92 problems.
- Nonsurjective maps into \(S^n\) are nullhomotopic
- Maps into \(S^n\) that are nowhere antipodal are homotopic
- Antipodal map on \(S^n\) is homotopic to the identity for \(n\) odd
- \(X\) is contractible iff \(\mathrm{id}_X\) is homotopic to a constant
- \(S^1 \times I \simeq\) the Möbius strip
- \(\mathbb{R}^3\setminus S^1\simeq S^1\vee S^2\)
- Letters of the alphabet up to homeomorphism and homotopy
- \(f\simeq g\) implies \(X\cup_f B^2 \simeq X\cup_g B^2\)
- Paths in a simply connected space are homotopic rel endpoints
- A path-connected space is simply connected iff every map \(S^1\to X\) extends over \(B^2\)
- The fundamental group of a product is the product of the fundamental groups
- \(\pi_1(S^n)=1\) for \(n\geq 2\)
- \(S^2 - \{p_0, p_1\} \simeq S^1\)
- \(S^3 - \{p_0, p_1\} \simeq S^2\)
- \(S^2\) is not homeomorphic to \(S^3\)
- Induced maps on \(\pi_1(S^1)\) for \(z^n\), antipodal, and \(e^{2\pi i\sin\theta}\)
- Winding number of \(8z^4+4z^3+2z^2+z^{-1}\) on \(S^1\)
- \(\pi_1\) of the Möbius strip and the class of the boundary
- Conjugation automorphism of \(\pi_1(T^2)\) induced by a loop
- Octagon pasting homeomorphic to \(\mathbb{R}^2/\mathbb{Z}^2\)
- Order-6 homeomorphism of the torus fixing the origin
- \(\pi_1\) of the Klein bottle as \(\ZZ\rtimes\ZZ\), and the torus as double cover
- \(\pi_1(S^n/\ZZ_2)\) for three \(\ZZ_2\)-actions
- The equatorial inclusion \(\mathbb{RP}^2\to\mathbb{RP}^3\) is not nullhomotopic
- No map \(S^2\to S^1\) commuting with the antipodal map
- Borsuk–Ulam theorem for maps \(S^2\to\mathbb{R}^2\)
- Ham Sandwich theorem
- \(K\) is not a topological group
- \(\pi_1(T^2)\cong\ZZ^2\)
- Fundamental groups of polygonal quotient spaces
- A punctured torus retracts to two circles but not to its boundary
- Fundamental groups of band-and-disc spaces
- Fundamental group of a face-identified cube
- Fundamental group of a face-identified prism
- Fundamental group of a torus-knot complement
- Three presentations of \(\pi_1(K)\) are isomorphic
- Presentation complex of a group
- Fundamental groups of punctured Euclidean spaces, spherical quotients, and the Hopf link complement
- The homophony group is trivial
- Mayer-Vietoris homology of \(\RP^2\), \(T^2\), the Klein bottle, and \(S^1\cup_{z^n}B^2\)
- \(\tilde H_i(\Sigma X)\cong\tilde H_{i-1}(X)\)
- For \(f: S^n\circlearrowleft\), show \(\deg f = \deg \Sigma f\)
- Space with prescribed finitely generated homology groups
- Reduced homology of a union of \(n\) sets with nested contractible intersections vanishes in degrees \(\geq n-1\)
- Homology of \(X\times S^n\) in terms of \(H_*(X)\), and of \(T^n\)
- Homology of two solid tori glued by the identity along the boundary torus
- Homology of the mapping torus of a degree-\(d\) map \(S^n\to S^n\)
- \(\widetilde{H}_*(X \cup CA) \cong \widetilde{H}_*(X, A)\)
- Naturality of the Mayer–Vietoris sequence
- \(S_m \vee S_n\)
