The law of a split Tiyomes/Center leaf of the Lulav:
What is the Tiyomes? In Halacha, special attention is given to the Tiyomus of the Lulav, and the criteria it must fulfill for the Lulav to be Kosher. The term Tiyomes comes from the word “twins”, and refers to the back area of the leaf of a Lulav, where the leaves are attached as pairs, or twins. More specifically, the Poskim explain that the Tiyomes refers to the top center leaf, which is higher than all the other leaves, and extends from the spine of the Lulav. [If a Lulav has two middle leaves of equal height, some Poskim rule that both leaves have the status of a Tiyomes.] This leaf, being that it grows in the center of the Lulav and is the highest leaf, is considered the head of the Lulav. Therefore, special laws and criteria are given to this top center leaf. In this Halacha, we will discuss the law if the Tiyomes leaf split by its back, and separated from its pair.
The law if the leaf split entirely: The Lulav naturally grows with paired leaves, in which each leaf is attached to a second leaf at its back. This likewise applies to the top center leaf, called the Tiyomes, which naturally grows as a pair of two leaves which it is attached by its back. If this center leaf grew without a second leaf to which it is paired with in the back, the Lulav is invalid [on the first day(s) of Sukkos]. This applies even if all the other leaves of the Lulav are double sided [and remain attached at the back]. Likewise, [even] if the Tiyomes grew as double sided attached leaf and afterwards the pair split into two leaves in its entire length, from the top of the leaf until the [top of the spine, which is the] area that the leaves sprout forth from, then the Lulav is invalid [on the first day(s) of Sukkos]. [If a Lulav has two middle leaves of equal height, then if any one of the two leaves completely splits, the Lulav is invalid.]
The law if the leaf partially split: The above invalidation only applies if the top center leaf split in its entire length [up until the top of the spine]. However, if it only partially split [even if it split in its majority], it remains Kosher. Nonetheless, this only applies after the fact [that one already said a blessing over the Lulav], or if one does not have another Lulav available, in which case one is not required to say a blessing over the Lulav of his friend [which has a complete Tiyomes, even] on the first day of Sukkos [and may use his Lulav that has a partially split Tiyomes]. However, Lechatchilah, it is a Mitzvah Min Hamvchar to search for [and buy] a Lulav whose top-center leaf has not split at all, and is completely attached to each other [at the back] from its top to the point the other leaves begin shooting out from the spine. [If a Lulav has two middle leaves of equal height, then it is a Mitzvah Min Hamuvchar for both leaves to be completely attached.]
On Chol Hamoed: All the above discussion of invalidation, and initial practice, regarding a split Tiyomes, only applies on the first day(s) of Sukkos, when the Mitzvah of Lulav is Biblical. However, on Chol Hamoed, a Lulav which has a deficiency of not being complete, is valid. Accordingly, during Chol Hamoed, a lulav which contains even a completely split Tiyomus, is Kosher.
On the second day of Sukkos in Chutz Laretz: It is debated in Poskim as to whether the second day of Sukkos in the Diaspora Rabbinically retains the stricter laws of the first day, regarding invalidations, or if it receives the lenses associated with Chol Hamoed regarding invalidations. Practically, Safek Brachos Lihakel, and therefore if one contains a Lulav with a completely split Tiyomes, he is to borrow another person’s Lulav on the second day and recite the blessing over it. He may use his own Lulav for the Nanuim of Hallel. If there is no other Lulav available, he is to use it without a blessing.
The Tiyomes is the middle upper most leaf [or leaves, if there are two middle leaves of the same height] which extends from the spine of the Lulav. Lechatchilah it is a Mitzvah Min Hamvchar to buy a Lulav whose double leafed Tiyomus is completely attached to (at the back) from its top, to the point the other leaves begin shooting out from the spine. If this is not available, or it split after buying it, then the Lulav remains valid so long as the leaf is not completely separated from the top until the spine [where the other leaves begin growing from it]. Furthermore, even if one has a friend which owns a Lulav with a complete Tiyomes, one is not required to use it and rather may use his Lulav that has a partially split Tiyomes. If the Tiyomes grew without a double-sided leaf, or if the leaf split entirely from the top until the spine, the Lulav is invalid on the first day of Sukkos, and may not be used with a blessing on the second day of Sukkos in the Diaspora. On Chol Hamoed, the Lulav remains Kosher even if the Tiyomus is completely split.
