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Choose distinct odd primes and take products of the nonisomorphic Gassmann equivalent regular subgroups, selecting one of the two groups at each prime. Product conjugacy-class counts give $2^n$ Gassmann equivalent subgroups of the same finite product of symmetric groups. Their fundamental groups are distinguished by whether the unique Sylow subgroup at each chosen prime is abelian. The closed-manifold realization of a finitely presented fundamental group and Sunada theorem therefore produce $2^n$ pairwise nonhomeomorphic, and hence nonisometric, isospectral closed four-manifolds. Gassmann equivalence alone, without a topology distinction, does not imply nonhomeomorphism.

On a compact smooth manifold without boundary, every smooth self-map sufficiently $C^1$-close to a diffeomorphism is a diffeomorphism. Reducing to a map near the identity, invertibility of its differential follows uniformly by compactness. It is then a proper local diffeomorphism, hence a finite covering map. Sufficient $C^0$ closeness gives a homotopy to the identity on each component; its induced map on the fundamental group is surjective. The covering therefore has one sheet on each component, proving the assertion. This lets a nearby Lagrangian submanifold transverse to cotangent fibers be represented by a single global differential one-form graph.

Give a discrete group $G$ a contractible free CW complex $EG$. The quotient $BG=EG/G$ has universal cover $EG$, so it has fundamental group $G$ and no higher homotopy groups. It is thus an Eilenberg–MacLane space $K(G,1)$, even when $G$ is nonabelian. The cohomology of $BG$ with the appropriate local coefficients computes group cohomology.

Every finitely presented group is the fundamental group of a closed connected smooth manifold in dimension at least four. Build a compact zero/one/two-handle manifold $W$ with the generators and relators of the presentation. Disjoint attaching loops can be embedded in its boundary of dimension at least three. The Seifert-van Kampen theorem gives ${\pi }_1(W)=T$, and the dual relative handle structure makes ${\pi }_1(\partial W)\to T$ surjective. Doubling $W$ along its boundary gives the amalgam of two copies of $T$ along that same surjection, which is again $T$. This construction underlies binary families of nonhomeomorphic Sunada quotients.

A rigid Skyrmion orientation is quantized on a cover of its Skyrmion collective-coordinate orbit. A lifted combined rotation in the static field's stabilizer subgroup identifies the same classical field and imposes ${\hat{D}}^J(R){\hat{D}}^I(A)\Psi ={\chi }_{\mathrm{FR}}\Psi$, where the sign is the chosen Finkelstein-Rubinstein constraints character of that configuration-space loop. Choosing the nontrivial character models unit baryons as fermions. A $2\pi$ spatial rotation or isorotation of charge $B$ then has sign $(-1{)}^B$: deforming the field to separated unit lumps adds the unit-lump loop classes in the fundamental group ${\mathbb{Z}}_2$, while the labelled orbital paths can be contracted in three dimensions. The resulting Finkelstein-Rubinstein constraints sign is the product of the unit-lump signs. Hence spin and isospin are half-integer for odd $B$ and integer for even $B$. Further stabilizer constraints restrict which pairs $(J,I)$ and which body-fixed states occur. The energy operator comes from the inertia tensors on the orbit; symmetry selects allowed states but does not by itself determine their energies. A bosonic choice of the trivial topological character is mathematically possible but does not model a fermionic nucleon.

A finite-dimensional commutative real algebra with a Euclidean multiplicative norm has dimension at most two. Automatic real bilinearity from a multiplicative norm supplies bilinearity if it was not part of the initial description. On its unit sphere, squaring has derivative $v\mapsto 2xv$, injective because there are no zero divisors. The only equal squares are antipodal pairs, so the map induces an injective immersion ${\mathbb{RP}}^{n-1}\to S^{n-1}$. Compactness and the inverse function theorem make it a diffeomorphism when $n\ge 2$. For $n\ge 3$, the fundamental groups $\mathbb{Z}/2$ and zero contradict this. The real and complex number algebras attain dimensions one and two.

