Group Of Self-homotopy Equivalences Research Articles

Smooth (non)rigidity of piecewise cusped locally symmetric manifolds

We define \emph{piecewise rank 1} manifolds, which are aspherical manifolds that generally do not admit a nonpositively curved metric but can be decomposed into pieces that are diffeomorphic to finite volume, irreducible, locally symmetric, nonpositively curved manifolds with $\pi_1$-injective cusps. We prove smooth (self) rigidity for this class of manifolds in the case where the gluing preserves the cusps' homogeneous structure. We compute the group of self homotopy equivalences of such a manifold and show that it can contain a normal free abelian subgroup and thus, can be infinite. Elements of this abelian subgroup are twists along elements in the center of the fundamental group of a cusp.

Nilpotency of self homotopy equivalences with coefficients

In this paper we study the nilpotency of certain groups of self homotopy equivalences. Our main goal is to extend, to localized homotopy groups and/or homotopy groups with coefficients, the general principle of Dror and Zabrodsky by which a group of self homotopy equivalences of a finite complex which acts nilpotently on the homotopy groups is itself nilpotent.

A quotient group of the group of self homotopy equivalences of SO(4)

The author studies the quotient group $\mathscr{E}$(SO(4))/$\mathscr{E}$#(SO(4)), where $\mathscr{E}$(SO(4)) is the group of homotopy classes of self homotopy equivalences of the rotation group SO(4) and $\mathscr{E}$#(SO(4)) is the subgroup of it consisting of elements that induce the identity on homotopy groups.

The group of self homotopy equivalences of some localized aspherical complexes

By studying the group of self homotopy equivalences of the localization (at a prime p and/or zero) of some aspherical complexes, we show that, contrary to the case when the considered space is a nilpotent, ℰm #(Xp ) is in general different from ℰm #(X)p. That is the case even when X = K (G, 1) is a finite complex and/or G satisfies extra finiteness or nilpotency conditions, for instance, when G is finite or virtually nilpotent. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Localization and completion of nilpotent groups of automorphisms

We study nilpotent subgroups of automorphism groups in the category of groups and the homotopy category of spaces. We establish localization and completion theorems for nilpotent groups of automorphisms of nilpotent groups. We then apply these algebraic theorems to prove analogous results for certain groups of self-homotopy equivalences of spaces.

$\boldsymbol {\pi _*}$-kernels of Lie groups

We study a filtration on the group of homotopy classes of self maps of a compact Lie group associated with homotopy groups. We determine these filtrations of $SU(3)$ and $Sp(2)$ completely. We introduce two natural invariants $lz_p(X)$ and $sz_p(X)$ defined by the filtration, where $p$ is a prime number, and compute the invariants for simple Lie groups in the cases where Lie groups are $p$-regular or quasi $p$-regular. We apply our results to the groups of self homotopy equivalences.

A group of self homotopy equivalences of SO(4)

A group of self homotopy equivalences of SO(4)

A bound for the nilpotency of a group of self homotopy equivalences

Let E Ω ( X ) \mathcal {E}_\Omega (X) be the group of homotopy classes of self-homotopy equivalences of X X such that Ω f ≃ 1 d Ω X \Omega f\simeq 1d_{\Omega X} . We prove that E Ω ( X ) \mathcal {E}_\Omega (X) is a nilpotent group and that nil ⁡ E Ω ( X ) ≤ cat ⁡ ( X ) − 1 \operatorname {nil} \mathcal {E}_{\Omega }(X)\le \operatorname {cat}(X)-1 .

Algebraic Invariants of Simple 4-Knots

We propose as an algebraic invariant for a simple 4-knot K with exterior X the triple (L, η, [λ]), where L = Z ⊕ π2(X)⊕π3(X) is a commutative graded ring with unit whose multiplication in positive degrees is determined by Whitehead product, η is composition with the Hopf map and [λ] is the orbit of the homotopy class of the longitude in π4(X) under the group of self homotopy equivalences of the universal covering space X′ which induce the identity on L. If K is fibred these invariants determine the fibre, and the natural Z[t,t-1]-module structures on the homotopy groups capture part of the monodromy. Every such triple with L finitely generated as an abelian group (and satisfying the order obviously necessary conditions) may be realized by some fibred simple 4-knot. In certain cases we can show that the triple determines the knot up to a finite ambiguity.

Finiteness properties of self-equivalence groups of rationalco-H-spaces

The group of self-homotopy equivalences of a finite complex which is rationally aco-H-space is studied. Some finiteness properties are obtained. Two subgroups consisting of elements which induce the identity on homotopy or homology groups are also studied. Examples are included showing these results are best possible.

Note on phantom phenomena and groups of self-homotopy equivalences

Note on phantom phenomena and groups of self-homotopy equivalences

Nilpotent subgroups of the group of self-homotopy equivalences

In this paper, we study subgroups of self-homotopy equivalences associated to generalized homology theories. We generalize Dror-Zabrodsky’s nilpotency theorem on the group of self-homotopy equivalences.

Rational homotopy equivalences of Lie type

Consider K⊂G, a proper pair of equal rank compact connected Lie groups. It is known (see [6], proof of theorem 1·1) that the group of self-homotopy equivalences of the rationalization of G/K is (anti)isomorphic, in a natural way, to the group of graded algebra automorphisms of H*(G/K; ℚ). On one side of the matter there is the fact that AutH*(G/K; ℚ) almost gives the integral picture of the group of homotopy classes of the self-homotopy equivalences of G/K, that is, up to a finite ambiguity and up to grading automorphisms of H*(G/K; ℚ) (i.e. those acting on H2i as λi.id, for some non-zero λ∈ℚ): see also [6]. On the other side there is the classical description by Borel[2] of H*(G/K; ℚ) in terms of invariants of Weyl groups, which gives hope for a satisfactory understanding of AutH*(G/K; ℚ).

The group of self-homotopy equivalences of principal three sphere bundles over the seven sphere

1. The set ℰ (X) of homotopy classes of self-homotopy equivalences of a space X forms a group under composition. In this note I complete the calculation of ℰ (X) when X is a torsion free rank 2 h-space, and give ℰ (X) up to extension for the remaining principal S3 bundles over S7.

Groups of self homotopy equivalences of induced spaces

Groups of self homotopy equivalences of induced spaces

(Co)homology self-closeness numbers of simply-connected spaces

The (co)homology self-closeness number of a simply-connected based CW-complexes $X$ is the minimal number $k$ such that any self-map $f$ of $X$ inducing an automorphism of the (co)homology groups for dimensions$\leq k$ is a self-homotopy equivalence. These two numbers are homotopy invariants and have a close relation with the group of self-homotopy equivalences. In this paper, we compare the (co)homology self-closeness numbers of spaces in certain cofibrations, define the mod $p$ (co)homology self-closeness number of simply-connected $p$-local spaces with finitely generated homologies and study some properties of the (mod $p$) (co)homology self-closeness numbers.