The behavior on fundamental group of a free pro-homotopy equivalence

We study the question: given a morphism $\text{?{(X}{}_{\mathrm{n}}\text{, x}{}_{\mathrm{n}}\text{)}→{(Y}{}_{\mathrm{n}}\text{, y}{}_{\mathrm{<mtext>n</mtext>}}\text{)}}$ in the category pro-(Poi nted. Homotopy) where the domain and range are inverse sequences of well-pointed CW complexes, and given that ? induces an isomorphism {X_n}→{Y_n} in pro-(Homotopy), what additional hypotheses force ? to be an isomorphism in pro-(Pointed Homotopy)? Conjecture. If the dimensions of the Y_n's are bounded, then ? is an isomorphism in pro-(Pointed Homotopy). We first prove the special case of this conjecture in which dim Y_n?d<∞ for all n, and $\text{lim}\text{{H}{}_{\mathrm{d}}\text{Y}{}_{\mathrm{n}}\text{}≠0}$, Y_n being the universal cover of Y_n. Then we deal with the general case: we show that there are certain elements of each π₁Y_n with the properties: (i) these elements commute if and only if ? is an isomorphism in pro-(Pointed Homotopy); (ii) if dim Y_n?d<∞ for all n, then powers of these elements commute. While (i) and (ii) are not incompatible, this result puts severe restrictions on the nature of any possible counter-example to the conjecture.Our method also gives pro-homotopy analogues of the well-known fact that if a K(π, 1) is N-dimensional, then π is torsion-free and contains no abelian subgroup of rank>N. The latter theorems apply to inverse sequences {Y_n} of CW complexes where dim Y_n is finite but not necessarily bounded, hence in particular to infinite-dimensional shape-aspherical compacta.     

Complete affine locally flat manifolds with a free fundamental group

Examples of free noncommutative subgroups of the affine group A(3), which act properly discontinuously on ³, are constructed in the paper. These examples refute a conjecture of Milnor to the effect that the fundamental group of any complete affine locally flat manifold contains a sòlvable subgroup of finite index.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 134, pp. 190–205, 1984.     

The fundamental group of a unitary orbit

The unitary orbit of a complex n × n matrix A is simply connected if and only if the portion of the commutant {A} which resides in the special unitary group is path connected.     

The fundamental group of a compact metric space

We give a forcing free proof of a conjecture of Mycielski that the fundamental group of a connected locally connected compact metric space is either finitely generated or has the power of the continuum.

     

The fundamental group scheme of a non-reduced scheme

We extend the definition of fundamental group scheme to non-reduced schemes over any connected Dedekind scheme. Then we compare the fundamental group scheme of an affine scheme with that of its reduced part.     

The fundamental group of a triangular algebra without double bypasses

Let A be a basic connected finite dimensional algebra over a field of characteristic zero. A fundamental group depending on the presentation of A has been defined by several authors [see R. Martínez-Villa, J.A. de La Peña, The universal cover of a quiver with relations, J. Pure Appl. Algebra 30 (1983) 277–292]. Assuming the quiver of A has no oriented cycles and no double bypasses, we show there exists a suitable presentation of A with quiver and admissible relations, with fundamental group denoted by ${\pi }_1(A)$, such that the fundamental group of any other presentation of A with quiver and admissible relations is a quotient of ${\pi }_1(A)$. To cite this article: P. Le Meur, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

The fundamental group of a locally finite graph with ends

We characterize the fundamental group of a locally finite graph G with ends combinatorially, as a group of infinite words. Our characterization gives rise to a canonical embedding of this group in the inverse limit of the free groups π₁(G^′) with G^′⊆G finite.     

The fundamental group of an algebraic link

We compute the fundamental group of an algebraic link. To cite this article: O. Neto, P.C. Silva, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

The algebraic fundamental group of a reductive group scheme over an arbitrary base scheme

We define the algebraic fundamental group π ₁(G) of a reductive group scheme G over an arbitrary non-empty base scheme and show that the resulting functor G? π₁(G) is exact.