[123] 3/11/23: The rational oriented cobordism ring is generated by the even dimensional complex projective spaces. [via Hirzebruch-Berger-Jung, "Manifolds and modular forms".]
[122] 3/10/23: The infinite loop space of the algebraic K-theory spectrum of connective K-theory, K(ku), can be expressed as ℤ × BGL1(ku). [via David Gepner's talk "Thom spectra and twisted umkehr maps".]
[121] 3/9/23: The endomorphism algebra of an irreducible finite-dimensional real representation is isomorphic to one of the three real division algebras: ℝ, ℂ, or ℍ. [via Constantin Teleman's "Representation theory" notes.]
[120] 3/8/23: The Thom spectrum functor is the counit of the adjunction: presheaves ⊣ Pic. [via David Gepner's talk "Thom spectra and twisted umkehr maps".]
[119] 3/7/23: The homotopy type of the Morava E-theory spectrum associated to a formal group law of height n is independent of the choice of formal group law. [via Luecke-Peterson, "There aren't that many Morava E-theories".]
[118] 3/6/23: The localization functors on spectra with repect to KU and KO are the same. [via Ravenel, "Localization with respect to certian periodic homology theories".]
[117] 3/5/23: The action of a Poincare duality group G on its cohomology with coefficients in ℤG is nontrivial, unless G=ℤ. [via Bieri-Eckmann, "Groups with homological duality generalizing Poincare duality".]
[116] 3/4/23: (Mahowald) If p=0 in an E∞-ring R, then R is an Eilenberg-Mac Lane spectrum. [via Mike Hill's talk "Equivariant commutative ring spectra".]
[115] 3/3/23: (Smash product theorem) At an odd prime, localizing with respect to complex K-theory (a.k.a. E(1)) is equivalent to smashing with the E(1)-local sphere. [via Ravenel, "Localization with respect to certian periodic homology theories".]
[114] 3/2/23: (Feit-Thompson) Every finite group of odd order is solvable. [via Wikipedia; and Atiyah, "Groups of odd order".]
[113] 3/1/23: Every nontrivial irreducible real representation of a finite group of odd order is a complex representation. [Yang, "On the coefficient groups of equivariant K-theory".]
February 2023[112] 2/28/23: The functor that takes a scheme to the set of vector bundles over it is not representable. [via Gabrielle La Nave; and Gómez, "Algebraic stacks".]
[111] 2/27/23: Every G-vector bundle on a G-space is a direct summand of a trivial G-vector bundle (i.e. a G-representation). [via Ningchuan Zhang's "Equivariant K-theory" notes; and Segal's "Equivariant K-theory".]
[110] 2/26/23: The clutching construction for vector bundles on a sphere is a special case (modulo a trivial excision) of Atiyah-Bott-Shapiro's "difference bundle" construction applied to the pair (disk, boundary).
[109] 2/23/23: The ring spectrum KO is not complex orientable. [via Agnes Beaudry's talk " An introduction to chromatic homotopy theory - Lecture 1".]
[108] 2/22/23: The Morava stabilizer groups are profinite groups. [via Vesna Stojanoska's talk "Deforming representations of the Morava stabilizer group".]
[107] 2/21/23: For n sufficiently large, MO-k(pt) ≅ MSO2n-k(ℝℙ2n). [via Atiyah, "Bordism and cobordism".]
[106] 2/20/23: At the prime 2, the Morava K-theories are not homotopy commutative ring spectra. [via Ravanel's organe book.]
[105] 2/19/23: The loop space of an (n+1)-sphere is homotopy equivalent to a CW-complex with one cell in each dimension divisible by n. [via Ravenel's orange book.]
[104] 2/18/22: For p an odd prime, n an odd positive integer, and k a positive integer, there are no elements of order pn+1 in πn+k(Sn)(p). [via Ravenel's orange book.]
[103] 2/14/23: The endofunctor on the bounded derived category of coherent sheaves on projective space that tensors with the canonical line bundle has no categorical entropy. [via Lucy Yang.]
[102] 2/13/23: The Picard group of KO is isomorphpic to the super Brauer group of ℝ. Is there a simple proof that doesn't use the fact that both groups are ℤ/8? [via Zach Halladay.]
