterwards, we study completeness-type properties of minimal topologies; minimal topologies are exactly the Hausdorff locally solid topologies in which uo-convergence implies topological convergence. [ABSTRACT FROM AUTHOR]

In this paper we consider homeomorphisms of the torus ℝ2/ℤ2, homotopic to the identity, and their rotation sets. Let f be such a homeomorphism, ...: ℝ2 → ℝ2 be a fixed lift and ρ(...) ⊂ ℝ2 be its rotation set, which we assume to have interior. We also assume that the frontier of ρ(...) contains a rational vector ρ ∈ ℚ2 and we want to understand how stable this situation is. To be more precise, we want to know if it is possible to find two different homeomorphisms f1 and f2, arbitrarily small C0-perturbations of f, in a way that ρ does not belong to the rotation set of ... but belongs to the interior of the rotation set of ..., where ... and ... are the lifts of f1 and f2 that are close to .... We give two examples where this happens, supposing ρ = (0, 0). The first one is a smooth diffeomorphism with a unique fixed point lifted to a fixed point of .... The second one is an area preserving version of the first one, but in this conservative setting we only obtain a C0 example. We also present two theorems in the opposite direction. The first one says that if f is area preserving and analytic, we cannot find f1 and f2 as above. The second result, more technical, implies that the same statement holds if f belongs to a generic one parameter family (ft)t∈[0,1] of C2-diffeomorphisms of T2 (in the sense of Brunovsky). In particular, lifting our family to a family ... of plane diffeomorphisms, one deduces that if there exists a rational vector ρ and a parameter t∗ ∈ (0, 1) such that ρ(...) has non-empty interior, and ρ ∉ ρ(...) for t < t∗ close to t∗, then ρ ∉ int(ρ(...)) for all t > t∗ close to t∗. This kind of result reveals some sort of local stability of the rotation set near rational vectors of its boundary. [ABSTRACT FROM AUTHOR]

In this paper, we study the compatibility between Talagrand's inequality and the concentration topology; i.e., if a sequence of mm-spaces satisfying Talagrand's inequality converges with respect to the observable distance, then the limit space satisfies Talagrand's inequality. [ABSTRACT FROM AUTHOR]

We show that any (norm-closed two-sided) ideal of a unital C*-algebra that comes equipped with certain natural quantum metric structure is a metrized quantum vector bundle, when canonically viewed as a module over A. Next, given a unital AF-algebra A equipped with a faithful tracial state, we equip each ideal of A with a metrized quantum vector bundle structure using previous work of the first author and Latrémolière. Moreover, we show that convergence of ideals in the Fell topology implies convergence of the associated metrized quantum vector bundles in the modular Gromov-Hausdorff propinquity of Latrémolière. In a similar vein but requiring a different approach, given a compact metric space (X,d), we equip each ideal of C(X) with a metrized quantum vector bundle structure, and show that convergence in the Fell topology implies convergence in the modular Gromov-Hausdorff propinquity. [ABSTRACT FROM AUTHOR]

In 1974, S.-S. Chern and J. Simons published a paper where they defined a new type of characteristic class, one that depends not just on the topology of a manifold but also on the geometry. The goal of this paper is to investigate what kinds of geometric information is contained in these classes by studying their behavior under the Ricci flow. In particular, it is shown that the Chern-Simons class corresponding to the first Pontryagin class is invariant under the Ricci flow on the warped products S² ×f S¹ and S¹ ×f S² but that this class is not invariant under the Ricci flow on a generalized Berger sphere. [ABSTRACT FROM AUTHOR]

