# Heegaard Floer homology and splicing homology spheres

## TL;DR

This paper establishes a fundamental inequality for d-invariants in knot splicing within homology spheres, leading to new insights into Floer homology ranks and conditions for homotopy equivalences of Seifert fibered homology spheres.

## Contribution

It introduces a new inequality for d-invariants in knot splicing and improves understanding of Floer homology rank relations and homotopy classifications of Seifert fibered homology spheres.

## Key findings

- Proved a basic inequality for d-invariants of spliced knots.
- Established a new relation on the rank of reduced Floer homology.
- Characterized when a degree one map between Seifert fibered homology spheres is a homotopy equivalence.

## Abstract

We prove a basic inequality for the d-invariants of a splice of knots in homology spheres. As a result, we are able to prove a new relation on the rank of reduced Floer homology under maps between Seifert fibered homology spheres, improving results of the first and second authors. As a corollary, a degree one map between two aspherical Seifert homology spheres is homotopic to a homeomorphism if and only if the Heegaard Floer homologies are isomorphic.