##
**Test map characterizations of local properties of fundamental groups.**
*(English)*
Zbl 1454.55015

The classical theory of covering maps \(p:(Y,y_0)\to(X,x_0)\) of basepointed spaces
requests that \(X\) be path-connected, locally path-connected, and semilocally
simply connected. Questions arise when only path-connectedness and local path-connectedness
(generally speaking local path-connectedness is not necessarily stipulated for spaces
under study herein) among these three are in play, and there are significant types of
examples that fall into this classification. Numerous studies of such classes of spaces have been made;
the authors provide many historical references to this in their introduction.

On page 38 is a list of six properties that a space might or might not have:

On page 38 is a list of six properties that a space might or might not have:

- 1
- Homotopically Hausdorff and its relative version,
- 2
- (transfinite products) Every homomorphism \(f_{\#}:\pi_1(\mathbb{H},b_0)\to\pi_1(X,x_0)\) induced by a map \(f:\mathbb{H}\to X\) on the Hawaiian earring is uniquely determined by its values \(f_{\#}([l_n])\) on the initial loops \([l_n]\),
- 3
- Existence of generalized universal and intermediate coverings,
- 4
- Homotopically path Hausdorff and its relative version,
- 5
- (1-\(UV_0\)) For every \(x\in X\) and every neighborhood \(U\) of \(x\) there is an open set \(V\) in \(X\) with \(x\in V\subseteq U\) and such that for every map \(f:D^2\to X\) from the unit disk with \(f(\partial D^2)\subseteq V\), there is a map \(g:D^2\to U\) with \(f:\partial D^2= f|\partial D^2\),
- 6
- (\(\pi_1\)-shape injectivity) The canonical homomorphism \(\pi_1(X,x_0)\to
\check\pi_1(X,x_0)\) to the first shape homotopy group is injective.

A diagram comparing properties (1), (3), (4), and (6) is provided in [H. Fischer et al., Topology Appl. 158, No. 3, 397–408 (2011; Zbl 1219.54028)].

The primary purpose of this paper is to provide a unified approach to comparing such properties as (1)–(6). We quote: “We are particularly motivated by the fact that even when \(X\) fails to admit a traditional universal covering, it is often the case that \(X\) admits a generalized universal covering […] which acts in many ways as a suitable replacement. A generalized covering map is characterized only by its lifting properties and need not be a local homeomorphism.”

A “diagram (on page 40) may serve as a reference for many of the results and definitions in this paper. It connects the relevant properties of a path-connected metric space \(X\) and the closure properties of a subgroup \(H\leq\pi_1(X,x_0)\) […] Equipped with this chart, we identify new types of subgroups that correspond to intermediate generalized coverings (Theorem 5.4 and Corollaries 7.14, 7.15) and shed more light on the relative position of the commutator subgroup of \(\pi_1(\mathbb{H},b_0)\) (Example 3.10). We also extend the existence of generalized universal coverings for Peano continua with residually \(n\)-slender fundamental group to all metric spaces (Corollary 6.5).

Property (5) is not an invariant of homotopy type, but is an important property held by one-dimensional and planar spaces and is known to imply the homotopically Hausdorff property for metric spaces. We improve the latter result by showing that every metric space with the \(1\)-\(UV_0\) property admits a generalized universal covering space (Theorem 6.9).”

Reviewer: Leonard R. Rubin (Norman)

### MSC:

55Q52 | Homotopy groups of special spaces |

57M12 | Low-dimensional topology of special (e.g., branched) coverings |

06A15 | Galois correspondences, closure operators (in relation to ordered sets) |

54D05 | Connected and locally connected spaces (general aspects) |

55Q07 | Shape groups |

57M05 | Fundamental group, presentations, free differential calculus |

### Keywords:

generalized covering space; homotopically Hausdorff; homotopically path Hausdorff; subgroup lattice of fundamental group; closure operator### Citations:

Zbl 1219.54028
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\textit{J. Brazas} and \textit{H. Fischer}, J. Topol. Anal. 12, No. 1, 37--85 (2020; Zbl 1454.55015)

### References:

[1] | Arhangelskii, A. and Tkachenko, M., Topological Groups and Related Structures, , Vol. 1 (Atlantic Press, 2008). |

[2] | W. A. Bogley and A. J. Sieradski, Universal path spaces, Unpublished notes (1998). |

[3] | Brazas, J., The fundamental group as a topological group, Topology Appl.160 (2013) 170-188. · Zbl 1264.57001 |

[4] | Brazas, J., Generalized covering space theories, Theory Appl. Categ.30 (2015) 1132-1162. · Zbl 1331.55009 |

