[Galaxy Note 3 Q&A, Help & Troubleshooting] StarTAC Fan:
Are you referring to GSM/EDGE, not GSM/UMTS? Or are you saying you want info for GSM (2G) and also info for UMTS (3G)?

Part 1: General Non-4G Support

http://www.att.com/shop/wireless/devices/samsung/galaxy-note-3-jet-black.html

2G: 800, 850, 1800, 1900

3G: 850, 1900, 2100

http://www.t-mobile.com/cell-phones/samsung-galaxy-note-3.html

GSM frequencies supported said as being "quad band" without needing to specify what they are, no doubt same as AT&T's 2G

UMTS/HSPA same as AT&T's 3G with addition of 1700 MHz

Verizon and Sprint have not yet posted 2G and 3G frequency specifications, indeed any details of the Galaxy Note 3, on their websites. However, FCC documents show that 2G 850 and 1900 MHz have been tested.

Part 2: 2G

GSM/GPRS/EDGE is 2G and therefore can't really provide usable data speed. So I'm unsure if this is what you want or if you just want to use it for voice (without GPRS and EDGE extensions). However, general rule of thumb, nowadays all GSM capable phones are "quad band" (850, 900, 1800 and 1900 MHz). Indeed, AT&T and T-Mobile which have posted specs on their website confirm this. 850 and 1900 are frequencies used in North America among other places while 900 and 1800 are used in Europe and other places as you are probably already aware.

Whether they have posted specs on their website or not already, for all 4 major American carriers, the FCC documents show testing was performed only for 2G 850 and 1900 MHz. That's a little misleading because to meet American regulatory requirements, the 900 and 1800 European frequencies don't have to be tested although support for them may be present in the device. That's why carrier and manufacturer specs can claim additional frequencies not of interest to the US FCC - not any of the other regulated frequencies.

Part 3: 3G

Your were concerned about 3G support in Europe. The Band 1 2100 MHz is universal for 3G for all phones, or should be. With insignificant exception, Band 1 is used by all European carriers as well as pretty much the rest of the world for 3G as you seem to be aware (you specifically mentioned looking for Band 1 support). The FCC documents don't specify and don't need to whether 3G Band 1 testing was carried out for transmission and reception. However, 2100 MHz is used as half of the Band 4 supported by the AT&T and T-Mobile models. That said, 3G Band 1 support should be present for both up and downlink.

Indeed the specs posted at AT&T and T-Mobile sites confirm that there is support for 3G Band 1 (implicitly up and down). With Sprint and Verizon, they have not yet confirmed support for 3G Band 1. But if they want their phones roamable on 3G networks elsewhere in the world, they must include Band 1 support at minimum. FCC documents submitted by the manufacturer, Samsung in this case, don't have to indicate support for 3G Band 1 because, again, it doesn't exist in the FCC's jurisdiction.

UMTS (also called WCDMA) is an early form of 3G theoretically capable of up to 384 kb/s. HSDPA and HSUPA were backward compatible improvements (D for downlink and U for uplink, and where HSDPA + HSUPA = HSPA) to which pretty much all the original UMTS carriers have upgraded. A further improvement is HSPA+ to which many HSPA carriers have upgraded. Although not the exact same, UMTS, WCDMA, HSPA and HSPA+ are interchangeably referred to as 3G. Modern phones are HSPA+ capable, therefore automatically backward compatible with UMTS (WCDMA) and HSPA carriers.

Part 4: Other Remarks

We'll know in a few weeks which phones are bootloader locked or unlocked. As mentioned, T-Mobile has historically not locked it.

LTE frequencies throughout the world are fragmented like no other. And we used to think quad band 2G and penta band 3G were excessive. But you're interested in quad band GSM and 3G at 2100 MHz, not universal 4G support - that's good.

I cannot tell you which carrier's GN3 or the International to get or not get particularly for US 4G + at minimum 2G/3G for international travel, only provide information for you to arrive at your own informed decision. That said, here is a summary:

AT&T and T-Mobile have already explicitly published that their Note 3 phones will be 2G quad-band.

AT&T and T-Mobile have published their phones will be 3G Band 1 2100 MHz capable in addition to their native 3G network frequencies.

Sprint and Verizon should also be quad-band 2G and, at minimum, Band 1 3G capable for roaming outside the US. FCC test reports cannot verify this for reason cited. It would be foolish for these 2 carriers to not to support minimal international roaming but we just have to wait.

As long as whatever variant you get has quad 2G and 3G @ 2100, you'll be fine for voice and data pretty much anywhere outside the Americas.