[101] 2/12/23: The Johnson-Wilson spectrum BP❬1❭ is a summand of the p-localization of connective complex K-theory. [via Ravenel's green book.]
[100] 2/11/23: Not every integral homology class on a space X is the fundamental class of a map from a manifold to X. [via Ravenel's green book.]
[99] 2/10/23: The unoriented cobordism spectrum, MO, is a wedge sum of ℤ/2 Eilenberg-Mac Lane spectra. [via Ravenel's green book.]
[98] 2/9/23: The localizations of the cobordism spectra MSO, MSU, MSp, and MU at odd primes are all wedge sums of suspensions of the Brown-Peterson spectrum. [via Ravenel's orange and green books.]
[97] 2/8/23: There exists a finite CW complex whose reduced K-theory is trivial and whose reduced singular cohomology is not trivial. [via Charles Rezk.]
[96] 2/7/23: There exists a (unique) closed symmetric monoidal structure on the category of all topological spaces. [via Riehl, "Categorical homotopy theory".]
[95] 2/6/23: Isomorphism classes of complex line bundles are the same as degree 2 integral cohomology classes (i.e. they have the same classifying space BU(1) ≃ K(𝕫,2)). The first Chern class realizes this identification.
[94] 2/5/23: If G is a p-group or a compact connected Lie group, then the map from the representation ring R(G) to its completion by the augmentation ideal is injective. [via Atiyah, "Characters and cohomology of finite groups"; Atiyah-Hirzebruch, "Vector bundles and homogeneous spaces".]
[93] 2/4/23: The reduced Morava K-theory of a non-contractible p-local finite CW-complex is nonzero for some n. [via Ravenel's orange book.]
[92] 2/3/23: The completion of modules over a Noetherian ring A with respect to an ideal of A is an exact functor. [via Atiyah-Hirzebruch, "Vector bundles and homogeneous spaces".]
[91] 2/2/23: A simply connected CW-complex whose (reduced) homology is torsion is homotopy equivalent to a wedge sum of its p-localizations. [via Ravenel's orange book.]
[90] 2/1/23: A KU-class on a finite CW-complex is nilpotent (or is invertible) if and only if the 0th component of its Chern character vanishes (or is = ± 1). [via Atiyah-Hirzebruch, "Vector bundles and homogeneous spaces".]
January 2023[89] 1/31/23: The universal cover of SLn(ℝ) is not a matrix group for n ≥ 2. [via Rui Loja Fernandes]
[88] 1/30/23: A function between Hilbert C*-modules that has an adjoint is necessarily linear, bounded, and a module homomorphism. [via Vincent Villalobos, and Blackadar, "K-theory for operator algebras".]
[87] 1/29/23: Every small abelian category is equivalent to a full subcategory of the category of modules over a ring. [via Ravenel's orange book; Mitchell, "The full imbedding theorem".]
[86] 1/28/23: Isomorphism classes of central extensions of (a compact Lie group) G by a circle bijectively correspond to degree 3 (integral) cohomology classes of BG. [via Freed-Hopkins-Lurie-Teleman, "TQFTs from compact Lie groups".]
[85] 1/27/23: The p-adic integers form a compact Hausdorff space that is not homotopy equivalent to a CW-complex. [via Ravenel's orange book, and Wikipedia.]
[84] 1/26/23: If X is a simply connected space with no homology in degrees less than n, then there exists a map X → K(πn(X),n) whose images under Hn and πn are isomorphisms. [via Ravenel's green book.]
[83] 1/25/23: An arbitrary sum of orthogonal closed subspaces of a Hilbert space is a closed subspace. The orthogonality condition is necessary even in the finite case. [via Halmos, "Introduction to Hilbert space".]
[82] 1/24/23: An exotic 7-sphere can be immersed in a standard 8-sphere, despite the fact that the (topological) inclusion map as the unit sphere in ℝ8 is not an immersion. [via Rui Loja Fernandes and Oscar Randal-Williams's answer.]
[81] 1/23/23: Homotopy (co)products agree with (co)products in the homotopy category. [via Riehl, "Categorical homotopy theory".]