In this paper we consider some open questions concerning ?- convergence in complete CAT(0) metric spaces (i.e. Hadamard spaces). Suppose (X, d) is a Hadamard space such that the sets {z ? X| d(x, z) = d(z, y)} are convex for each x, y ? X. We introduce a so-called half-space topology such that convergence in this topology is equivalent to ?-convergence for any sequence in X. For a major class of Hadamard spaces, our results answer positively open questions nos. 1, 2 and 3 in [W. A. Kirk and B. Panyanak, A concept of convergence in geodesic spaces, Nonlinear Anal., 68 (2008) 3689- 3696]. Moreover, we give a new characterization of ?-convergence and a new topology that we call the weak topology via a concept of a dual metric space. The relations between these topologies and the topology which is induced by the distance function have been studied. The paper concludes with some examples. [ABSTRACT FROM AUTHOR]

In this paper we prove that for every unbounded convex closed set C in c0 there exists a nonexpansive mapping T : C → C which is fixed point free. This result solves in a negative sense a question that has remained open for some time in Metric Fixed Point Theory. [ABSTRACT FROM AUTHOR]

This paper compares two models of the equivariant homotopy type of the smash powers of a spectrum, namely the "Bökstedt smash product" and the Hill-Hopkins-Ravenel norm. [ABSTRACT FROM AUTHOR]

In \mathbb {R}^d, a closed, convex set has zero Lebesgue measure if and only its interior is empty. More generally, in separable, reflexive Banach spaces, closed and convex sets are Haar null if and only if their interior is empty. We extend this facts by showing that a closed, convex set in a separable Banach space is Haar null if and only if its weak^* closure in the second dual has empty interior with respect to the norm topology. It then follows that, in the metric space of all nonempty, closed, convex and bounded subsets of a separable Banach space, converging sequences of Haar null sets have Haar null limits. [ABSTRACT FROM AUTHOR]

In this paper, we consider the self-dual Chern-Simons CP(1) model in the whole plane R². After reducing to a single equation, we prove the uniqueness of topological multivortex solutions for the model if the Chern-Simons coupling parameter is sufficiently large. [ABSTRACT FROM AUTHOR]

In this paper, we study the topology of complete noncompact Riemannian manifolds with asymptotically nonnegative Ricci curvature and large volume growth. We prove that they have finite topological types under some curvature decay and volume growth conditions. We also generalize it to the manifolds with kth asymptotically nonnegative Ricci curvature by using extensions of Abresch-Gromoll's excess function estimate. [ABSTRACT FROM AUTHOR]

Sz.-Nagy's famous theorem states that a bounded operator T which acts on a complex Hilbert space H is similar to a unitary operator if and only if T is invertible and both T and T-1 are power bounded. There is an equivalent reformulation of that result which considers the self-adjoint iterates of T and uses a Banach limit L. In this paper first we present a generalization of the necessity part in Sz.-Nagy's result concerning operators that are similar to normal operators. In the second part we provide a characterization of all possible strong operator topology limits of the self-adjoint iterates of those contractions which are similar to unitary operators and act on a separable infinite-dimensional Hilbert space. This strengthens Sz.-Nagy's theorem for contractions. [ABSTRACT FROM AUTHOR]

We show that the space RpRc(Wg2n) of metrics with positive Ricci curvature on the manifold Wg2n := ♯g (Sn × Sn) has nontrivial rational homology if n ≢ 3 (mod 4) and g are both sufficiently large. The same argument applies to RpRc(Wg2n ♯ N) provided that N is spin and Wg2n ♯ N admits a Ricci positive metric. [ABSTRACT FROM AUTHOR]

A classical theorem due to R. Phelps states that if C is a weakly compact set in a Banach space E, the strongly exposing functionals form a dense subset of the dual space E'. In this paper, we look at the concrete situation where C ⊂ L1(ℝd) is the closed convex hull of the set of random variables Y ∊ L1(ℝd) having a given law ν. Using the theory of optimal transport, we show that every random variable X ∊ L∞(ℝd), the law of which is absolutely continuous with respect to the Lebesgue measure, strongly exposes the set C. Of course these random variables are dense in L∞(ℝd). [ABSTRACT FROM AUTHOR]

