# Connected sums of almost complex manifolds, products of rational homology spheres, and the twisted spin^c Dirac operator

## TL;DR

This paper investigates conditions under which connected sums of almost complex manifolds retain almost complex structures, extending results to rational homology spheres using obstruction theory and index theory of the twisted spin^c Dirac operator.

## Contribution

It provides new criteria for the existence of almost complex structures on connected sums and products involving rational homology spheres, extending previous results with novel obstruction and index-theoretic methods.

## Key findings

- Connected sum of two closed almost complex manifolds may not always be almost complex.
- Criteria are established for when connected sums of multiple almost complex manifolds are almost complex.
- Extension of nonexistence results to products of rational homology spheres using index theory.

## Abstract

We record an answer to the question "In which dimensions is the connected sum of two closed almost complex manifolds necessarily an almost complex manifold?". In the process of doing so, we are naturally led to ask "For which values of l is the connected sum of l closed almost complex manifolds necessarily an almost complex manifold?". We answer this question, along with its non-compact analogue, using obstruction theory and Yang's results on the existence of almost complex structures on (n-1)-connected 2n-manifolds. Finally, we partially extend Datta and Subramanian's result on the nonexistence of almost complex structures on products of two even spheres to rational homology spheres by using the index of the twisted spin^c Dirac operator.