Must the Tiyomes be double leafed throughout its entire length?
It is implied from Admur that it is a Mitzvah Min Hamuvchar for the Tiyomes to be entirely double leafed from the top of the Tiyomes until the area of the spine, and so rule some Rabbanim. However, others learn that it is not necessary for the Tiyomes to be entirely double leafed and so is the ruling of many great Rabbanim.
What if the Tiyomes is split only at its tip as is common to occur?
It is a Mitzvah Min Hamuvchar to purchase a Tiyomes that is completely closed, including its tip.
If the split of the Tiyomes is only noticeable after close examination is it initially invalid?
Some Poskim rule that initially one is to purchase a Lulav that its Tiyomes is fully closed even after proper contemplation. [It is not necessary to use a magnifying glass.]
What if the Tiyomes is split in the middle, but attached on top?
Some Poiskim rule the Lulav remains valid.
Does it help to glue a split Tiyomes leaf back together?
Some Poskim rule it is valid to glue the Tiyomes together.
 Sukkah 32b “What is the law if the Tiyomes split”
 Rashi Sukkah ibid
 Admur 645:10 [to note however that he does not mention the term Tiyomes]; Rama 645:3 in name of Yeish Mefarshim and that so is custom; Terumos Hadeshen 96; Rashi Sukkah ibid “It refers to the two middle leaves, from where the spine ends”; Tosafus Sukkah 32b; Or Zarua 2:306
Other opinions: Some Poskim rule the Tiyomes is a general term for the back of every pair of leaf of the Lulav, in which the two leaves are attached to each other, and does not specifically refer to the most top center leaf. According to this approach, a split Tiyomes means that the majority of leaves of the Lulav contain split pairs of leaves. [Michaber 645:3; Taz 645:4 that so rule majority of Poskim] Other Poskim rule the Tiyomes refers to the top leaves of the Lulav, and that are attached with the brown/reddish chaff in both its front and back. [Teshuvas Hageonim, brought in Beis Yosef 645; Taz 645:4; P”M 645 M”Z 4; Implication of Bahag, brought in Tosafus Sukkah 32b]
 Admur ibid; Rashi Bava Kama 96a; Ran Sukkah 15; Ritva Sukkah 31b; Maggid Mishneh Sukkah 8:4
 Taz 645:4; Bach 645; Kneses Hagedola 645:11; Elya Raba 645:4; M”B 645:15; ; Kaf Hachaim 645:23
 Admur ibid
 Admur 645:10; Rama 645:3 in name of Yeish Mefarshim and that so is custom; Rashi Sukkah ibid “And the spine split until the lower leaves”; Tosafus Sukkah 32b; Terumos Hadeshen 96; Or Zarua 2:306;
Other opinions: Some Poskim rule the Tiyomes is a general term for the back of every pair of leaf of the Lulav, in which the two leaves are attached to each other, and does not specifically refer to the most top center leaf. According to this approach, there is no difference between the middle leaf and any of the other leaves of the Lulav, and therefore the Lulav remains Kosher even if it completely split. [Implication of Michaber 645:3; and so explains: Beis Yosef 645; M”B 645:12; Kaf Hachaim 645:18; Taz 645:4 that so rule majority of Poskim]
 Admur 645:9; Michaber 645:3
 Admur ibid; M”A 645:3; Darkei Moshe 645:5; Terumos Hadeshen 96; Seemingly, this is coming to negate the explanation of the Geonim [brought above] who explain it refers to the chaff that attaches the top leaves to each other.
