Surgeries modify manifolds by removing and replacing submanifolds. Properly constructed surgeries preserve invariants like homology groups and fundamental groups.
Theorem 3.1 (Invariant Preservation) If φ: M → M' is a continuous, bijective transformation with continuous inverse, then all topological invariants of M are preserved in M'.
Towards A Proof
Since φ is a homeomorphism, open sets map to open sets bijectively. Therefore, properties like compactness, connectivity, and genus are preserved by the continuity and invertibility of the map.
Surgeries modify manifolds by removing and replacing submanifolds. Properly constructed surgeries preserve invariants like homology groups and fundamental groups.
Theorem 3.1 (Invariant Preservation)
If φ: M → M' is a continuous, bijective transformation with continuous inverse, then all topological invariants of M are preserved in M'.
Towards A Proof
Since φ is a homeomorphism, open sets map to open sets bijectively. Therefore, properties like compactness, connectivity, and genus are preserved by the continuity and invertibility of the map.