Gemara

31b - Mishnah: If the top of a Lulav was cut off, it is Pasul.

Rav Huna: This is only if was cut off, but if it was Nisdak (cleft) it is Kosher.

Question (Beraisa): If a Lulav is bent like a hook, has prickly points along its spine (alternatively - is shriveled), Saduk (cleft) or bent like a sickle it is Pasul. If it is as hard as wood it is Pasul. If it is (beginning to get) hard like wood it is Kosher.

Answer (Rav Papa): (It is Pasul if it is) like a Hemnek (a Keli with two heads).

Mishnah: If the leaves were Nifretzu (separated) it is Pasul.

Rav Papa: This is when they are like a broom.

Question (Rav Papa): If the Tiyomes (this will be defined) was Nechlak (divided), what is the law?

Version #1 - Answer: R. Yehoshua ben Levi taught that if the Tiyomes was removed it is Pasul.

Suggestion: The same applies when it is Nechlak!

Rejection: No, it is worse if it is removed, for then it is Chaser (incomplete).

Version #2 - Answer: R. Yehoshua ben Levi taught that if the Tiyomes is Nechlak it is as if it was removed, and it is Pasul.

Rishonim

Rif: If it is Nisdak it is Kosher. It is Pasul if it is Nisdak like a Hemnek, i.e. a two-headed tool like this ץ. Nifretzu is Pasul; this is when the leaves were detached from the spine. If the Tiyomes is Nechlak, it is as if it was removed, and it is Pasul. The Tiyomes is the back of the Lulav. The leaves of a Lulav are doubled; they are joined at the back. If the leaves are separated from each other but the Tiyomes is intact, it is Kosher. If the Tiyomes is Nechlak, it is as if the leaves were Nifretzu (detached), and it is Pasul.

Ran (DH Niktam): The Ra'avad explains that 'top' refers to the top of the spine. This is like R. Yochanan, who says that 'Lulav' does not include the leaves. Shmuel disagrees, but the Halachah follows R. Yochanan against Shmuel. Others say that 'top' refers to the top of the spine. One should be stringent like both opinions to disqualify whether the top of the spine or the top leaves are cut, bent or Saduk.

Rambam (Hilchos Lulav 8:3): If the top of the Lulav is Nisdak and the two sides are distanced from each other and appear like two, it is Pasul. Nifretzu is Pasul; this is when the leaves fall off the spine.

Rambam (4): Normally, the leaves of a Lulav grow in pairs. They are joined at the back. The back of every two leaves is called Tiyomes. If the Tiyomes is Nechlak it is Pasul.

Rosh (3:3): Rashi says that Nifretzu is when the leaves were detached from the spine. This is difficult. Rather, the double leaves were bent outwards to make a broom.

Rosh (3:4): Nisdak is Kosher, but if it is Nisdak like a Hemnek, it is Pasul. This is a Keli with two heads, like this ?.

Rosh (3:6) and Tosfos (Bava Kama 96a DH Nechlekah): If the Tiyomes is Nechlak, it is as if it was removed, and it is Pasul. Rashi says that the Tiyomes is the top middle leaves where the spine ends. If they were separated and the spine is split until the leaves below them it is Pasul. The Ge'onim say that it is the uppermost leaf. It is like two leaves stuck to each other. We disqualify when the Tiyomes is Nechlak, i.e. they are not stuck together and they look like two. According to this, not one out of 500 of our Lulavim is Kosher! Perhaps they refer to when it is not like it was created. I.e. at first it was connected, and later it was split. However, if it was never connected, it is Kosher.

Beis Yosef (OC 645 DH veha'Tosfos and v'Nir'eh): Here (in Eretz Yisrael) we find red matter on Lulavim that makes the entire top of the Lulav like one piece of wood without any separation.

Ri (in Tosfos, ibid.): Some explain Nechlekah ha'Tiyomes to be when all the leaves are doubled, and two double leaves come out of the top of the spine, and they are separated slightly on the spine.

Beis Yosef (ibid.): According to this the two leaves look like two and need not stick together, but there may not be a gap between them.

Taz (4): The Beis Yosef says that the Ri and the Ge'onim define Tiyomes the same way; they argue only about whether or not they may look two leaves. This is wrong. Tosfos brought the Ge'onim's definition, then the Ri's. Also, Tosfos said that not one Lulav out of 500 is like the Ge'onim require; he says that most do not have a Tiyomes like the Ri discusses.

Rosh (ibid.): The Rif and Rambam say that the Tiyomes is junctures of the leaves in back. Surely, they hold that Nechlak disqualifies only if the majority of the leaves are Nechlakim. This is how Tosfos explains 'Nifretzu Alav', therefore Tosfos needed to explain Nechlekah ha'Tiyomes differently. (Rav Papa explained Nifretzu Alav, but asked about Nechlekah ha'Tiyomes.) BaHaG says that the Tiyomes is what joins the leaves in back and makes them like one. Tosfos says that this must refer to the top leaves. This is not so; perhaps he means that it makes each pair of leaves one. Rav Papa asks whether a split Tiyomes is like leaves that were Nifretzu, or merely bent away from the spine.

