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Papers by Krzysztof Pawałowski

Fundamenta Mathematicae

In the theory of transformation groups, it is important to know what kind of isotropy subgroups o... more In the theory of transformation groups, it is important to know what kind of isotropy subgroups of G do occur at points of the space upon which the given group G acts. In this article, for a finite group G, we prove the Equivariant Bundle Subtraction Theorem (Theorem 2.2) which allows us to construct smooth G-manifolds with prescribed isotropy subgroups around the G-fixed point sets. In Theorem 0.1, we restate Oliver's result about manifolds M and G-vector bundles over M that occur, respectively, as the G-fixed point sets and their equivariant normal bundles for smooth G-actions on disks. In Theorems 0.2 and 0.3, we prove the corresponding results for smooth G-actions on disks with prescribed isotropy subgroups around M. In Theorems 0.4 and 0.5, for large classes of finite groups G, we explicitly describe manifolds M that occur as the G-fixed point sets for such actions on disks. These actions are expected to be useful for answering the question of which manifolds occur as the G-fixed points sets for smooth G-actions on spheres. 0. Introduction. In the theory of transformation groups, it often happens that a solution of a particular problem depends on the family of the isotropy subgroups that we allow to occur at points in the space upon which a given group G acts. In [O4], for a finite group G not of prime power order, Oliver describes necessary and sufficient conditions under which a smooth manifold M occurs as the G-fixed point set and a smooth G-vector ν over M stably occurs as the equivariant normal bundle of M in D (resp., E

Proceedings of the Steklov Institute of Mathematics

For a p-toral group G, we answer the question which compact (respectively, open) smooth manifolds... more For a p-toral group G, we answer the question which compact (respectively, open) smooth manifolds M can be diffeomorphic to the fixed point sets of smooth actions of G on compact (respectively, open) smooth manifolds E of the homotopy type of a finite ℤ-acyclic CW complex admitting a cellular map of period p, with exactly one fixed point. In the case where the CW complex is contractible, E can be chosen to be a disk (respectively, Euclidean space).

Bounds on the torus rank.- The equivariant wall finiteness obstruction and Whitehead torsion.- Ho... more Bounds on the torus rank.- The equivariant wall finiteness obstruction and Whitehead torsion.- Homotopy actions and cohomology of finite groups.- Normally linear Poincare complexes and equivariant splittings.- Free (?/2)k-actions and a problem in commutative algebra.- Verschlingungszahlen von Fixpunktmengen in Darstellungsformen. II.- An algebraic approach to the generalized Whitehead group.- Almost complex S1-actions on cohomology complex projective spaces.- A product formula for equivariant Whitehead torsion and geometric applications.- Balanced orbits for fibre preserving maps of S1 and S3 actions.- Involutions on 2-handlebodies.- Normal combinatorics of G-actions on manifolds.- Topological invariance of equivariant rational Pontrjagin classes.- On the existence of acyclic ? complexes of the lowest possible dimension.- Unstable homotopy theory of homotopy representations.- Duality in orbit spaces.- Cyclic homology and idempotents in group rings.- ?2 surgery theory and smooth involutions on homotopy complex projective spaces.- Proper subanalytic transformation groups and unique triangulation of the orbit spaces.- A remark on duality and the Segal conjecture.- On the bounded and thin h-cobordism theorem parameterized by ?k.- Algebraic and geometric splittings of the K- and L-groups of polynomial extensions.- Coherence in homotopy group actions.- Existence of compact flat Riemannian manifolds with the first Betti number equal to zero.- Which groups have strange torsion?.

Osaka Journal of Mathematics, 1999

Contemporary Mathematics, 2000

K-Monographs in Mathematics, 2002

Some applications of shifted subgroups in transformation groups.- Equivariant finiteness obstruct... more Some applications of shifted subgroups in transformation groups.- Equivariant finiteness obstruction and its geometric applications - A survey.- On conic spaces.- Computations of stable pseudoisotopy spaces for aspherical manifolds.- The fundamental groups of algebraic varieties.- Invariants of graphs and their applications to knot theory.- Morse theory of closed 1-forms.- Morava K-theories: A survey.- Examples of lack of rigidity in crystallographic groups.- Sur la Topologie des Bras Articules.- Semicontractible link maps and their suspensions.- The KO-assembly map and positive scalar curvature.- Equivariant splittings associated with smooth toral actions.- Lefschetz numbers of C*-complexes.- On the homotopy category of Moore spaces and an old result of Barratt.- An additive basis for the cohomology of real Grassmannians.- On the topology of the space of reachable symmetric linear systems.- Homotopy ring spaces and their matrix rings.- Homotopy colimits on E-I-categories.- On bordism rings with principal torsion ideal.- "Localization and the sullivan fixed point conjecture".- Characteristic numbers and group actions.- Remarks on one fixed point A 5-actions on homology spheres.- A note on the mod 2 cohomology of SL(?).- Characteristic classes and 2-modular representations for some sporadic simple groups - II.- The abelianization of the theta group in low genus.

