Drouin, Joshua2023-04-132023-04-132023https://hdl.handle.net/2097/43035We study the difference between the homotopy groups of spaces of smooth embeddings and spaces topological embeddings of a sphere into four-manifolds. In particular, we show that: [see PDF file for equation] may have an arbitrarily high-rank summand for a some 4-manifolds. Here, [see PDF file] represents the component to the embedding space containing a specific embedding S. This behavior is found for spheres of arbitrary self-intersection. We also establish an analogous result for homology groups.en-US© the author. This Item is protected by copyright and/or related rights. You are free to use this Item in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you need to obtain permission from the rights-holder(s).http://rightsstatements.org/vocab/InC/1.0/Embedding spacesGauge theoryLow-dimensional topologyFamilies of embeddingsDissertation