Poskim

Shulchan Aruch (OC 645:3): Normally, the leaves of a Lulav grow in pairs. They are joined at the back. The back of every two leaves is called Tiyomes. If (Rema - the majority of leaves of) the Tiyomes is Nechlak it is Pasul.

Source (Gra DH Briyas): Rav Papa asked about Nechlekah ha'Tiyomes right after explaining Nifretzu Alav. This suggests that they are similar (unlike Rashi, who says that the Tiyomes is the top double leaf).

Magen Avraham (3): This is when the majority of each of these leaves is Nechlak.

Sha'ar ha'Tziyon (11): It is as if the leaves are missing, so it is Pasul all seven days. However, Rashi disqualifies Nifretzu Alav due to Hadar, and the Rambam and others say that this is only on the first day.

Rema: Some say that Nechlekah ha'Tiyomes is when the top center leaf on the spine is Nechlak until the spine. This is our custom.

Taz (4): This is like Rashi according to SMaK (unlike our text of Rashi). The Ran brings an opinion that allows a cleft unlike a Hemnek only if it is widthwise (horizontal when the Lulav is upright). The Gemara connotes otherwise, but it is good to be stringent. We cannot say that we are concerned for any gap at all, for this was not taught amidst things which must be so smooth that a fingernail would not get caught on it. It is unreasonable to be concerned for less than a Tefach, especially since most Poskim have no special concern for the middle leaf, only for the majority of leaves.

Mishnah Berurah (15): If two leaves come out of the top of the spine, both are considered Tiyomos. If either is Nechlak, it is Pasul.

Chazon Ish (145:4): Presumably it is Kosher because a Tiyomes remains, just like if one of them was cut it is Kosher.

Mishnah Berurah (16,19): If the majority until the spine is Nechlak it is as if it is totally Nechlak, since shaking it is prone to make it split fully. One need not be stringent about the minority unless it is split like a Hemnek.

Chazon Ish (145:4): Since we are stringent like Rashi's opinion, it is proper to be stringent if the majority is Nechlak. If no other is available one may bless on it, since most Rishonim hold like the Rif.

Note: If a Kosher Lulav has a small split, one may glue the sides together to prevent the split from growing.

Kaf ha'Chayim (26): R. Yerucham says that Nisdak and Nechlekah ha'Tiyomes are Kosher after the first day, but other Pesulim are due to Hadar and apply all seven days. The Shulchan Aruch (649:5) holds that even Hadar is only on the first day.

Rema (ibid.): However, the ideal Mitzvah l'Chatchilah is a Lulav in which the top leaf is not Nechlak at all, for some are stringent if it is Nechlak even a little.

Mishnah Berurah (18): One need not bless on a different Lulav due to this stringency.

Kaf ha'Chayim (24): The Rosh suggested that the Ge'onim disqualify a split of any size only if it did not grow this way. Therefore, when inspecting the middle leaves one should not separate them. If red matter or something else makes them stick together, surely the middle leaf is not Nechlak, or at most only a bit (and the Lulav is Kosher).

Bechori Yakov (1): It is improper to take a Lulav covered with bark. A Tefach of the Lulav must be untied so the leaves can rustle. The leaves do not rustle in a closed Lulav!

Sedei Chemed (Arba'ah Minim 2:1): Sefardiim prefer that the leaves are totally attached and do not separate at all. Some Ashkenazim are not concerned for this, and others require them to be separate, so the leaves can rustle.

Shulchan Aruch (7): If it (the top of the Lulav) is split and the two sides are distanced from each other and appear like two, it is Pasul.

Beis Yosef (DH Nisdak): Rashi explains that the Lulav grew with two spines. The Tur says that 'Nisdak' connotes that is became split later. This is not difficult, for Rashi explains the Beraisa, which says 'Siduk'.

Beis Yosef (ibid.): It appears that Rashi disqualifies only if the spine is split until the leaves below the top ones. This is unlike R. Yerucham, who says that Rashi disqualifies if the top two leaves are split, for this is a divided Tiyomes.

Beis Yosef (DH veha'Ran): The Ran holds that a split disqualifies only if it is the majority of the Tiyomes, or if it is like a Hemnek. According to the Rif perhaps it is never Pasul unless it is like a Hemnek, and even this is Pasul only if it did not grow this way.

Beis Yosef (DH Nechlekah): The Rif and Rambam did not specify the Shi'ur for a divided Tiyomes. It seems that it is the majority of the leaves, like the Rosh and Magid Mishnah understand them.

Rema: This is even if the top Tiyomes is not split enough to disqualify because Nechlekah ha'Tiyomes.

Taz (9): This means that even if the split does not extend to the spine, if it is like a Hemnek it is Pasul.

Magen Avraham (7): This means that only the leaves by the top of the Tiyomes are split.

Levushei Serad (9) and Mishbetzos Zahav (9): Many people are not concerned for a split less than a Tefach. This is improper. Usually, the sides are distanced, and it is like a Hemnek!

Maharsham (on Orchos Chayim ha'Chadash 645, cited in Arba'as ha'Minim ha'Shalem p.234): It is considered like a Hemnek only if the sides are a fair distance apart and do not come together when the Lulav is placed down or shaken.

Chazon Ish (145:8): The Rif and Rosh hold that a Hemnek disqualifies only if all the leaves on the right are bent away from all those on the left.