Proceedings of the Edinburgh Mathematical Society

For any finite group G, we impose an algebraic condition, the Gnil-coset condition, and prove tha... more For any finite group G, we impose an algebraic condition, the Gnil-coset condition, and prove that any finite Oliver group G satisfying the Gnil-coset condition has a smooth action on some sphere with isolated fixed points at which the tangent G-modules are not isomorphic to each other. Moreover, we prove that, for any finite non-solvable group G not isomorphic to Aut(A6) or PΣL(2, 27), the Gnil-coset condition holds if and only if rG ≥ 2, where rG is the number of real conjugacy classes of elements of G not of prime power order. As a conclusion, the Laitinen Conjecture holds for any finite non-solvable group not isomorphic to Aut(A6).

Lecture Notes in Mathematics, 1986

Topology, 2003

In this article, we deal with the following two questions. For smooth actions of a given finite g... more In this article, we deal with the following two questions. For smooth actions of a given finite group G on spheres S, which smooth manifolds F occur as the fixed point sets in S ,a nd which realG-vector bundles ν over F occur as the equivariant normal bundles of F in S ?Wef ocus on the caseG is an Oliver

Proceedings of the American Mathematical Society, 2009

For smooth actions of G on spheres with exactly two fixed points, the Laitinen Conjecture propose... more For smooth actions of G on spheres with exactly two fixed points, the Laitinen Conjecture proposed an answer to the Smith question about the G-modules determined on the tangent spaces at the two fixed points. Morimoto obtained the first counterexample to the Laitinen Conjecture for G = Aut(A 6). By answering the Smith question for some finite solvable Oliver groups G, we obtain new counterexamples to the Laitinen Conjecture, presented for the first time in the case where G is solvable. 0. The Laitinen Conjecture In 1960, P. A. Smith [17] posed the following question. If a finite group G acts smoothly on a sphere with exactly two fixed points a and b, is it true that the real G-modules determined on the tangent spaces at a and b are always isomorphic? Smith Equivalence. For a finite group G, two real G-modules U and V are called Smith equivalent if as real G-modules, U ∼ = T a (S) and V ∼ = T b (S) for a smooth action of G on a homotopy sphere S with exactly two fixed points a and b.

Mathematica Slovaca, 2012

We construct smooth circle actions on symplectic manifolds with non-symplectic fixed point sets o... more We construct smooth circle actions on symplectic manifolds with non-symplectic fixed point sets or cyclic isotropy sets. All such actions are not compatible with any symplectic form on the manifold in question. In order to cover the case of non-symplectic fixed point sets, we use two smooth 4-manifolds (one symplectic and one non-symplectic) which become diffeomorphic after taking the products with the 2-sphere. The second type of actions is obtained by constructing smooth circle actions on spheres with non-symplectic cyclic isotropy sets, which (by the equivariant connected sum construction) we carry over from the spheres on products of 2-spheres. Moreover, by using the mapping torus construction, we show that periodic diffeomorphisms (isotopic to symplectomorphisms) of symplectic manifolds can provide examples of smooth fixed point free circle actions on symplectic manifolds with non-symplectic cyclic isotropy sets.

Algebraic & Geometric Topology, 2002

In 1960, Paul A. Smith asked the following question. If a finite group G acts smoothly on a spher... more In 1960, Paul A. Smith asked the following question. If a finite group G acts smoothly on a sphere with exactly two fixed points, is it true that the tangent G-modules at the two points are always isomorphic? We focus on the case G is an Oliver group and we present a classification of finite Oliver groups G with Laitinen number a G = 0 or 1. Then we show that the Smith Isomorphism Question has a negative answer and a G ≥ 2 for any finite Oliver group G of odd order, and for any finite Oliver group G with a cyclic quotient of order pq for two distinct odd primes p and q. We also show that with just one unknown case, this question has a negative answer for any finite nonsolvable gap group G with a G ≥ 2. Moreover, we deduce that for a finite nonabelian simple group G, the answer to the Smith Isomorphism Question is affirmative if and only if a G = 0 or 1.

Mathematische Annalen, 2008

We construct for the first time smooth circle actions on highly symmetric manifolds such as disks... more We construct for the first time smooth circle actions on highly symmetric manifolds such as disks, spheres, and Euclidean spaces which contain two points with the same isotropy subgroup whose representations determined on the tangent spaces at the two points are not isomorphic to each other. This allows us to answer negatively