[80] 1/22/23: The category of super (left) Cl(n)-modules is isomorphic to the category of super (left) Cl(-n)-modules, but Cl(n) is not super Morita equivalent to Cl(-n) when n is not a multiple of 4. [via Dan Berwick-Evans and Theo Johnson-Freyd.]
[79] 1/21/23: If two commutative rings are Morita equivalent, then they are isomorphic. [via Doron Grossman-Naples.]
[78] 1/20/23: Two rings R and S are Morita equivalent if and only if the rings M∞(R) and M∞(S) are isomorphic, where M∞(A) is the ring of countably infinite matrices with finitely many nonzero entries in A. [via Abrams-Ruiz-Tomforde, "Morita equivalence for graded rings".]
[77] 1/19/23: An equivalence of 2-categories C → D does not necessarily induce an equivalence of lax functor 2-categories out of C and D. [via Mike Shulman's post .]
[76] 1/18/23: (Devinatz-Hopkins-Smith) Complex cobordism theory detects nilpotence of self-maps on finite spectra, i.e. if F is a finite spectrum, f : ΣkF → F, and MU*f = 0, then f is nilpotent. [via Vesna Stojanoska.]
[75] 1/17/23: Every (not necessarily smooth) left-invariant vector field on a Lie group is smooth. [via Rui Loja Fernandes.]
[74] 1/16/23: The forgetful functor from simplicial spaces to semi-simplicial spaces has a left adjoint which preserves the weak homotopy type of geometric realizations. [via Ebert–Randal-Williams, "Semi-simplicial spaces".]
[73] 1/15/23: Every uncountable set of nonzero vectors in a Hilbert space (e.g. complex numbers) is not summable. [via Halmos, "Introduction to Hilbert space".]
[72] 1/14/23: A pullback square in which one of the two "legs" is a (quasi)fibration is a homotopy pullback square. [via McDuff, "Configuration spaces".]
[71] 1/13/23: The space of finite subsets of ℝn and the n-fold (based) loop space of the n-sphere have homotopy equivalent classifying spaces. [via McDuff, "Configuration spaces"; Segal, "Configuration-spaces and iterated loop-spaces".]
[70] 1/6/23: Every path-connected space has the homology of an Eilenberg-Mac Lane space; specifically, a K(π,1). [via Adams, "Infinite loop spaces"; Kan-Thurston.]
[69] 1/5/23: The "thin" geometric realization of a degree-wise homotopy equivalence between simplicial spaces need not be a homotopy equivalence. (Segal's "fat" realization fixes this defect.) [via Riehl, "Categorical homotopy theory"; nLab; Segal, "Categories and cohomology theories."]
[68] 1/4/23: The classifying space of the 3-sphere is the infinite quaternionic projective space, i.e. BS3 ≃ ℍℙ∞. [via Adams, "Infinite loop spaces".]
[67] 1/3/23: A morphism in a model category is a weak equivalence iff its image in the homotopy category is an isomorphism. (This statement is false for general "homotopical categories".) [via Riehl, "Categorical homotopy theory".]
[66] 1/2/23: The inclusion of O(n) into the diffeomorphism group of an (n-1)-sphere is a homotopy equivalence iff 0 ≤ n ≤ 4. (The n = 4 case is not yet published.) [via Lennart Meier's comment; Smale; Hatcher; Wikipedia; Watanabe, "Some exotic nontrivial elements..."]
[65] 1/1/23: A path-connected, commutative, associative H-space with a strict identity is weakly equivalent to a product of Eilenberg-Mac Lane spaces. [via Sullivan's 1970 MIT notes; Hatcher, "Algebraic topology".]
December 2022[64] 12/31/22: Not every Poincaré duality space is (weakly) homotopy equivalent to a smooth manifold. [via Madsen-Milgram, "The classifying spaces for surgery and cobordism of manifolds."]
[63] 12/30/22: Spin structures on a compact complex manifold bijectively correspond to holomorphic square roots of the canonical line bundle. [via Atiyah, "Riemann surfaces and spin structures".]
[62] 12/29/22: Every KU-module is a KO-module via complexification : KO → KU. In particular, KU is equivalent to KO ∧ Σ-2ℂℙ2 as a KO-module. [via Adams's "Blue book"; Wolbert, "Classifying modules over K-theory spectra"; Snaith, "A descent theorem for Hermitian K-theory".]
