Extending families of homeomorphisms over 4-dimensional handlebodies

Abstract Let upper H Subscript g $H_g$ H g denote the 4-dimensional handlebody of genus g and upper U Subscript g $U_g$ U g its boundary. We show that for all g greater than or equals 0 $g \ge 0$ g ≥ 0 the map from upper B backslash exclamation mark backslash operatorname upper H o m e o left parenthesis upper H Subscript g Baseline right parenthesis $B\!\operatorname {\mathrm {Homeo}}(H_g)$ B \! \operatorname H o m e o ( H g ) to upper B backslash exclamation mark backslash operatorname upper H o m e o left parenthesis upper U Subscript g Baseline right parenthesis $B\!\operatorname {\mathrm {Homeo}}(U_g)$ B \! \operatorname H o m e o ( U g ) induced by restriction to the boundary admits a section.