[5] | Brazas, J. and Fabel, P., Thick Spanier groups and the first shape group, Rocky Mountain J. Math.44 (2014) 1415-1444. · Zbl 1306.57004 |

[6] | Brazas, J. and Fabel, P., On fundamental groups with the quotient topology, J. Homotopy Relat. Struct.10 (2015) 71-91. · Zbl 1311.55018 |

[7] | Brodskiy, N., Dydak, J., Labuz, B. and Mitra, A., Covering maps for locally path-connected spaces, Fund. Math.218 (2012) 13-46. · Zbl 1260.55013 |

[8] | Cannon, J. W. and Conner, G. R., The combinatorial structure of the Hawaiian earring group, Topology Appl.106 (2000) 225-271. · Zbl 0955.57002 |

[9] | Cannon, J. W. and Conner, G. R., On the fundamental groups of one-dimensional spaces, Topology Appl.153 (2006) 2648-2672. · Zbl 1105.55008 |

[10] | Conner, G. R., Meilstrup, M., Repovs̆, D., Zastrow, A. and Z̆eljko, M., On small homotopies of loops, Topology Appl.155 (2008) 1089-1097. · Zbl 1148.57030 |

[11] | S. Corson, On subgroups of first homology, preprint (2016), arXiv:1610.05422. · Zbl 1436.54025 |

[12] | Curtis, M. L. and Fort, M. K. Jr., The fundamental group of one-dimensional spaces, Proc. Amer. Math. Soc.10 (1959) 140-148. · Zbl 0089.38802 |

[13] | Eda, K., Free \(\sigma \)-products and noncommutatively slender groups, J. Algebra148 (1992) 243-263. · Zbl 0779.20012 |

[14] | Eda, K., Free subgroups of the fundamental group of the Hawaiian earring, J. Algebra.219 (1999) 598-605. · Zbl 0951.20016 |

[15] | Eda, K., The fundamental groups of one-dimensional spaces and spatial homomorphisms, Topology Appl.123 (2002) 479-505. · Zbl 1032.55013 |

[16] | K. Eda, Algebraic topology of Peano continua, Topol. Appl.153 (2005) 213-226; Correction Topol. Appl.154 (2007) 771-773. · Zbl 1225.55006 |

[17] | Eda, K., Homotopy types of one-dimensional Peano continua, Fund. Math.209 (2010) 27-42. · Zbl 1201.55002 |

[18] | Eda, K., Singular homology groups of one-dimensional Peano Continua, Fund. Math.232 (2016) 99-115. · Zbl 1335.55005 |

[19] | Eda, K. and Fischer, H., Cotorsion-free groups from a topological viewpoint, Topology Appl.214 (2016) 21-34. · Zbl 1396.20062 |

[20] | Eda, K. and Kawamura, K., The fundamental groups of one-dimensional spaces, Topology Appl.87 (1998) 163-172. · Zbl 0922.55008 |

[21] | Fischer, H. and Guilbault, C. R., On the fundamental groups of trees of manifolds, Pacific J. Math.221 (2005) 49-79. · Zbl 1097.55011 |

[22] | Fischer, H., Repovš, D., Virk, Z. and Zastrow, A., On semilocally simply connected spaces, Topology Appl.158 (2011) 397-408. · Zbl 1219.54028 |

[23] | Fischer, H. and Zastrow, A., The fundamental groups of subsets of closed surfaces inject into their first shape groups, Algebr. Geom. Topol.5 (2005) 1655-1676. · Zbl 1086.55009 |

[24] | Fischer, H. and Zastrow, A., Generalized universal covering spaces and the shape group, Fund. Math.197 (2007) 167-196. · Zbl 1137.55006 |

[25] | Fischer, H. and Zastrow, A., Combinatorial \(\mathbb{R} \)-trees as generalized Caley graphs for fundamental groups of one-dimensional spaces, Geom. Dedicata163 (2013) 19-43. · Zbl 1297.20044 |

[26] | Spanier, E., Algebraic Topology (McGraw-Hill, 1966). · Zbl 0145.43303 |

[27] | Virk, Z. and Zastrow, A., A homotopically Hausdorff space which does not admit a generalized universal covering, Topology Appl.160 (2013) 656-666. · Zbl 1420.54049 |

[28] | Virk, Z. and Zastrow, A., The comparison of topologies related to various concepts of generalized covering spaces, Topology Appl.170 (2014) 52-62. · Zbl 1296.54042 |

[29] | A. Zastrow, Generalized \(\pi_1\)-determined covering spaces, Unpublished notes (2002). |

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