We prove the existence of lattice isomorphic line arrangements having π1-equivalent or homotopy-equivalent complements and non-homeomorphic embeddings in the complex projective plane. We also provide two explicit examples: one is formed by real-complexified arrangements, while the second is not. [ABSTRACT FROM AUTHOR]

We settle all problems concerning the additivity of the Gerlits- Nagy property and related additivity numbers posed by Scheepers in his tribute paper to Gerlits. We apply these results to compute the minimal number of concentrated sets of reals (in the sense of Besicovitch) whose union, when multiplied with a Gerlits-Nagy space, need not have Rothberger's property. We apply these methods to construct a large family of spaces whose product with every Hurewicz space has Menger's property. Our applications extend earlier results of Babinkostova and Scheepers. [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce a polynomial invariant fK(t) for virtual knots. We show that fK(t) can be used to distinguish some virtual knot from its inverse and mirror image. The behavior of fK(t) under a connected sum is also given. Finally, we discuss which kinds of polynomials can be realized as fK(t) for some virtual knot K. [ABSTRACT FROM AUTHOR]

In this paper we will establish some semi-uniform ergodic theorems for skew-product transformations with discontinuity from the point of view of topology. The main assumptions are that the discontinuity sets of transformations are neglected in some measure-theoretical sense. The theorems have extended the classical results which have been established for continuous dynamical systems. [ABSTRACT FROM AUTHOR]

Let X be a Noetherian scheme, K(FlatX) be the homotopy category of flat quasi-coherent OX-modules and Kp(FlatX) be the homotopy category of all flat complexes. It is shown that the pair (Kp(FlatX), K (dg- CofX)) is a complete cotorsion theory in K(FlatX), where K (dg-CofX) is the essential image of the homotopy category of dg-cotorsion complexes of flat modules. Then we study the homotopy category K(dg-Cof X). We show that in the affine case, this homotopy category is equal with the essential image of the embedding functor j* : K(ProjR) -? K(FlatR) which has been studied by Neeman in his recent papers. Moreover, we present a condition for the inclusion K(dg-Cof X) ⊆ K(Cof X) to be an equality, where K(Cof X) is the essential image of the homotopy category of complexes of cotorsion flat sheaves. [ABSTRACT FROM AUTHOR]

Assume that X is a metric space and B ⊂ X is compact. Let f : B → X be a continuous map homotopic to g, the selfmap of B. The aim of this paper is to present a method for detecting fixed points of f. It is based on the notion of the Wa?zewski set for the homotopy F between g and f. [ABSTRACT FROM AUTHOR]

In this paper we investigate multinormed W*-algebras in terms of the central topologies of W*-algebras. The main result asserts that each multinormed W*-algebra can be realized as a local von Neumann algebra on a certain domain in a Hilbert space. Moreover, it admits the predual (unique up to an isometry), which is the ℓ1-normed space. In the normed case the assertion is reduced to the known Sakai theorem. [ABSTRACT FROM AUTHOR]

This paper records two results which were inexplicably omitted from the paper on Pin structures on low dimensional manifolds in the LMS Lecture Note Series, volume 151, by Kirby and this author. Kirby declined to be listed as a coauthor of this paper. par A $Pin^{-}$-structure on a surface $X$ induces a quadratic enhancement of the mod $2$ intersection form, $qcolon H_1(X;mathbb {Z}/2mathbb {Z})to mathbb {Z}/4mathbb {Z}$. par Theorem 1.1 says that $q$ vanishes on the kernel of the map in homology to a bounding $3$-manifold. This is used by Kreck and Puppe in their paper in Homology, Homotopy and Applications, volume 10. The arXiv version, arXiv:0707.1599 [math.AT], referred to an email from the author to Kreck for the proof. A more polished and public proof seems desirable. par In Section 6 of the paper with Kirby, a $Pin^{-}$-structure is constructed on a surface $X$ dual to $w_2$ in an oriented 4-manifold, $M^4$. Theorem 2.1 says that $q$ vanishes on the Poincaré dual to the image of $H^1(M;mathbb {Z}/2mathbb {Z})$ in $H^1(X;mathbb {Z}/2mathbb {Z})$. [ABSTRACT FROM AUTHOR]

