Marek Golasiński

Spherical Space Forms — Homotopy Types and Self-equivalences

Let Q 4m be a generalized quaternion group and X (n) an n-dimensional CW-complex with the homotopy type of an n-sphere. We compute the number of distinct homotopy types of spherical space forms with respect to all Q 4m-actions on all CW-complexes X (4n − 1) and deduce an existence of finite space forms given by some free cellular Q 4m-actions on Q CW-complexes X (3) which do not have the homotopy type of a Clifford-Klein form provided that m is not a power of 2. We show that homotopy types of (2mn − 1)-space forms with respect to G-actions are exhausted by homotopy types of orbit spaces of joins of CW-complexes X (2n − 1) with appropriate G-actions

Polynomial and Regular Maps into Grassmannians

We find a regular deformation retractionn,r (K) :I dem n,r (K) → Gn,r (K) from the manifold Idemn,r (K) of idempotent n × n matrices with rank r to the Grassmannian manifold Gn,r (K) over K the reals, complex numbers or quaternions. Then we derive an injection P C(VC, Idemn,r (K)) → RR(V, Gn,r (K)) from the sets of homotopy classes of complex-valued polynomial to such a set of real-valued regular maps, where VC denotes the Zariski closure in the affine space C n of a subset V ⊆ R n . Furthermore, we list complex-valued polynomial maps S 2 → S 2 of any Brouwer degree and deduce that the map � 2,1(C) :I dem2,1(C) → G2,1(C) yields an isomorphism PC(S 2 , S 2 ) ∼ −→ RR(S 2 , S 2 ) of cyclic infinite homotopy groups. Finally, we show that every nonzero even Brouwer degree of the spheres S n and S n cannot be realized by a real-valued (resp. complex-valued) homogeneous polynomial map provided that n is even.

Generalized Eilenberg–Zilber Type Theorem and its Equivariant Applications

Abstract We state the Eilenberg–Zilber type theorem for the product of two small categories and consider the Bredon homology of an equivariant space with local coefficients in the light of homologies of small categories. Then, in particular, an exterior product, Kunneth formula and cap-product for the Bredon Homology of equivariant spaces with local coefficients are established.

Gottlieb and Whitehead Center Groups of Spheres, Projective and Moore Spaces

Introduction.- Gottlieb groups of Spheres.- Gottlieb and Whitehead Center Groups of Projective Spaces.- Gottlieb and Whitehead Center Groups of Moore Spaces.

Automorphism Groups of Generalized (Binary) Icosahedral, Tetrahedral and Octahedral Groups

Let G be any of the (binary) icosahedral, generalized octahedral (tetrahedral) groups or their quotients by the center. We calculate the automorphism group Aut(G).

On automorphisms of finite Abelian p-groups

Let A be a finitely generated abelian group. We describe the automorphism group Aut(A) using the rank of A and its torsion part p-part Ap.For a finite abelian p-group A of type (k1, ..., kn), simple necessary and sufficient conditions for an n × n-matrix over integers to be associated with an automorphism of A are presented. Then, the automorphism group Aut(A) for a finite p-group A of type (k1, k2) is analyzed.

Gottlieb and Whitehead Center Groups of Projective Spaces

By the use of Siegel’s method and the classical results of homotopy groups of spheres and Lie groups, we determine in this chapter some Gottlieb groups of projective spaces or give the lower bounds of their orders. Furthermore, making use of the properties of Whitehead products, we determine some Whitehead center groups of projective spaces.

On the higher Whitehead product

Porter’s approach is used to derive some properties of higher order Whitehead products, similar to those ones for triple products obtained by Hardie. Computations concerning the higher order Whitehead product for spheres and projective spaces are presented as well.

On path-components of the mapping spaces \(M(\mathbb {S}^m,\mathbb {F}P^n)\)

Faculty of Mathematics and Computer Science University of Warmia and Mazury, Sloneczna 54 Street

On the spectralization of affine and perfectly normal spaces

Abstract We show that for a field K and n ≥ 1 {n\geq 1} , the soberification 𝒮 ⁢ ( 𝔸 n ⁢ ( K ) ) {\mathcal{S}(\mathbb{A}^{n}(K))} of the affine n-space 𝔸 n ⁢ ( K ) {\mathbb{A}^{n}(K)} over K is homeomorphic to its spectralization ℬ ⁢ 𝒮 ⁢ ( 𝔸 n ⁢ ( K ) ) {\mathcal{BS}(\mathbb{A}^{n}(K))} , and it can be embedded into the spectrum Spec ⁢ ( K ⁢ [ X 1 , … , X n ] ) {\mathrm{Spec}(K[X_{1},\dots,X_{n}])} . Moreover, if the field K is algebraically closed, then there are homeomorphisms 𝒮 ⁢ ( 𝔸 n ⁢ ( K ) ) ≈ ℬ ⁢ 𝒮 ⁢ ( 𝔸 n ⁢ ( K ) ) ≈ Spec ⁢ ( K ⁢ [ X 1 , … , X n ] ) \mathcal{S}(\mathbb{A}^{n}(K))\approx\mathcal{BS}(\mathbb{A}^{n}(K))\approx% \mathrm{Spec}(K[X_{1},\dots,X_{n}]) . We also show that for a space X, the subspace z ⁢ Spec ⁢ ( C ⁢ ( X ) ) ⊆ Spec ⁢ ( C ⁢ ( X ) ) z\mathrm{Spec}(C(X))\subseteq\mathrm{Spec}(C(X)) of prime z-ideals of the ring C ⁢ ( X ) {C(X)} of real-valued continuous functions on X is homeomorphic to the space z ⁢ 𝒮 ⁢ ℛ ⁢ ( X ) {z\mathcal{SR}(X)} of prime z-filters with an appropriate topology and there is a homeomorphism ℬ ⁢ 𝒮 ⁢ ( X ) ≈ z ⁢ Spec ⁢ ( C ⁢ ( X ) ) {\mathcal{BS}(X)\approx z\mathrm{Spec}(C(X))} provided X is perfectly normal.