[61] 12/28/22: The category of super vector spaces and grading preserving linear maps has: algebra objects = super algebras, and internal hom(V,W) = all linear maps V → W. [via Dan Berwick-Evans.]
[60] 12/26/22: If Fp is a field of prime order p ≠ 2 such that either a or -a has a square root for all a ∊ Fp×, then p ≡ 3 mod 4. [via Lawson-Michelsohn, "Spin geometry".]
[59] 12/25/22: The spin group, Spin(n), is contained in both Clifford algebras Cl(± ℝn), where +/- corresponds to the positive/negative definite inner product on ℝn. [Lawson-Michelsohn, "Spin geometry".]
[58] 12/23/22: The Picard group of the category of spectra is the integers (represented by the spheres). [via Richard Wong, "Picard groups"; and Hopkins-Mahowald-Sadofsky, "Constructions of elements in Picard groups".]
[57] 12/22/22: The category of simplicially enriched categories is the full subcategory of simplicial (objects in) categories on the simplicial categories whose object sets are constant. [via kerodon.net]
[56] 12/21/22: The map from degree 3 cohomology (with integer coefficients) to twists of K-theory can be seen as a delooping of the Atiyah-Bott-Shapiro construction. [via Ando-Blunberg-Gepner, "Twists of K-theory and TMF".]
[55] 12/20/22: An endofunctor is the composite of an adjoint pair if and only if it carries the structure of a monad. [via Awodey, "Category theory".]
[54] 12/19/22: The (integral) homology of the 2kth space in the complex cobordism spectrum has no torsion. [via Hopkins's "Universal Hopf rings" talk; and Ravenel-Wilson's "The Hopf ring for complex cobordism".]
[53] 12/18/22: The forgetful functor Rings → Sets is the affine scheme 𝔸1 ≅ spec(𝕫[t]). [via Strickland, "Formal schemes and formal groups".]
[52] 12/17/22: The classifying space of a symmetric monoidal category is an E∞-space. [via Adams, "Infinite loop spaces".]
[51] 12/16/22: The ∞-category of presheaves of spaces on a space X is equivalent to the ∞-category of spaces over X. [via Ando-Blumberg-Gepner-Hopkins-Rezk, "An ∞-categorical approach to R-line bundles, R-module Thom spectra, and twisted R-homology".]
[50] 12/15/22: The homotopy category of combinatorial (a.k.a. simplicial/semisimplicial) spectra is equivalent to the homotopy category of spectra of simplicial sets. [via nLab, and Kan's "Semisimplicial spectra".]
[49] 12/14/22: The category of abelian groups is equivalent to a full subcategory of the ∞-category of spectra given by the Eilenberg-Mac Lane spectra. [via Lurie's "Finiteness and ambidexterity in K(n)-local stable homotopy theory (Part 4)" talk; and "Higher algebra".]
[48] 12/13/22: The first and second Stiefel-Whitney classes of a vector bundle V are equal to the first and second Stiefel-Whitney classes of Cl(V), the associated bundle of Clifford algebras. [via Donavan-Karoubi, "Graded Brauer groups and K-theory with local coefficients".]
[47] 12/12/22: The 2nd homotopy group of a Lie group is trivial. [via Dan Berwick-Evans; and this answer.]
[46] 12/11/22: The empty space is ambidextrous with respect to a category C if and only if C is pointed. [via Lurie's "Finiteness and ambidexterity in K(n)-local stable homotopy theory (Part 3)" talk.]
[45] 12/10/22: The configuration space of k points in infinite dimensional Euclidean space is a classifying space for the symmetric group on k elements. [via McDuff, "Configuration spaces" in "K-theory and operator algebras".]
[44] 12/9/22: Every (degree zero) K-homology class on a compact Hausdorff space X is represented by a Fredholm operator between Hilbert C(X)-modules which is C(X)-linear modulo compact operators. [via Atiyah, "A survey of K-theory" in "K-theory and operator algebras".]
[43] 12/8/22: A pointwise weak equivalence of bisimplicial sets (abelian groups) induces a weak equivalence of the associated diagonal simplicial sets (abelian groups). [via Goerss-Jardine, "Simplicial homotopy theory".]
