  We show that a $k$-uniform hypergraph on $n$ vertices has a spanning subgraph
homeomorphic to the $(k - 1)$-dimensional sphere provided that $H$ has no
isolated vertices and each set of $k - 1$ vertices supported by an edge is
contained in at least $n/2 + o(n)$ edges. This gives a topological extension of
Dirac's theorem and asymptotically confirms a conjecture of Georgakopoulos,
Haslegrave, Montgomery, and Narayanan.
  Unlike typical results in the area, our proof does not rely on the Absorption
Method, the Regularity Lemma or the Blow-up Lemma. Instead, we use a recently
introduced framework that is based on covering the vertex set of the host graph
with a family of complete blow-ups.
