135 Proofs of Bott periodicity 2009-12-13T20:48:03.670

122 What non-categorical applications are there of homotopical algebra? 2014-06-06T06:50:41.107

92 What do the stable homotopy groups of spheres say about the combinatorics of finite sets? 2011-09-27T18:18:27.343

91 What properties make $[0,1]$ a good candidate for defining fundamental groups? 2012-03-25T22:21:08.580

70 What is the mistake in the proof of the Homotopy hypothesis by Kapranov and Voevodsky? 2016-03-25T11:47:32.607

65 Is there a high-concept explanation for why "simplicial" leads to "homotopy-theoretic"? 2011-03-15T02:24:29.977

60 What are the uses of the homotopy groups of spheres? 2010-04-28T11:27:13.473

58 third stable homotopy group of spheres via geometry? 2010-11-04T19:56:50.020

51 Do we still need model categories? 2011-10-17T23:57:59.800

51 What is Homology anyway? 2017-07-23T11:43:55.147

48 What are surprising examples of Model Categories? 2010-03-16T04:36:08.120

44 What are the higher homotopy groups of Spec Z ? 2009-10-28T18:10:43.877

44 Total spaces of $TS^2$ and $S^2 \times R^2$ not homeomorphic 2012-03-27T01:52:20.043

43 Is $\mathbb{R}^3 \setminus \mathbb{Q}^3$ simply connected? 2015-08-29T18:46:53.497

41 Cochains on Eilenberg-MacLane Spaces 2010-12-03T05:00:40.533

40 What is modern algebraic topology(homotopy theory) about? 2016-01-14T07:45:29.343

39 Non-examples of model structures, that fail for subtle/surprising reasons? 2010-05-02T18:07:59.907

39 Spaces with same homotopy and homology groups that are not homotopy equivalent? 2011-01-26T20:32:47.927

39 Homotopy as a general organizing principle 2013-07-15T14:40:28.223

37 Why does one think to Steenrod squares and powers? 2009-11-21T16:58:36.800

37 Why the "W" in CGWH (compactly generated weakly Hausdorff spaces)? 2010-11-29T18:20:48.953

37 Homotopy type of TOP(4)/PL(4) 2012-09-12T02:38:57.530

36 Are there any "homotopical spaces"? 2010-04-17T15:00:07.727

36 Homotopy groups of $S^2$ 2011-05-06T15:34:45.893

36 Is there an analog of Sperner's lemma for the Hopf invariant? 2014-01-24T21:59:44.020

35 Finite complexes whose homotopy groups are not "finitely generated" 2014-05-29T01:32:43.870

34 What is the "intuition" behind "brave new algebra"? 2009-11-07T21:33:31.750

34 Are there pairs of highly connected finite CW-complexes with the same homotopy groups? 2009-11-08T19:17:15.200

34 Why do finite homotopy groups imply finite homology groups? 2009-12-20T11:41:02.430

34 Definition of an E-infinity algebra 2010-08-23T13:20:53.233

34 Functor that maps to both $KO^n$ and $KO^{-n}$ 2011-04-22T19:35:56.653

34 Why should I care about topological modular forms? 2014-01-01T04:27:14.980

34 Why the Dold-Thom theorem? 2014-10-07T17:48:24.690

33 Analogue to covering space for higher homotopy groups? 2009-10-17T23:13:42.673

33 What (if anything) unifies stable homotopy theory and Grothendieck's six functors formalism? 2014-06-08T00:08:25.230

33 Why is it so hard to compute $\pi_n(S^n)$? 2014-10-20T23:27:44.863

33 Maps which induce the same homomorphism on homotopy and homology groups are homotopic 2015-02-05T22:24:05.950

33 What can topological modular forms do for number theory? 2015-05-28T02:40:33.313

33 Why is the definition of the higher homotopy groups the "right one"? 2017-10-21T17:39:31.130

33 Defining $SU(n)$ in HoTT 2018-01-02T00:15:59.947

32 Integral cohomology (stable) operations 2010-12-27T23:02:06.913

32 What do loop groups and von Neumann algebras have to do with elliptic cohomology? 2012-07-06T01:01:01.097

32 Is there a "simplification" functor in algebraic topology? 2015-01-08T22:10:06.993

31 How to think about model categories? 2009-10-23T21:00:31.737

31 Topological Langlands? 2009-11-30T15:32:20.337

31 How can I visualize the nontrivial element of $\pi_4(S^3)$ and $\pi_5(S^3)$ ? 2010-09-11T03:28:53.813

31 `Naturally occuring' $K(\pi, n)$ spaces, for $n \geq 2$. 2010-10-29T02:13:33.973

31 Classifiying sphere eversions 2015-09-08T09:23:26.207

31 Is there anything to the obvious analogy between Joyal's combinatorial species and Goodwillie calculus? 2017-04-04T01:22:54.683