- Hexagon with identifications \(a+b+c-a-b-c\)
- Word for the genus-\(g\) orientable surface
- Nonorientable surface of genus \(g\)
- \(S^1\vee S^1\) with 2-cells along \((ab)^3\) and \((ab)^6\)
- Cellular homology of a tetrahedron with pairwise face identifications
- Cellular homology of a triangular prism with identified faces
- Cellular homology of a cube with paired faces
- CW structure, \(\pi_1\), and homology of the lens space \(L(p,1)\)
- Topological degree of a polynomial on the Riemann sphere
- Topological degree of \(p(z)=\frac{\prod_i^n(z-a_i)}{\prod_j^m(z-b_j)}\)
- A map \(S^m\to S^n\) homeomorphic onto an open set is a surjection of \(S^n\)
- Tensor products \(\ZZ_m\otimes\ZZ_n\), \(\ZZ_{60}^4\otimes(\ZZ_{24}^3\oplus\ZZ_8^4\oplus\ZZ_{120})\), \(\ZZ_n\otimes\QQ\), and \((\ZZ\oplus\ZZ_n)\otimes(\QQ/\ZZ)\)
- \(\tor(\ZZ\oplus\ZZ_2\oplus\ZZ_8,\ZZ\oplus\ZZ_4\oplus\ZZ_4)\) and \(\ext(\ZZ\oplus\ZZ_2\oplus\ZZ_3,\ZZ\oplus\ZZ_4\oplus\ZZ_5)\)
- \(H_*(\RP^n; \ZZ_2)\), \(H_*(\RP^n; \ZZ_3)\), and \(H^*(\RP^n; \ZZ_6)\)
- \(H^1(X)\) is free abelian
- \(H_*(X; \QQ) = H_*(X;\ZZ)\otimes\QQ\) and \(H^*(X; \ZZ) = \operatorname{Hom}(H_*(X; \ZZ), \QQ)\)
- A space with \(H_*=(\ZZ,\ZZ_6,\ZZ_{12},\ZZ\oplus\ZZ_4)\) in degrees \(0\)–\(3\) and vanishing above, and its cohomology
- \(H_*(\RP^2 \times \RP^2; \ZZ_2)\)
- \(H_*(\Sigma\RP^2 \times \RP^2; \ZZ)\)
- \(H_*(\RP^2 \times \RP^3; \ZZ)\)
- \(H_*(G)\) is an algebra, with \(G\) acting through \(\pi_0(G)\)
- The kernel of \(A\to A\otimes\QQ\), \(a\mapsto a\otimes 1\), is the torsion subgroup of \(A\)
- \(A\hookrightarrow B\) implies \(A\otimes\mathbb{Q}\hookrightarrow B\otimes\mathbb{Q}\)
- A free resolution of \(\QQ\) as a \(\ZZ\)-module
- \(\operatorname{Tor}(\mathbb{Q},A)\) and \(\operatorname{Tor}(\mathbb{Q}/\mathbb{Z},A)\)
- \(\operatorname{Ext}\) over \(\ZZ[x,y]\) of \(\ZZ[x,y]/(x-y)\) and \(\ZZ[x,y]/(x,y)\)
- Acyclicity of the exterior algebra under wedging by a \(1\)-form
- \(\operatorname{Tor}\) of the sign module over \(\ZZ[\ZZ_2]\)
- \(\mathrm{Tor}_R^*\) is symmetric
- \(\operatorname{Ext}(C,A)\) classifies extensions of \(C\) by \(A\)
- Cohomology ring of the torus via Poincaré duals of singular chains
- Cohomology ring of the closed orientable surface of genus g
- Cohomology rings of a disjoint union and a wedge in terms of the summands
- The torus and the wedge S^1 vee S^1 vee S^2 are not homotopy-equivalent
- Cohomology of the suspension and vanishing of positive-degree cup products
- Vanishing cup products under a cover by n contractible sets; covering projective spaces
- The cohomology ring of the n-torus is an exterior algebra on H^1
- No retraction from CP^n onto CP^k for k < n, and the mod-2 analogue for RP^n
- Compactly-supported cohomology of X x R is a suspension isomorphism
- Direct limit is an exact functor
- Taking homology commutes with direct limits
- Direct limits commute with the associated long exact sequence