The group of self-homotopy equivalences ofA_n²-polyhedra

Abstract

<jats:title>Abstract</jats:title><jats:p>Let <jats:italic>X</jats:italic> be a finite type <jats:inline-formula id="j_jgth-2018-0203_ineq_9998_w2aab3b7d334b1b6b1aab1c15b1b3Aa"><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_jgth-2018-0203_eq_0129.png" /><jats:tex-math>{A_{n}^{2}}</jats:tex-math></jats:alternatives></jats:inline-formula>-polyhedron, <jats:inline-formula id="j_jgth-2018-0203_ineq_9997_w2aab3b7d334b1b6b1aab1c15b1b5Aa"><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_jgth-2018-0203_eq_0363.png" /><jats:tex-math>{n\geq 2}</jats:tex-math></jats:alternatives></jats:inline-formula>. In this paper, we study the quotient group <jats:inline-formula id="j_jgth-2018-0203_ineq_9996_w2aab3b7d334b1b6b1aab1c15b1b7Aa"><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_jgth-2018-0203_eq_0261.png" /><jats:tex-math>{\mathcal{E}(X)/\mathcal{E}_{*}(X)}</jats:tex-math></jats:alternatives></jats:inline-formula>, where <jats:inline-formula id="j_jgth-2018-0203_ineq_9995_w2aab3b7d334b1b6b1aab1c15b1b9Aa"><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_jgth-2018-0203_eq_0264.png" /><jats:tex-math>{\mathcal{E}(X)}</jats:tex-math></jats:alternatives></jats:inline-formula> is the group of self-homotopy equivalences of <jats:italic>X</jats:italic> and <jats:inline-formula id="j_jgth-2018-0203_ineq_9994_w2aab3b7d334b1b6b1aab1c15b1c13Aa"><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_jgth-2018-0203_eq_0265.png" /><jats:tex-math>{\mathcal{E}_{*}(X)}</jats:tex-math></jats:alternatives></jats:inline-formula> the subgroup of self-homotopy equivalences inducing the identity on the homology groups of <jats:italic>X</jats:italic>. We show that not every group can be realised as <jats:inline-formula id="j_jgth-2018-0203_ineq_9993_w2aab3b7d334b1b6b1aab1c15b1c17Aa"><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_jgth-2018-0203_eq_0264.png" /><jats:tex-math>{\mathcal{E}(X)}</jats:tex-math></jats:alternatives></jats:inline-formula> or <jats:inline-formula id="j_jgth-2018-0203_ineq_9992_w2aab3b7d334b1b6b1aab1c15b1c19Aa"><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_jgth-2018-0203_eq_0261.png" /><jats:tex-math>{\mathcal{E}(X)/\mathcal{E}_{*}(X)}</jats:tex-math></jats:alternatives></jats:inline-formula> for <jats:italic>X</jats:italic> an <jats:inline-formula id="j_jgth-2018-0203_ineq_9991_w2aab3b7d334b1b6b1aab1c15b1c23Aa"><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_jgth-2018-0203_eq_0129.png" /><jats:tex-math>{A_{n}^{2}}</jats:tex-math></jats:alternatives></jats:inline-formula>-polyhedron, <jats:inline-formula id="j_jgth-2018-0203_ineq_9990_w2aab3b7d334b1b6b1aab1c15b1c25Aa"><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_jgth-2018-0203_eq_0364.png" /><jats:tex-math>{n\geq 3}</jats:tex-math></jats:alternatives></jats:inline-formula>, and specific results are obtained for <jats:inline-formula id="j_jgth-2018-0203_ineq_9989_w2aab3b7d334b1b6b1aab1c15b1c27Aa"><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_jgth-2018-0203_eq_0361.png" /><jats:tex-math>{n=2}</jats:tex-math></jats:alternatives></jats:inline-formula>.</jats:p>