The battery of laptop - use or not with AC adapter?

RCinFLA said:

LiIon battery longevity is degraded by heat. In a laptop, the heat generated by the running computer, not the charging process or leaving charger connected to power the laptop, is the most degrading effect on battery.

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A "properly designed" laptop will not significantly increase the temperature of the battery! A lot of effort is made to thermally insulate the battery from the electronics. Feel your laptop's batteries - if they feel hot there is something wrong.

RCinFLA said:

Voltage greater then about 3.9 vdc per cell does have some degradation in longevity so storage should be done at about 50% state of charge but not allowed to self discharge below about 3.0 vdc per cell. Self discharge of LiIon is very low. Frequent topping charge to 4.2 vdc per cell reduces battery longevity.

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Yes, FREQUENT topping will reduce the battery life. But we are talking about in-service batteries here. It's a trade-off of battery life versus run-time, and for this application the user's expectation is that the battery will power the laptop for the longest time possible. Therefore, it is typically recommended that the battery be topped-off when the voltage falls below 4.00V.