[42] 12/7/22: An n-fold loop space is an En-space. [via Adams, "Infinite loop spaces"; also May and Lurie.]
[41] 12/6/22: Every path-connected topological space is weakly equivalent to the classifying space of a monoid. [via Dusa McDuff's "On the classifying spaces of discrete monoids".]
[40] 12/5/22: The rational homotopy category of simply connected spaces is equivalent to the homotopy category of reduced differential graded Lie algebras over ℚ. [via Lurie's "Lie algebras and homotopy theory" talk, and Quillen's "Rational homotopy theory".]
[39] 12/4/22: A category with binary products (or coproducts) is contractible. [via Omar Antolín Camarena's website.]
[38] 12/3/22: There is an element [f : S4m-1 → S2m] ∈ π4m-1(S2m) of infinite order for all m > 0. [via Ravenel, "Complex cobordism and stable homotopy groups of spheres".]
[37] 12/2/22: The cohomology of a simply connected 4-manifold has no torsion. [via Robert Dicks.]
[36] 12/1/22: The Thom space of a vector bundle V → X can be expressed as a quotient of projective bundles P(V ⊕ 1)/P(V). [via Davis Deaton; and Daniel Dugger's "A geometric introduction to K-theory".]
November 2022[35] 11/30/22: A map between simple spaces that induces an isomorphism on cohomology is a weak equivalence. [via Langwen Hui.]
[34] 11/29/22: (Dold-Thom) The homotopy groups of the infinite symmetric product of a connected CW complex X are isomorphic to the reduced homology groups of X. [via nLab and Hatcher, "Algebraic Topology".]
[33] 11/28/22: The algebraic K-theory groups of a field of order k are isomorphic to the homotopy groups of the homotopy fixed points of the Adams operation ψk. [via Quillen, "On the cohomology and K-theory of the general linear groups over a finite field".]
[32] 11/27/22: The Thom space of a sphere bundle is its homotopy cofiber. [via Eric Peterson, "Formal geometry and bordism operations".]
[31] 11/26/22: The term "loop space" is equivalent to "A∞-space whose path-components form a group". [via Adams, "Infinite loop spaces".]
[30] 11/25/22: Every finitely presented group is the fundamental group of a compact n-manifold, for all n ≥ 4. [via Jonathan Rosenberg, "Surgery theory today" in "Surveys on surgery theory, vol. 2".]
[29] 11/23/22: A loop space ΩX is homotopy equivalent to the space Ω'X of (Moore) loops of length t, t ≥ 0. [via Adams, "Infinite loop spaces".]
[28] 11/22/22: For an 8k-dimensional Riemannian spin manifold X: S⊗S ≅ Cl(X) as Cl(X)-Cl(X)-bimodules, where S is the irreducible real spinor bundle of X, and Cl(X) is the bundle of Clifford algebras of the tangent spaces of X. [via Lawson-Michelsohn, "Spin geometry".]
[27] 11/21/22: If D is the Dirac operator of a Dirac bundle, then ker(D) = ker(D2). [via Lawson-Michelsohn, "Spin geometry".]
[26] 11/20/22: The map BO → BGL1(S) → BGL1(KO) induced by the J-homomorphism is a 2-equivalence. [via Eric Peterson's "Formal geometry and bordism operations", with help from Charles Rezk and Eric Peterson.]
[25] 11/19/22: A loop space is an H-space. [via Adams, "Infinite loop spaces".]
[24] 11/18/22: For a topological group G, not every contractible space with free G-action is the total space of a universal G-bundle EG → BG. [via Charles Rezk's Math 526 (algebraic topology II) class.]
[23] 11/17/22: The ring of symmetric functions is the Grothendieck group of the category Schur, whose objects are finite direct sums of finite dimensional representations of distinct symmetric groups. [via Baez-Moeller-Trimble, "Schur functors and categorified plethysm"]
[22] 11/16/22: The sphere spectrum is the first Weiss-derivative of the functor which takes an inner product space V to BO(V), where O(V) is the orthogonal group of V. [via Weiss, "Orthogonal calculus".]