In the present paper, slightly modifying the topological KKM Theorem of Park and Kim (1996), we obtain a new existence theorem for generalized vector equilibrium problems related to an admissible multifunction. We work here under the general framework of G-convex space which does not have any linear structure. Also, we give applications to greatest element, fixed point and vector saddle point problems. The results presented in this paper extend and unify many results in the literature by relaxing the compactness, the closedness and the convexity conditions. [ABSTRACT FROM AUTHOR]

The effect of matrix perturbations on the polar decomposition has been studied by several authors and various results are known. However, for operators between infinite-dimensional spaces the problem has not been considered so far. Here, we prove in particular that the partial isometry in the polar decomposition of an operator is stable under perturbations, given that kernel and range of original and perturbed operator satisfy a certain condition. In the matrix case, this condition is weaker than the usually imposed equal-rank condition. It includes the case of semi-Fredholm operators with agreeing nullities and deficiencies, respectively. In addition, we prove a similar perturbation result where the ranges or the kernels of the two operators are assumed to be sufficiently close to each other in the gap metric. [ABSTRACT FROM AUTHOR]

Examples show that in three dimensions the natural topology on the space of equivalence classes of CR structures need not be Hausdorff. However, the space of equivalence classes of compact aspherical CR structures is Hausdorff. Thus the non-Hausdorff property is due to spherical or Levi-flat points. [ABSTRACT FROM AUTHOR]

We prove a filtered version of the Homotopy Transfer Theorem which gives an A-infinity algebra structure on any page of the spectral sequence associated to a filtered dg-algebra. We then develop various applications to the study of the geometry and topology of complex manifolds, using the Hodge filtration, as well as to complex algebraic varieties, using mixed Hodge theory. [ABSTRACT FROM AUTHOR]

Let XR be the zero locus in RPn of one or two independently and Kostlan distributed random real quadratic forms (this is equivalent to the corresponding symmetric matrices being in the Gaussian Orthogonal Ensemble). Denoting by b(XR) the sum of the Betti numbers of XR, we prove that (1)...The methods we use combine random matrix theory, integral geometry and spectral sequences: for one quadric hypersurface it is simply a corollary of Wigner's semicircle law; for the intersection of two quadrics it is related to the (intrinsic) volume of the set of singular symmetric matrices of norm one. [ABSTRACT FROM AUTHOR]

A classical theorem due to R. Phelps states that if C is a weakly compact set in a Banach space E, the strongly exposing functionals form a dense subset of the dual space E'. In this paper, we look at the concrete situation where C ⊂ L1(ℝd) is the closed convex hull of the set of random variables Y ∊ L1(ℝd) having a given law ν. Using the theory of optimal transport, we show that every random variable X ∊ L∞(ℝd), the law of which is absolutely continuous with respect to the Lebesgue measure, strongly exposes the set C. Of course these random variables are dense in L∞(ℝd). [ABSTRACT FROM AUTHOR]

Iwase [Topology 42 (2003), pp. 701–713] has constructed two 16-dimensional manifolds M_2 and M_3 with LS-category 3 which are counter-examples to Ganea's conjecture: \operatorname {cat} (M\times S^n)=\operatorname {cat} M+1. We show that the manifold M_3 is a counter-example to the logarithmic law for the LS-category of the square of a manifold: \operatorname {cat}(M\times M)=2\operatorname {cat} M. Also we construct a map of degree one \begin{equation*} f:2(M_3\times S^2\times S^{14})\#-(M_2\times S^2\times S^{14})\to M_2\times M_3 \end{equation*} which reduces Rudyak's conjecture to the question whether \operatorname {cat}(M_2\times M_3)\ge 5 and show that \operatorname {cat}(M_2\times M_3)\ge 4. [ABSTRACT FROM AUTHOR]