Glycolysis is really the conversion of Glucose to Pyruvate, which takes place in the cytosol. Now, this portion doesn't require any oxygen whatsoever. Think about it, when oxygen is scarce, this pathways is still running: Glucose is being converted to pyruvate, which then undergoes either alcoholic fermentation or lactic fermentation.
However, some books may refer to the entire catabolism of glucose (including the TCA cycle and the electron transport chain) as glycolysis. Under such circumstances, then you can look at glycolysis as an aerobic process.
But I think it's more accurate to look at glycolysis as only conversion of glucose to pyruvate, which is indeed anaerobic.