 See continuation for law on Chol Hamoed and 2nd day of Yom Tov in the Diaspora
 Admur ibid; Rama ibid; Kol Bo 72
 (However, lower than this area, is not called the middle leaf [Tiyomes] but rather is part of the actual spine.) [Admur ibid, parentheses in original]
 See continuation for law on Chol Hamoed and 2nd day of Yom Tov in the Diaspora
 Admur 645:10; Rama ibid; Rashi ibid; Tosafus Sukkah ibid; Terumos Hadeshen ibid; Or Zarua ibid
The reason: The invalidation of this middle leaf, the Tiyomes, is not due to a blemish in the Hadar/beauty of the Lulav, but rather due to [incompletion as] the Torah states “Ulikachtem Lachem” and the Sages expounded that it must be complete and whole. Now, since this middle leaf is considered the centerpiece of the Lulav, as one sees it in his first sight of the Lulav, and it naturally grows double-sided in all the Lulavim, therefore, if it is [completely] split to two leaves, being that it is readily apparent and visible that the Lulav is not complete in the area of the split [therefore it is invalid]. [Admur ibid; M”A 645:6; Rabbeinu Yerucham Nesiv 8 3 p. 58]
 Taz 645:4; Bach 645; Kneses Hagedola 645:11; Elya Raba 645:4; M”B 645:15
 Admur 645:11; Rama 645:3; Siddur Yaavetz; Bikureiy Yaakov 645:10; Aruch Hashulchan 645:10; M”B 645:18; Kaf Hachaim 645:27; Chazon Ish 145:1; Piskeiy Teshuvos 645:4
 Implication of Admur ibid, Rama ibid and Poskim ibid who constantly write “Split in its entire length”; Bikureiy Yaakov 645:10 in name opf Gedolei Chachmei Ashkenaz, which is the Terumas Hadeshen, Or Zarua and Semak; Chazon ish 145:1; Piskeiy Tehsuvos 645:4 footnote 17
Other opinions: Some Poskim rule that if majority of the length of the Tiyomes split, it is considered as if it split in its entirety, and the Lulav is invalid. [M”B 645:15; Taz 645:4; Bach 645; Kneses Hagedola 645:11; Elya Raba 645:4; Biur Hagr”a 645; Ran Sukkah 15; Ritva Sukkah 31b; Maggid Mishneh Sukkah 8:4; Kaf Hachaim 645:23 and 25]
Opinion of Admur: Although Admur constantly writes in 645:10 and 11 the words “in all its length” when discussing the invalidation. On the other hand, when discussing the validation in 645:11, Admur writes “If only Miktzas/part” was split. Some desired to
 And certainly, on Chol Hamoed, and the second day of Sukkos, one is not required to borrow another person’s Lulav, as on Chol Hamoed the entire invalidation of a split Tiyomes is not applicable.
 Admur ibid; M”B 645:18; Kaf Hachaim 645:27
 Admur ibid; Rama ibid that so is custom; Chayeh Adam 149:10; Kaf Hachaim 645:28
The reason: The reason for this is because there are opinions [brought in Ran ibid] who rule that the Lulav is invalid even if the middle leaf is only partially split. Now, although we do not rule like this opinion, nevertheless it is proper to suspect for their opinion when possible. [Admur ibid; Rama ibid; Ran Sukkah 15; Ritva Sukkah 31b; Kol Bo 72] The reason for this is because once part of the leaf has split, we suspect that the entire leaf will split through the shaking of the Lulav. [See Biur Hagr”a 645]
Other opinions: Some Poskim rule that a leaf whose Tiyomes has split on its top less than a Tefach worth [8 cm] is valid according to all opinions, even according to the Ran ibid, and accordingly, there is no need to search for a Lulav with a completely attached Tiyomes, so long as it is not a Tefach split. [Taz 645:4; Elya Raba 645:4; Beir Heiytiv 645:5; Machatzis Hashekel 645:4
 Elya Raba 645:4; Shaar Hatizyon 645:16; Kaf Hachaim 645:23
 Admur 649:17; Rama 649:5; Tosafus Sukkah 29b; Rosh 3:3
 Admur 649:19; M”A 645:6; M”B 645:17; Rabbeinu Yerucham Nesiv 8 3 p. 58; Piskeiy Teshuvos 645:4
Other opinions: Some Poskim rule a split Tiyomes is invalid also on Chol Hamoed. [Rav Akiva Eiger]
 649:21; Michaber 649:5
 Kashrus Daled Minim [Shtern] in name of Rav Elyashiv based on Admur 645:9
 Rav SZ”A and others mentioned in Piskeiy Teshuvos 645:5 footnote 24
 Piskeiy Teshuvos 645:9
 Piskeiy Teshuvos 645:9; Shut Rav Akiva Eiger
 Bikureiy Yaakov 645:6; Piskeiy Teshuvos 645:4
 Piskeiy Teshuvos 645:4 in name of Rav YY Fisher, as the entire reason for the invalidation is due to it continuing to split, which is prevented with glue