[21] 11/15/22: The homotopy fiber of the map BO(n) → BO(n+1) induced by the inclusion O(n) → O(n+1) is the n-sphere. [via Weiss, "Orthogonal calculus".]
[20] 11/14/22: (Edited on 11/20/22:) A given KO-theory Thom isomorphism (for a vector bundle) may not arise from any particular spin structure. [via Charles Rezk.] Example?
[19] 11/13/22: "KO-theory is the first Weiss-derivative of the K-theory of Clifford algebras." [via Charles Rezk's answer here .]
[18] 11/12/22: For a finite p-group, the correspondence between representaions and characters can be expressed as a relationship between K(BG) and H(Loops(BG)). [via Jacob Lurie's lecture on "Loop spaces, p-divisible groups, and character theory", Notre Dame, 2012.]
[17] 11/11/22: A spin structure on a manifold X is not necessary for the existence of a bundle of irreducible real Clifford modules. Example: X = the complex projective plane. [via Xuan Chen's thesis, "Bundles of irreducible Clifford modules and the existence of spin structures".]
[16] 11/10/22: The homotopy groups of a simplicial abelian group A are naturally isomorphic to the homology groups of A viewed as a chain complex with boundary map = alternating sum of face maps. [via Goerss-Jardine, "Simplicial homotopy theory".]
[15] 11/9/22: The KO-theory suspension class cannot be represented by a (locally trivial) bundle of Clifford modules. [via Dan Berwick-Evans.]
[14] 11/8/22: A continuous map is a weak equivalence if and only if its homotopy fiber is contractible. [via Miguel Barata.]
[13] 11/7/22: Every abelian von Neumann algebra is isomorphic to the algebra of L-infinity functions on a measure space. [via Connor Grady.]
[12] 11/5/22: A continuous map with contractible fibers need not be a (weak) homotopy equivalence. (This is analogous to the fact that a continuous bijection need not be a homeomorphism.) [via Dan Freed's lecture notes on K-theory.]
[11] 11/4/22: The degree 1 shift between the Bott periodicity of the orthogonal/unitary group and the KO/KU-theory of spheres can be seen by the clutching construction of vector bundles: the equator of a sphere is a sphere of one dimension lower. [via Dan Freed's lecture notes on K-theory.]
[10] 11/3/22: Every invertible super (real/complex) algebra is Morita equvalent to a (real/complex) Clifford algebra. [via various lecture notes of Dan Freed.]
[9] 11/2/22: The Todd class of a vector bundle measures the difference between the Thom classes in K-theory and ordinary cohomology (related by the Chern character). [via Lawson-Michelsohn, "Spin geometry".]
[8] 11/1/22: The index of a (self-adjoint) Dirac operator D is the supertrace of the projection onto kernel(D). [via Berline-Getzler-Vergne, "Heat kernels and Dirac operators".]
October 2022[7] 10/31/22: Geometric sheaves on manifolds are manifolds. [via Dan Berwick-Evans.]
[6] 10/30/22: The prestack (on spaces) represented by a topological groupoid is not always a stack. [via Carchedi, "Categorical properties of topological and differentiable stacks" and Connor Grady.]
[5] 10/29/22: For any map of spectra, f, the homotopy fiber of f is weakly equivalent to loops of the homotopy cofiber of f. [via Barnes-Roitzheim, "Foundations of stable homotopy theory".]
[4] 10/28/22: There exist exotic spheres of arbitrarily large dimension. In particular, in dimensions 8k + 1 and 8k + 2, for all k. [via Lawson-Michelsohn, "Spin geometry"; due to Adams and Milnor.]
[3] 10/27/22: The spin cobordism group in dimension n is isomorphic to the nth stable homotopy group of MSpin. [via Lawson-Michelsohn, "Spin geometry".]
[2] 10/26/22: The Adams operations can be defined by composing the total K-theory power operation (of Atiyah) with the function on the representation ring of the symmetric group that takes the trace of the "long cycle". [via Davis Deaton.]
[1] 10/25/22: [I turn 25.] The (standard) Clifford algebra contains the Lie algebra so(n) of Spin(n) as well as the group Spin(n) itself. Exponentiation can then be seen as taking place entirely inside of a Clifford algebra. [via Berline-Getzler-Vergne, "Heat kernels and Dirac operators".]