Good morning (rchkilme...) First of all is there any chance you could have another person take a look at the cracked pipe? This would be quite easy to view since you've indicated it is a top section. The chimney "cap" will lift off easily for viewing. Reason to check this is, often a crack is not significant enough to have to do anything and even if it is there is a special caulking that can be applied to the crack rather than a full replacement of the pipe. Now, if it is badly cracked and needs replacement it is not going to be too big of a job. Again, since it's a top section it will be readily accessible to make the repair or replacement.
I'm suggesting "another person" to do this checking because it's not a bad idea to have a second opinion on something like this.
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A crack in a flue can be very dangerious . My suggestion would be a flue liner. The stainless steel flue liners are a lot safer and can easily be installed. You will never have to worry about them cracking and they are easier to keep clean. More information can be found by clicking here.
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Please note in my previous answer I explained that the crack being in the "top" section will make the repair or replacement quite easy. This pipe will actually be protruding above the top of the bricks. If it is badly cracked and can't be repaired it can be broken and removed from right there. No bricks will need to be removed. Although this pipe could be stabilized with mortar between it and the bricks. This is still a relatively simple repair.
Mr. Booker, with our JA site has indicated the danger with a cracked flue pipe and replacing with a new stainless steel liner for the clay pipe. Sometimes this could indeed be a real concern, however. I believe Mr. Booker may have overlooked your indication that the crack was the "top" section. With this being the location it is unlikely there would be any danger at this time.
Also, please remember before having this repaired or replaced to obtain a second opinion of the severity of the crack.