It is possible, but only in certain conditions. I live in a tropical country, so, 20 degrees centigrade is considered cold here. My conmute to work is almost flat, with only one climb, something a very steep 300 meters. If it were not for that, I'd be able to get to the office almost completely dry.
What's the trick? I use a hardtail mountain bike with a small rack in the seatpost. The bike has a sporty position, but the trick is to pedal in the easy gears and making very little effort, accelerating very slowly. The effort should feel comparable to a moderate walk. Also, take as much advantage as I can from the downhills (Update: later on, I changed the tires to slick ones, that reduces the effort needed to move).
However, ideal conditions are an exception. I have already told you that I have to climb a very steep hill, so I just put the easiest gear and pedal like a grandmother for that hill (While in "sports mode" I can climb the same hill in 4 gears higher and a quarter of the time). I sweat the minimum and I manage to avoid clothes changes, but there are other things I do to minimize body odor:
- Drink more water than actually necessary: urine then turns colorless which is a practical indication that the urinary system has got rid of most waste substances. This helps because some of these substances make their way into sweat and contribute to body odor. The practical point is that even if you sweat, it won't stink that much. *(1, 2, 3)
- Apply a little deodorant in other parts of the body besides armpits. (I recommend a little in the chest and in the back)
- Carry a clean hand towel and use it to dry the sweat when you get to the workplace. First the face, then the hair, then the neck, arms and last, armpits. This helps because part of body odor is caused by bacteria(2) that reproduce and feed in the warm, humid environment provided by the sweaty body, so cutting the time the humidity lasts reduce the growing of bacteria. The towel can be pre-scented with a suitable cologne if you want. Rinse the towel thoroughly and twist dry it as much as you can. It won't be smelly if you do it right, so you can discretely hang-dry it near your desk without disturbing anyone.
- Leave home early enough so you can take your time getting to the office, also, plan to get there 20-15 minutes prior to the time you have to be there, so you can spend those "cooling off". Sprinkle your face with fresh water, moisten hair (and apply hair gel, if you use to do so) which help against "helmet hair".
- Avoid carrying backpacks or anything similar. Any strap that makes your clothing tight against your upper body will wrinkle the clothes, will restrict airflow around your torso, preventing sweat evaporation thus avoiding its cooling effect(1), so it will make you sweat more. They will also cause curious sweat marks in your shirt.
All those tips are easy to implement, and can keep you from needing to change clothes. After you cool off and clean yourself with the towel, most people won't even notice a difference on you if you hadn't commuted by bike.
As per anecdotal stuff, my grandfather used to commte exclusively by bike, in a small city located in the Valley of Sula (Valle de Sula) near the city of San Pedro Sula in Honduras. A valley in a tropical country has a very hot weather all year round. He used a bike similar to the "Flying Pigeon" (Google Flying Pigeon Bike for images). That is a heavy steel bike with 28" wheels, a very high gearing and a very comfortable seating position. I remember that he was never sweaty, even if he commuted from workplace to his home for lunch, right under caribean sun at noon. Two elements had a big role in it: 1) the city is completely flat, no climbs, no downhills. 2) The bike he used is almost impossible to pedal in an sporty way, it kind of 'forced' you to pedal in a smooth fashion, with slow cadence, but after a while, you get some speed that is really easy to keep. It makes you feel you can go for hours without getting tired.