@CihanPostsThms

Posting some theorems, and occasionally other stuff. By @bahran_cihan

Joined July 2022
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Some theorems retweeted
sloth12@slothanalystSep 8
Replying to @plain_simon
This might also stem from them having only 16 references which is absurd for a paper of this length
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[Brehm–Kühnel 1987]: For fixed d ∈ℤ⁺, every combinatorial d-manifold with strictly less than 3⌈d/2⌉ + 3 vertices is a d-sphere.
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Up to oriented diffeomorphism, - There is a unique smooth structure on the 61-sphere. - There are exactly 24 smooth structures on the 62-sphere. - There are exactly 142211872163171481167115958878208 smooth structures on the 63-sphere.
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The "look and say sequence" {L[n]} goes 1, 11, 21, 1211, 111221, 312211, 13112221, ... [Conway 1986]: Writing d[n] for the number of digits in L[n], the limit of the ratio d[n+1]/d[n] exists, and is an algebraic integer of degree 71.
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The quadratic integer ring ℤ[√14] is a Euclidean domain, but 𝐍𝐎𝐓 with respect to the natural norm N(a + b√14) := a² - 14b². The first unconditional proof of the Euclidean-ness of ℤ[√14] was by Harper in 2004.
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The maximum number of colors necessary to properly color a map drawn on a closed connected surface with Euler characteristic χ is ⌊½(7 + sqrt(49-24χ))⌋, with a single exception: for the Klein bottle it is 6 (not 7). The 2-sphere case is the Four Color Theorem.
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[Kirby–Siebenmann]. TOP/PL = K(ℤ/2, 3).
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42 = (-80538738812075974)³ + (80435758145817515)³ + (12602123297335631)³
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[Antonyan–Dobrowolski 2015]. For a locally compact Hausdorff group G with a closed subgroup H, TFAE: (1) G/H is locally contractible. (2) G/H is a manifold.
Some theorems@CihanPostsThms19 Mar 2023
[Antonyan 2011]: For a compact Hausdorff group G with a closed subgroup H, TFAE: (1) The coset space G/H is contractible. (2) H = G.
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Some theorems retweeted
Some theorems@CihanPostsThmsJan 31
From the introduction of https://arxiv.org/abs/1902.04011v1

ALT From the introduction of https://arxiv.org/abs/1902.04011v1

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[Bosboom–Demaine–Hesterberg–Lynch–Waingarten 2015]. It is PSPACE-hard to decide whether Player 1 has a forced win in a two-player Mario Kart race from given starting positions for the players.
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Every position of Rubik's cube can be solved in at most 20 half-turn moves. Moreover there are positions for which 19 moves is not enough.
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[Anderson 1977]. For a commutative ring R, every Artinian R-module is countably generated.
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[Elkies–Klagsbrun 2024]. The elliptic curve y² + xy = x³ – 27006183241630922218434652145297453784768054621836357954737385x + 55258058551342376475736699591118191821521067032535079608372404779149413277716173425636721497 has rank ≥ 29.
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[Cohen–Moore–Neisendorfer]: For every odd prime p, the least power of p that annihilates the p-primary part of all the homotopy groups of S²ⁿ⁺¹ is pⁿ.
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The ℤ-rank of the homotopy groups of a simply-connected finite CW-complex of dimension d either become zero in degrees ≥ 2d, or they exhibit exponential growth.
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[Basu–Basu 2015]: The homotopy groups of a simply-connected closed topological 4-manifold are completely determined by its second Betti number.
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[Gorshkov 2013]. Given n≥5, TFAE: (1) Every finite group G with ω(G) = ω(A_n) is isomorphic to A_n. (2) n ∉ {6, 10}.
Some theorems@CihanPostsThms23 Sep 2023
Given an element g of a group G, write |g| ∈ ℤ⁺ ⋃ {∞} for its order, and set ω(G) := {|g| : g∈G}. [Lytkina–Kuznetsov 2007]. Given a group G, TFAE: (1) ω(G) = {1, 2, 3, 4, 7}. (2) G ≅ PSL₂(7).
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It is consistent with ZF that there is an 𝔽₂-vector space V with GL(V) = C₃. Note that GL(𝔽₂⊕𝔽₂) = S₃ .
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[Tachtsis 2018]: ZF has a model in which the Boolean prime ideal theorem holds, but also there is a free group with a non-free subgroup.
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