31 What are the potential applications of perfectoid spaces to homotopy theory? 2017-06-30T06:05:15.757

31 Why do we need model categories? 2017-11-27T14:50:46.313

30 For which spaces is homology (or cohomology) determined by the Eilenberg-Steenrod axioms 2009-10-22T01:19:24.943

30 "Homotopy-first" courses in algebraic topology 2010-03-20T15:40:17.460

30 In which situations can one see that topological spaces are ill-behaved from the homotopical viewpoint? 2011-09-27T13:11:07.103

29 Homotopy Type Theory: What is it? 2015-03-18T00:49:41.323

28 How should a homotopy theorist think about sheaf cohomology? 2010-07-20T20:19:43.310

28 Is the space of diffeomorphisms homotopy equivalent to a CW-complex? 2013-04-08T07:35:28.937

28 Unstable homotopy groups of spheres beyond Toda's range 2014-12-17T16:31:06.333

28 What is the fewest number of points you must delete from $\mathbb{R}^3$ to make it not simply connected? 2015-08-31T21:33:34.020

27 On the (derived) dual to the James construction. 2011-02-04T20:42:51.473

27 K(r)-localization and monochromatic layers in the chromatic spectral sequence 2012-10-06T20:29:58.810

27 Is every ''group-completion'' map an acyclic map? 2012-10-14T11:51:18.433

27 What is an $(\infty,1)$-topos, and why is this a good setting for doing differential geometry? 2013-02-10T07:15:35.643

27 Why not a Roadmap for Homotopy Theory and Spectra? 2013-11-15T20:49:47.550

27 Are the higher homotopy groups of the Hawaiian earring trivial? 2014-04-20T01:12:15.557

27 What is a triangle? 2016-12-18T05:14:38.977

27 Quillen + construction for finite groups 2017-10-07T18:04:36.037

26 Spectrum of the Grothendieck ring of varieties 2009-10-11T22:10:05.750

26 What is 'formal' ? 2010-01-26T03:02:53.733

26 Identifying the stacks in Devinatz-Hopkins-Smith 2010-09-29T06:10:14.797

26 Poincare duality and the $A_\infty$ structure on cohomology 2010-12-23T11:32:48.923

26 Surveys of Goodwillie Calculus 2012-01-23T17:27:58.317

26 Is there a (discrete) monoid M injecting into its group completion G for which BM is not homotopy equivalent to BG? 2013-02-18T19:34:34.677

26 Is there a general theory of fiber theorems? 2013-03-19T02:53:25.647

26 Why should have Peter May worked with CGWH instead of CGH in "The Geometry of Iterated Loop Space"? 2015-04-28T14:19:29.270

26 What is the relation between sphere spectrum and supersymmetry? 2016-04-14T21:48:42.200

26 Is a filtered colimit of rational spaces again rational? 2016-08-12T23:07:49.560

25 triangulated vs. dg/A-infinity 2009-10-17T00:21:51.583

25 What is so "spectral" about spectra? 2010-05-10T12:08:05.143

25 A question about the topological proofs of Bott periodicity 2011-01-27T23:26:23.843

25 H-principle and PDE's 2012-02-15T12:34:57.400

25 The homotopy category is not complete nor cocomplete 2016-05-20T15:52:55.387

24 (∞, 1)-categorical description of equivariant homotopy theory 2009-10-28T22:28:24.860

24 Do the signs in Puppe sequences matter? 2009-11-18T02:17:28.240

24 Noncommutative rational homotopy type 2009-12-17T01:59:42.367

24 Classifying Space of a Group Extension 2010-01-17T21:18:38.403

24 What's the current state of the classification of not-fully-extended TQFTs? 2010-07-09T05:35:19.413

24 Smooth homotopy theory 2010-08-11T07:26:49.510

24 Modern Source for Spectra (including Ring Spectra) 2011-11-23T19:13:18.537

24 What is the algebraic geometry version of the spheres? 2012-12-07T00:09:05.683

24 origin of spectral sequences in algebraic topology 2014-01-31T13:36:52.137

24 From the perspective of bordism categories, where does the ring structure on Thom spectra come from? 2014-07-09T17:27:23.587

24 Morava K-theories for dummies? 2016-11-21T19:28:11.210

24 Atiyah-Singer style index theorem for elliptic cohomology? 2017-06-29T06:06:14.370

23 Failure of smoothing theory for topological 4-manifolds 2009-12-05T23:10:28.147

23 Is super-vector spaces a "universal central extension" of vector spaces? 2011-12-08T17:39:12.210

23 Conjectures in Grothendieck's "Pursuing stacks" 2012-12-05T21:54:19.843

23 Is the counit of geometric realization a Serre fibration? 2013-06-24T11:39:51.910

23 Conceptual explanation for the relationship between Clifford algebras and KO 2013-08-07T00:00:39.007