In a strictly planar nematic liquid crystal, a charge-one nematic disclination is a winding-two map into the circle ${\mathbb{RP}}^1$. Its nonzero fundamental group class prevents complete unwinding while the surrounding loop stays fixed, but permits splitting into two charge-$1/2$ defects. For separation $r_0\ll d\ll L$, the two near fields contribute $\pi K{\lambda }_0^2(1/4+1/4)\log (d/r_0)$ while their common charge-one far field contributes $\pi K{\lambda }_0^2\log (L/d)$. Compared with the unsplit defect, the change isIt is negative at sufficiently large scale separation. Thus integer-defect fission is energetically favorable despite conservation of total topological charge. This leading-log argument proves failure of global energetic stability, not absence of kinetic barriers or pinning. If escape into the third dimension is allowed, an integer winding can also become topologically trivial in ${\mathbb{RP}}^2$.

If a connected complete Riemannian manifold has $\mathrm{Ric}\ge (n-1)\kappa g$ with $\kappa >0$, its universal cover with the lifted metric is complete and satisfies the same bound. The Bonnet-Myers theorem makes that cover compact. A covering fiber is closed and discrete and hence finite; the number of its elements equals the order of the fundamental group. Applying the diameter theorem only to the base would not prove this conclusion: the positive lower bound must also be used on the cover.

The first homology group $H_1(X)$ is the group of one-dimensional cycles modulo boundaries. For a path-connected space it is the abelianization of the fundamental group.

Every finitely presented group is the fundamental group of a closed connected smooth oriented four-manifold. Build a compact four-dimensional handle decomposition with one zero-handle, a one-handle for each generator of a group and a two-handle for each relator. Turning the handle decomposition upside down shows that its boundary surjects on the fundamental group. Its double of a manifold has the same fundamental group by the Seifert-van Kampen theorem.

A free homotopy between maps is an ordinary homotopy with no specified basepoint held fixed. Two loops at the same basepoint are freely homotopic precisely when their elements of the fundamental group are conjugate: a homotopy traces a connecting basepoint path, and the boundary of its parameter square gives conjugation. Conversely that connecting path can be slid around the loop to realize the conjugation.

Every finitely generated group acts freely and properly discontinuously by Riemannian isometries on a connected manifold in any dimension $d\ge 3$. Choose a surjection $F_r\to T$ and the regular covering of ${\#}_r(S^1\times S^{d-1})$ associated with its kernel. Pulling back a metric makes the deck transformation group isometric. Uniqueness of lifts makes its action free. The cover can be noncompact and need not be simply connected; no claim about realizing every group as a closed three-manifold fundamental group is involved.

The fundamental group of a bouquet of $r$ circles is the free group on $r$ generators, represented by the oriented circles. For finite $r$, iterated Seifert-van Kampen theorem proves the claim by attaching one circle at a time. Nontrivial reduced words in a free group distinguish based loops, while free homotopy classes correspond to conjugacy classes.

A generalized Heegaard diagram permits compression bodies and a nonempty negative boundary. Equivalently, a suitable handlebody with two-handles attached along only part of a compressing system can retain torus boundary components. Disk crossings supply generators and attaching-curve words supply relators for its fundamental group.

Every finitely presented group is the fundamental group of a closed symplectic four-manifold. The construction starts from products of surfaces, realizes relator words by embedded loops after stabilization, and takes symplectic sums with copies of the rational elliptic surface along square-zero tori. Since the complement of a regular elliptic fiber in the rational elliptic surface is simply connected, van Kampen's theorem imposes the desired generators and relators.

The group manifold is the underlying smooth manifold of a Lie group, with smooth multiplication and inversion. Its global topology need not be determined by its Lie algebra: the SU(2) group and SO(3) group have isomorphic Lie algebras but different fundamental groups.

The quotient of the real Heisenberg group by the Integer Heisenberg group is the mapping torus of the torus automorphismIts universal cover is the contractible real Heisenberg group, so its fundamental group is the integer Heisenberg group.

For a marked closed hyperbolic surface, the representation of its fundamental group whose image acts as the deck group on the hyperbolic plane. It is discrete and faithful, determined up to conjugation by the marking and the metric, and gives $X={\mathbb{H}}^2/\rho ({\pi }_1{\Sigma }_g)$. For a closed orientable surface it admits determinant-one lifts. It is different from the tangent-space action of the Riemannian holonomy group.